PATENT DOCUMENT

Publication Number: US-7308517-B1
Application Number: US-74979103-A
Country: US
Kind Code: B1

Title: Gap count analysis for a high speed serialized bus

Abstract:
A method of optimizing communication over a high-speed serial bus by minimizing the delay between packets transmitted over the bus is disclosed. The method comprises: calculating the round trip delay between PHYs connected on the bus by pinging; a bus manager sending a configuration packet to all PHYs connected on the bus, the configuration packet containing a minimum gap_count parameter value; and all PHYs connected on the bus sending packets over the bus using the minimum gap_count parameter value as a delay between packets.

Claims:
1. A device for use in a first node of a serial bus, the device comprising:
 a first module adapted to ping a second node; 
 a second module adapted to receive a ping response from the second node; 
 a third module adapted to calculate a maximum round trip delay between a first PHY associated with the first node and a second PHY associated with the second node based at least in part upon a jitter value, and further based at least in part on the ping response sent to the second module; and 
 a fourth module adapted to send a configuration packet to all PHYs on the serial bus, the configuration packet containing a gap count, the gap count derived from the maximum round trip delay between the first PHY and the second PHY; 
 wherein at least one of said first PHY and second PHY comprises a first pair of ports and a second pair of ports and the jitter value is defined as being greater than or equal to the absolute value of a total quantity, the total quantity defined as the difference between a first quantity and a second quantity; and 
 wherein the first quantity comprises the sum of a first sub-quantity and a second sub-quantity, the first sub-quantity consisting of a PHY delay between the first ordered pair of ports divided by two, the second sub-quantity consisting of an arbitration response delay between the first ordered pair of ports divided by two. 
 
     
     
       2. The device of  claim 1 , wherein the second quantity comprises the sum of a third subquantity and a fourth subquantity, the third subquantity consisting of a PHY delay between the second ordered pair of ports divided by two, and the fourth subquantity consisting of an arbitration response delay between the second ordered pair of ports divided by two. 
     
     
       3. The device of  claim 1 , wherein the response comprises a self-identification packet. 
     
     
       4. The device of  claim 1 , wherein all PHYs of the high-speed serial bus comprise a subaction gap detection time that is greater than a maximum idle value that can occur within a subaction. 
     
     
       5. The device of  claim 1 , wherein all PHYs of the high-speed serial bus comprise an arbitration reset gap timeout value that is greater than the largest subaction gap that can occur over the high-speed serial bus. 
     
     
       6. A method of optimizing communication over a high-speed serial bus by minimizing the delay between packets transmitted over the bus, the method comprising:
 sending a ping from a first node to a second node; 
 sending a response from the second node to the first node after receiving the ping; 
 calculating a maximum round trip delay between a first PHY of the first node and a second PHY of the second node based at least in part upon a jitter, and further based at least in part upon the response sent to the first node; 
 sending via a bus manager a configuration packet to all PHYs connected on the bus, the configuration packet containing a minimum gap_count parameter value, the minimum gap_count parameter value derived from the maximum round trip delay between the first PHY and the second PHY; and 
 sending via all PHYs connected on the packets over the bus, using the minimum gap_count parameter value as a delay between packets; 
 wherein at least one of said first PHY and second PHY comprises a first pair of ports and a second pair of ports, wherein the jitter is defined as being greater than or equal to the absolute value of a total quantity, the total quantity defined as the difference between a first quantity and a second quantity; and 
 wherein the first quantity comprises the sum of a first subquantity and a second subquantity, the first subquantity consisting of a PHY delay between the first ordered pair of ports divided by two, the second subquantity consisting of an arbitration response delay between the first ordered pair of ports divided by two. 
 
     
     
       7. The method of  claim 6 , wherein the second quantity comprises the sum of a third subquantity and a fourth subquantity, the third subquantity consisting of a PHY delay between the second ordered pair of ports divided by two, and the fourth subquantity consisting of an arbitration response delay between the second ordered pair of ports divided by two. 
     
     
       8. The method of  claim 6 , further comprising preserving an ack/iso gap between packets, wherein a first PHY sent a most recently-sent packet and a second PHY is responding to the first PHY. 
     
     
       9. The method of  claim 8 , wherein the second PHY is adapted to respond with an ack packet. 
     
     
       10. The method of  claim 8 , wherein the second PHY is adapted to respond with an isochronous arbitration packet. 
     
     
       11. The method of  claim 6 , wherein the first PHY sends an isochronous packet, observes a sub action gap, and initiates an arbitration indication. 
     
     
       12. The method of  claim 6 , wherein the first PHY sends an asynchronous packet, observes an arbitration reset gap, and initiates an arbitration indication. 
     
     
       13. The method of  claim 6 , wherein calculating the round trip delay comprises executing a ping command at a link layer on said first node directed at a link layer on said second node. 
     
     
       14. The method of  claim 13 , wherein calculating the round trip delay comprises calculating a round trip delay from a first link on the first node and a second link on the second node. 
     
     
       15. The method of  claim 6 , wherein the second PHY has a subaction gap timeout value that is greater than an IDLE value that can occur within a subaction and an isochronous interval on the high-speed serial bus. 
     
     
       16. The method of  claim 6 , wherein all PHYs of the high-speed serial bus comprise a subaction gap detection time that is greater than a maximum idle value that can occur within a subaction. 
     
     
       17. The method of  claim 6 , wherein all PHYs of the high-speed serial bus comprise an arbitration reset gap timeout value that is greater than the largest subaction gap that can occur over the high-speed serial bus. 
     
     
       18. The method of  claim 6 , wherein the response comprises a self-ID packet. 
     
     
       19. A computer-readable medium containing instructions which, when executed by a processor, minimize the delay between packets transmitted over a high-speed serial bus, by performing the method comprising:
 sending a ping from a first node to a second node; 
 sending a response from the second node to the first node after receiving the ping; 
 calculating a maximum round trip delay between a first PHY of the first node and a second PHY of the second node based at least in part upon a jitter, and further based at least in part upon the response sent to the first node; 
 sending via a bus manager a configuration packet to all PHYs connected on the bus, the configuration packet containing a minimum gap_count parameter value, the minimum gap_count parameter value derived from the maximum round trip delay between the first PHY and the second PHY; and 
 sending via all PHYs connected on the packets over the buss using the minimum gap_count parameter value as a delay between packets; 
 wherein at least one of said first PHY and second PHY comprises a first pair of ports and a second pair of ports, wherein the jitter is defined as being greater than or equal to the absolute value of a total quantity, the total quantity defined as the difference between a first quantity and a second quantity; and 
 wherein the first quantity comprises the sum of a first subquantity and a second subquantity, the first subquantity consisting of a PHY delay between the first ordered pair of ports divided by two, the second subquantity consisting of an arbitration response delay between the first ordered pair of ports divided by two. 
 
     
     
       20. The method of  claim 19 , wherein the second quantity comprises the sum of a third subquantity and a fourth subquantity, the third subquantity consisting of a PHY delay between the second ordered pair of ports divided by two, and the fourth subquantity consisting of an arbitration response delay between the second ordered pair of ports divided by two. 
     
     
       21. The method of  claim 19 , further comprising preserving an ack/iso gap between packets, wherein a first PHY sent a most recently-sent packet and a second PHY is responding to the first PHY. 
     
     
       22. The method of  claim 21 , wherein the second PHY is adapted to respond with an ack packet. 
     
     
       23. The method of  claim 21 , wherein the second PHY is adapted to respond with an isochronous arbitration packet. 
     
     
       24. The method of  claim 19 , wherein the first PHY sends an isochronous packet, observes a sub action gap, and initiates an arbitration indication. 
     
     
       25. The method of  claim 19 , wherein the first PHY sends an asynchronous packet, observes an arbitration reset gap, and initiates an arbitration indication. 
     
     
       26. The method of  claim 19 , wherein calculating the round trip delay comprises executing a ping command at a link layer on said first node directed at a link layer on said second node. 
     
     
       27. The method of  claim 26 , wherein calculating the round trip delay comprises calculating a round trip delay from a first link on the first node and a second link on the second node. 
     
     
       28. The method of  claim 19 , wherein the second PHY has a subaction gap timeout value that is greater than an IDLE value that can occur within a subaction and an isochronous interval on the high-speed serial bus. 
     
     
       29. The method of  claim 19 , wherein all PHYs of the high-speed serial bus comprise a subaction gap detection time that is greater than a maximum idle value that can occur within a subaction. 
     
     
       30. The method of  claim 19 , wherein all PHYs of the high-speed serial bus comprise an arbitration reset gap timeout value that is greater than the largest subaction gap that can occur over the high-speed serial bus. 
     
     
       31. The method of  claim 19 , wherein the response comprises a self-ID packet. 
     
     
       32. A device for use in a first node of a serial bus, the device comprising:
 means for pinging a second node; 
 means for receiving a ping response from the second node; 
 means for calculating a maximum round trip delay between a first PHY associated with the first node and a second PHY associated with the second node based at least in part upon a jitter value, and further based at least in part on the ping response sent to the second module; and 
 means for sending a configuration packet to all PHYs on the serial bus, the configuration packet containing a gap count, the gap count derived from the maximum round trip delay between the first PHY and the second PHY; 
 wherein at least one of said first PHY and second PHY comprises a first pair of ports and a second pair of ports and the jitter value is defined as being greater than or equal to the absolute value of a total quantity, the total quantity defined as the difference between a first quantity and a second quantity; and 
 wherein the first quantity comprises the sum of a first sub-quantity and a second sub-quantity, the first sub-quantity consisting of a PHY delay between the first ordered pair of ports divided by two, the second sub-quantity consisting of an arbitration response delay between the first ordered pair of ports divided by two. 
 
     
     
       33. A device for use in a first node of a serial bus, the device comprising:
 a first module adapted to send a first signal to a second node, said first signal adapted to elicit a response from said second node; 
 a second module adapted to calculate a maximum round trip delay between the first node and the second node based at least in part upon a jitter value, and further based at least in part on the response elicited from the second node; and 
 a third module adapted to send a packet to all nodes on the serial bus, the packet containing a gap count, the gap count derived from the maximum round trip delay between the first node and the second node; 
 wherein at least one of said first node and said second node comprises a first pair of ports and the jitter value is defined as being greater than or equal to the absolute value of a total quantity, the total quantity defined as the difference between a first quantity and a second quantity; and 
 wherein the first quantity comprises the sum of a first sub-quantity and a second sub-quantity, the first sub-quantity consisting of a delay between the first pair of ports divided by two, the second sub-quantity consisting of an arbitration response delay between the first pair of ports divided by two. 
 
     
     
       34. The device of  claim 33 , wherein at least one of said first node and said second node further comprises a second pair of ports, wherein the second quantity comprises the sum of a third subquantity and a fourth subquantity, the third subquantity consisting of a delay between the second pair of ports divided by two, and the fourth subquantity consisting of an arbitration response delay between the second pair of ports divided by two. 
     
     
       35. The device of  claim 33 , wherein the response comprises a self-identification packet. 
     
     
       36. A method of optimizing communication over a serial bus, the method comprising:
 sending a first message from a first node to a second node; 
 sending a response from the second node to the first node after receiving the first message; 
 calculating a maximum round trip delay between the first node and the second node based at least in part upon a jitter, and further based at least in part upon the response sent to the first node; 
 sending a packet to all nodes connected on the bus, the packet containing a minimum gap count parameter value derived from the maximum round trip delay between the first node and the second node; and 
 sending via all nodes connected on the bus the minimum gap count parameter value as a delay between packets; 
 wherein at least one of said first node and second node comprises a first pair of ports, and wherein the jitter is defined as being greater than or equal to the absolute value of the difference between a first quantity and a second quantity; and 
 wherein the first quantity comprises the sum of a first subquantity and a second subquantity, the first subquantity comprising a PHY delay related to the first pair of ports, the second subquantity comprising an arbitration response delay related to the first pair of ports. 
 
     
     
       37. The method of  claim 36 , wherein at least one of said first node and said second node further comprises a second pair of ports, wherein the second quantity comprises the sum of a third subquantity and a fourth subquantity, the third subquantity comprising a PHY delay related to the second pair of ports, and the fourth subquantity comprising an arbitration response delay related to the second pair of ports. 
     
     
       38. The method of  claim 36 , wherein calculating the round trip delay comprises executing a ping command at a link layer on said first node directed at a link layer on said second node. 
     
     
       39. The method of  claim 38 , wherein calculating the round trip delay comprises calculating a round trip delay from a first link on the first node and a second link on the second node. 
     
     
       40. The method of  claim 36 , wherein the second node has a subaction gap timeout value that is greater than an idle value that can occur within a subaction and an isochronous interval on the serial bus. 
     
     
       41. The method of  claim 36 , wherein all nodes of the serial bus comprise a subaction gap detection time that is greater than a maximum idle value that can occur within a subaction. 
     
     
       42. The method of  claim 36 , wherein all nodes of the high-speed serial bus comprise an arbitration reset gap timeout value that is greater than the largest subaction gap that can occur over the serial bus. 
     
     
       43. The method of  claim 36 , wherein the response comprises a self-identification packet.

Description:
FIELD OF THE INVENTION 
     The present invention relates broadly to serial bus performance. Specifically, the present invention relates to improving bus performance by calculating the optimal gap_count parameter for a given topology utilizing a high-speed serial bus to connect devices. 
     BACKGROUND OF THE INVENTION 
     The Institute of Electrical and Electronic Engineers (IEEE) has promulgated a number of versions of a high-speed serial bus protocol falling under the IEEE 1394 family of standards (referred to herein collectively as “1394”). A typical serial bus having a1394 architecture interconnects multiple node devices via point-to-point links, such as cables, each connecting a single node on the serial bus to another node on the serial bus. Data packets are propagated throughout the serial bus using a number of point-to-point transactions, such that a node that receives a packet from another node via a first point-to-point link retransmits the received packet via other point-to-point links. A tree network configuration and associated packet handling protocol ensures that each node receives every packet once. The 1394-compliant serial bus may be used as an alternate bus for the parallel backplane of a computer system, as a low cost peripheral bus, or as a bus bridge between architecturally compatible buses. Bus performance is gauged by throughput, or the amount of data that can be transmitted over the bus during a period of time. 
     There are several ways to improve bus performance. Devices connected to the bus can be arranged to minimize the longest round-trip delay between any two leaf nodes. This may involve either minimizing the number of cable connections between the farthest devices, reducing cable lengths, or both. Another way to improve bus performance is to group devices with identical speed capabilities next to one another. This avoids the creation of a “speed trap” when a slower device lies along the path between the two faster devices. Finally, bus performance can be improved by setting the PHY gap count parameter to the lowest workable value for a particular topology. However, determining this lowest workable value is problematic in that all of the variables affecting this value are unknown. Gap count parameters have been configured in the past using a subset of all possible variables, and the result is that the gap count is not optimal. 
     SUMMARY OF THE INVENTION 
     The present invention provides an optimal gap count that allows a high-speed serial bus to run faster and thus realize superior performance over prior buses. In an embodiment, bus management software sends a special PHY configuration packet that is recognized by all PHYs on the bus. The configuration packet contains a gap count value that all PHYs on the bus can use. As this gap count value decreases the time interval between packets that are transmitted, more real data can be transmitted over the bus per unit of time. 
     In an embodiment, the bus manager pings a PHY. The PHY sends a response to the ping, and a flight time value of the response from the PHY to the bus manager is added to calculate a round trip delay value. The ping command runs at the link layer level, from the link layer of one node to the link layer of another node. All flight time between link layer and PHY is ignored, and just the flight time from one PHY to another PHY is calculated. The ping time measured shows the link-to-link delay. The delay between the bus and the link is specified in the bus standard with minimum and maximum values. The PHY and link layer of a node is designed to be within that range specified by the standard. The round trip delay between nodes can be calculated as: 
     
       
         
           
             
               Round_Trip 
               ⁢ 
               
                 _Delay 
                 max 
                 
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                                 Round_Trip 
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                                   _Delay 
                                   
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                                     max 
                                   
                                   
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                                   · 
                                   
                                     Jitter 
                                     n 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
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                                 ⁢ 
                                 
                                   _Delay 
                                   
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                                     , 
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                                     n 
                                   
                                 
                               
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                                 ARB_RESPONSE 
                                 ⁢ 
                                 
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                                     N 
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                               PHY_DELAY 
                               
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                               ⁢ 
                               
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                                     N 
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                                     N 
                                     BM 
                                   
                                 
                               
                             
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                               PHY_DELAY 
                               
                                 N 
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     This value can be communicated as the gap count parameter contained in the configuration packet, thus setting the gap between packets to an optimal value and increasing bus performance. 
     Many other features and advantages of the present application will become apparent from the following detailed description considered in conjunction with the accompanying drawings, in which: 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates an intervening path model between two nodes, X &amp; Y, and denotes the reference points required for a full analysis; 
         FIG. 2  illustrates ack/iso gap preservation, in the case where PHY X originated the most recent packet and PHY Y is responding (either with an ack or the next isochronous arbitration/packet). 
         FIG. 3  illustrates the sequence PHY Y will follow in responding to a received packet. 
         FIG. 4  illustrates subaction gap preservation, in the case where PHY X originated the most recent packet and PHY Y is responding after a subaction gap with arbitration for the current fairness interval. 
         FIG. 5  illustrates consistent subaction gap detection, in the case where PHY X originates an isochronous packet, observes a subaction_gap, and begins to drive an arbitration indication. 
         FIG. 6  illustrates an internal gap detection sequence, by showing the timing reference for relating the external gap detection times to the internal gap detection times. 
         FIG. 7  illustrates consistent arbitration reset gap detection, in the case where PHY X originates an asynchronous packet, observes an arbitration reset gap, and begins to drive an arbitration indication. 
         FIG. 8  illustrates a ping subaction issued by the link in Node X and directed to Node Y. 
         FIG. 9  illustrates a Bus Manager Leaf to Leaf topology. 
         FIG. 10  illustrates a topology where the bus manager is not a leaf but is part of the connecting path between the two leaves. 
         FIG. 11  illustrates a topology where the bus manager is not a leaf but is not part of the connecting path between the two leaves. 
     
    
    
     DETAILED DESCRIPTION 
     Four well known limiting corner cases for gap count are examined in an effort to find the minimum allowable gap count for a given topology. Both the table method and pinging method of determining the optimal gap count are explored. 
     It is important to note that this analysis assumes that PHY_DELAY can never exceed the maximum published in the PHY register set. However, corner conditions have been identified in which it is theoretically possible to have PHY_DELAY temporarily exceed the maximum published delay when repeating minimally spaced packets. Although not a rigorous proof, this phenomena is ignored for this analysis on the basis that it is presumed to be statistically insignificant. 
     The path between any two given PHYs can be represented as a daisy chain connection of the two devices with zero or more intervening, or repeating, PHYs.  FIG. 1  illustrates such a path between two nodes, X &amp; Y, and denotes the reference points required for a full analysis. 
     
       
         
           
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 Variable Definitions 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 ARB_RE- 
                 Delay in propagating arbitration indication 
               
               
                 SPONSE_DE- 
                 received from port P n  of PHY n to port 
               
               
                 LAY n   P     n     →P′     n     
                 P′ n  of PHY n. 
               
               
                 BASERATE n   
                 Fundamental operating frequency of PHY n. 
               
               
                 cable_delay n   
                 One-way flight time of arbitration and data 
               
               
                   
                 signals through cable n . The flight-time is 
               
               
                   
                 assumed to be constant from one transmission 
               
               
                   
                 to the next and symmetric. 
               
               
                 DATA_END_TIME n   P     n     
                 Length of DATA_END transmitted on port 
               
               
                   
                 P n  of PHY n. 
               
               
                 PHY_DELAY n   P′     n     →P     n     
                 Time from receipt of first data bit at port P′ n  of 
               
               
                   
                 PHY n to re-transmission of same bit at port 
               
               
                   
                 P n  of PHY n. 
               
               
                 RESPONSE_TIME P n   P′     n     
                 Idle time at port P′ n  of PHY n between the 
               
               
                   
                 reception of a inbound packet and the 
               
               
                   
                 associated outbound arbitration indication 
               
               
                   
                 for the subsequent packet intended to occur 
               
               
                   
                 within the same isochronous interval or 
               
               
                   
                 asynchronous subaction. 
               
               
                   
               
            
           
         
       
     
     For any given topology, the gap count must be set such that an iso or ack gap observed/generated at one PHY isn&#39;t falsely interpreted as a subaction gap by another PHY in the network. Ack/Iso gaps are known to be at their largest nearest the PHY that originated the last packet. To ensure that the most recent originating PHY doesn&#39;t interrupt a subaction or isochronous interval with asynchronous arbitration, its subaction_gap timeout must be greater than the largest IDLE which can legally occur within a subaction or isochronous interval.  FIG. 2  illustrates the case in which PHY X originated the most recent packet and PHY Y is responding (either with an ack or the next isochronous arbitration/packet). 
     For all topologies, the idle time observed at point Px must not exceed the subaction gap detection time: 
     
       
         
           
             
               
                 
                   
                     Idle 
                     max 
                     
                       P 
                       X 
                     
                   
                   &lt; 
                   
                     subaction_gap 
                     min 
                     
                       P 
                       X 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     The idle time at point Px can be determined by examining the sequence of time events in the network. All timing events are referenced to the external bus (as opposed to some internal point in the PHY). 
     
       
         
           
               
               
             
               
                   
               
             
            
               
                 t 0   
                 First bit of packet sent at point P X   
               
               
                 t 1   
                 Last bit of packet sent at point P x , DATA_END begins. t 1  follows t 0   
               
               
                   
                 by the length of the packet timed in PHY X&#39;s clock domain. 
               
               
                 t 2   
                 DATA_END concludes at point P X , IDLE begins. t 2  follows t 1  by 
               
               
                   
                 DATA_END_TIME X   P     X     
               
               
                 t 3   
                 First bit of packet received at point P′ Y . t 3  follows t O  by all 
               
               
                   
                 intervening cable_delay and PHY_DELAY instances. 
               
               
                 t 4   
                 Last bit of packet received at point P′ Y . t 4  follows t 3  by the length of 
               
               
                   
                 the packet timed in PHY Y-1&#39;s clock domain. 
               
               
                 t 5   
                 DATA_END concludes at point P′ Y , gap begins. t 5  follows t 4  by 
               
               
                   
                 DATA_END_TIME Y−1   P     Y−1     
               
               
                 t 6   
                 PHY Y responds with ack packet, isoch packet, or isoch arbitration 
               
               
                   
                 within RESPONSE_TIME Y   P     Y    following t 5   
               
               
                 t 7   
                 Arbitration indication arrives at point P X . t 7  follows t 6  by the all 
               
               
                   
                 intervening cable_delay and ARB_RESPONSE_DELAY instances. 
               
               
                   
               
            
           
         
       
     
                                       t   1     =       t   0     +     packet_length     packet_speed   ·     BASERATE   X                   (   2   )                 t   2     =         t   1     +     DATA_END   ⁢     _TIME   X     P   X           =       t   0     +     packet_length     packet_speed   ·     BASERATE   X         +     DATA_END   ⁢     _TIME   X   Px                   (   3   )                 t   3     =       t   0     +     cable_delay   x     +       ∑     n   =     X   +   1         Y   -   1       ⁢     (       cable_delay   n     +     PHY_DELAY   n       P   n   ′     →     P   n                         (   4   )                 t   4     =         t   3     +     packet_length     packet_speed   ·     BASERATE     Y   -   1             =       t   0     +     cable_delay   X     +       ∑     n   =     X   +   1         Y   -   1       ⁢     (       cable_delay   n     +     PHY_DELAY   n       P   n   ′     →     P   n           )       +     packet_lenght     packet_speed   ·     BASERATE     Y   -   1                       (   5   )                 t   5     =         t   4     +     DATA_END   ⁢     _TIME     Y   -   1       P     y   -   1             =       t   0     +     cable_delay   X     +       ∑     n   =     X   +   1         Y   -   1       ⁢     (       cable_delay   n     +     PHY_DELAY   n       P   n   ′     →     P   n           )       +     packet_length     packet_speed   ·     BASERATE     Y   -   1           +     DATA_END   ⁢     _TIME     Y   -   1       P     y   -   1                       (   6   )                 t   6     =         t   5     +     RESPONSE_TIME   Y     P   Y   ′         =       t   0     +     cable_delay   X     +       ∑     n   =     X   +   1         Y   -   1       ⁢     (       cable_delay   n     +     PHY_DELAY   n       P   n   ′     →     P   n           )       +     packet_length     packet_speed   ·     BASERATE     Y   -   1           +     DATA_END   ⁢     _TIME     Y   -   1       P     y   -   1           +     RESPONSE_TIME   Y     P   Y   ′                   (   7   )                                     t   7     =         t   6     +       ∑     n   =     X   +   1         Y   -   1       ⁢     (       cable_delay   n     +     ARB_RESPONSE   ⁢     _DELAY   n       P   n   ′     →     P   n             )       +     cable_delay   X       =         t   0     ⁢       ∑     n   =     X   +   1         Y   -   1       ⁢     (       2   ·     cable_delay   n       +     PHY_DELAY   n       P   n   ′     →     P   n         +     ARB_RESPONSE   ⁢     _DELAY   n       P   n   ′     →     P   n             )         +     2   ·     cable_delay   X       +     packet_length     packet_speed   ·     BASERATE     Y   -   1           +     DATA_END   ⁢     _TIME     Y   -   1       P     y   -   1           +     RESPONSE_TIME   Y     P   Y   ′                   (   8   )                 
Given t 0  through t 7  above, the Idle time seen at point P x  is given as:
 
                           Idle     P   x       =         t   7     -     t   2       =         ∑     n   =     X   +   1         Y   -   1       ⁢     (       2   ·     cable_delay   n       +     
     ⁢     PHY_DELAY   n       P   n   ′     →     P   n         +     ARB_RESPONSE   ⁢   _   ⁢           ⁢     DELAY   n       P   n     →       P   ′     n             )       +     
     ⁢           ⁢     2   ·     cable_delay   X       +     RESPONSE_TIME   y     P   Y   ′       +     
     ⁢     DATA_END   ⁢     _TIME     Y   -   1       P     y   -   1           -     DATA_END   ⁢     _TIME   X     P   x         +       packet_length   packet_speed     ·     (       1     BASERATE     Y   -   1         -     1     BASERATE   X         )                   (   9   )               Let   ⁢     :                               DE_delta     [       P     y   -   1       ,     P   x       ]       =       DATA_END   ⁢     _TIME     Y   -   1       P     y   -   1           -     DATA_END   ⁢     _TIME   X     P   x                   (   10   )                 PPM_delta     [       Y   -   1     ,   X     ]       =       packet_length   packet_speed     ·     (       1     BASERATE     Y   -   1         -     1     BASERATE   X         )               (   11   )                 Round_Trip   ⁢     _Delay     [       P   X     ⁢     OP   Y       ]         =       ∑     n   =     X   +   1         Y   -   1       ⁢       (       2   ·     cable_delay   n       +     PHY_DELAY   n       P   n   ′     →     P   n         +     ARB_RESPONSE   ⁢     _DELAY   n       P   n     →       P   ′     n             )     ⁢     2   ·     cable_delay   X                   (   12   )               Then   ,                             Idle     P   x       =       Round_Trip   ⁢     _Delay     [       P   X     ⁢     OP   Y       ]         +     RESPONSE_TIME   Y     P   Y   ′       +     DE_delta     [       P     y   -   1       ,     P   x       ]       +     PPM_delta     [       Y   -   1     ,   X     ]                 (   13   )                 
Substituting into Equation (1), Ack and Iso gaps are preserved network-wide if and only if:
 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 Round_Trip 
                                 ⁢ 
                                 
                                   _Delay 
                                   
                                     [ 
                                     
                                       
                                         P 
                                         X 
                                       
                                       ⁢ 
                                       
                                         OP 
                                         Y 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               + 
                               
                                 RESPONSE_TIME 
                                 Y 
                                 
                                   P 
                                   Y 
                                   ′ 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 DE_delta 
                                 
                                   [ 
                                   
                                     
                                       P 
                                       
                                         y 
                                         - 
                                         1 
                                       
                                     
                                     , 
                                     
                                       P 
                                       x 
                                     
                                   
                                   ] 
                                 
                               
                               + 
                               
                                 PPM_delta 
                                 
                                   [ 
                                   
                                     
                                       Y 
                                       - 
                                       1 
                                     
                                     , 
                                     X 
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     max 
                   
                   &lt; 
                   
                     subaction_gap 
                     min 
                     
                       P 
                       x 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     The minimum subaction_gap at point P x  isn&#39;t well known. IEEE1394-1995, in Table 4-33, defines the minimum subaction_gap timeout used at a PHY&#39;s internal state machines, not at the external interface. It has been argued that the internal and external representations of time may differ by as much as ARB_RESPONSE_DELAY when a PHY is counting elapsed time between an internally generated event and an externally received event. However, the ARB_RESPONSE_DELAY value for a particular PHY isn&#39;t generally known externally. Fortunately, the ARB_RESPONSE_DELAY value for a PHY whose FIFO is known to be empty is bounded by the worst case PHY_DELAY reported within the PHY register map. This suggests a realistic bound for the minimum subaction_gap referenced at point P x : 
                     subaction_gap   min     P   X       ≥       subaction_gap   min     i   x       -     PHY_DELAY     X   ,   max       P   x                 (   15   )               
where
 
                     subaction_gap   min     i   x       =       27   +     gap_count   ·   16         BASERATE     X   ,   max                 (   16   )               
Combing Equations (14), (15), and (16):
 
                       [             Round_Trip   ⁢     _Delay     [       P   X     ⁢     OP   Y       ]         +                 RESPONSE_TIME   Y     P   Y   ′       +                 DE_delta     [       P     y   -   1       ,     P   x       ]       +               PPM_delta     [       Y   -   1     ,   X     ]             ]     max     &lt;     [           ⁢         27   +     gap_count   ·   16         BASERATE     X   ,   max         -     PHY_DELAY     X   ,   max       P   x         ]             (   17   )               
Solving for gap_count:
 
     
       
         
           
             
               
                 
                   gap_count 
                   &gt; 
                   
                     
                       
                         
                           BASERATE 
                           
                             X 
                             , 
                             max 
                           
                         
                         · 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     Round_Trip 
                                     ⁢ 
                                     
                                       _Delay 
                                       max 
                                       
                                         [ 
                                         
                                           
                                             P 
                                             x 
                                           
                                           ⁢ 
                                           
                                             OP 
                                             y 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                 RESPONSE_TIME 
                                                 Y 
                                                 
                                                   P 
                                                   Y 
                                                   ′ 
                                                 
                                               
                                               + 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 DE_delta 
                                                 
                                                   [ 
                                                   
                                                     
                                                       P 
                                                       
                                                         y 
                                                         - 
                                                         1 
                                                       
                                                     
                                                     , 
                                                     
                                                       P 
                                                       x 
                                                     
                                                   
                                                   ] 
                                                 
                                               
                                               + 
                                             
                                           
                                         
                                         
                                           
                                             
                                               PPM_delta 
                                               
                                                 [ 
                                                 
                                                   
                                                     Y 
                                                     - 
                                                     1 
                                                   
                                                   , 
                                                   X 
                                                 
                                                 ] 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                     max 
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   PHY_DELAY 
                                   
                                     X 
                                     , 
                                     max 
                                   
                                   
                                     P 
                                     x 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       - 
                       27 
                     
                     16 
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     Since RESPONSE_TIME, DE_delta, and PPM_delta are not independent parameters, the maximum of their sum is not accurately represented by the sum of their maximas. Finding a more accurate maximum for the combined quantity requires the identification of components of RESPONSE_TIME. 
     As specified in p1394a, RESPONSE_TIME includes the time a responding node takes to repeat the received packet and then drive a subsequent arbitration indication. (Note that by examination of the C code, RESPONSE_TIME is defined to include the time it takes to repeat a packet even if the PHY in question is a leaf node.)  FIG. 3  illustrates the sequence PHY Y will follow in responding to a received packet. i Y  denotes the timings as seen/interpreted by the PHY state machine. Note that P Y  can be any repeating port on PHY Y. Consequently, the timing constraints referenced to P Y  in the following analysis must hold worst case for any and all repeating ports. 
     Beginning with the first arrival of data at P′ Y  (t 3 ), the elaborated timing sequence for RESPONSE_TIME is: 
     
       
         
           
               
               
             
               
                   
               
             
            
               
                 t 3   
                 First bit of packet received at point P′ Y   
               
               
                 t 3 ′ 
                 First bit of packet repeated at point P Y . t 3 ′ lags t 3  by PHY_DELAY 
               
               
                 t 4   
                 Last bit of packet received at point P′ Y . t 4  follows t 3  by the length of 
               
               
                   
                 the packet timed in PHY N&#39;s clock domain. DATA_END begins 
               
               
                 t 4 ′ 
                 Last bit of packet repeated at point P Y . t 4 ′ lags t 3 ′ by the length of the 
               
               
                   
                 packet timed in PHY Y&#39;s clock domain. The PHY begins “repeating” 
               
               
                   
                 DATA_END 
               
               
                 t 5   
                 DATA_END concludes at point P′ Y . t 5  follows t 4  by 
               
               
                   
                 DATA_END_TIME Y−1   P     Y−1     
               
               
                 t 5a   
                 stop_tx_packet( ) concludes at point i Y  and the state machines 
               
               
                   
                 command the PHY ports to stop repeating DATA_END. t 5a  leads t 5 ′ 
               
               
                   
                 by any transceiver delay. 
               
               
                 t 5 ′ 
                 DATA_END concludes at point P Y . t 5 ′ follows t 4 ′ by 
               
               
                   
                 DATA_END_TIME Y   P     Y     
               
               
                 t 5b   
                 start_tx_packet( ) commences at point i Y  and the state machines 
               
               
                   
                 command the PHY ports to begin driving the first arbitration 
               
               
                   
                 indication of any response. t 5b  lags t 5a  by an IDLE_GAP and an 
               
               
                   
                 unspecified state machine delay herein called SM_DELAY. 
               
               
                 t 6   
                 PHY Y drives arbitration at points P′ Y . t 6  follows t 5b  by any 
               
               
                   
                 transceiver delay. 
               
               
                   
               
            
           
         
       
     
                           t     3   ′       =       t   3     +     PHY_DELAY   Y       P   Y   ′     →     P   Y                   (   19   )                 t     4   ′       =         t     3   ′       +     packet_length     packet_speed   ·     BASERATE   Y           =       t   3     +     PHY_DELAY   Y       P   Y   ′     →     P   Y         +     packet_length     packet_speed   ·     BASERATE   Y                     (   20   )                 t     5   ′       =         t     4   ′       +     DATA_END   ⁢     _TIME   Y     P   Y           =       t   3     +     PHY_DELAY   Y       P   Y   ′     →     P   Y         +     packet_length     packet_speed   ·     BASERATE   Y         +     DATA_END   ⁢     _TIME   Y     P   Y                     (   21   )                 t     5   ⁢   a       =         t     5   ′       -     transceiver_delay   Y     P   Y         =       t   3     +     PHY_DELAY   Y       P   Y   ′     →     P   Y         +     packet_length     packet_speed   ·     BASERATE   Y         +     DATA_END   ⁢     _TIME   Y     P   Y         -     transceiver_delay   Y     P   Y                   (   22   )                 t     5   ⁢   b       =         t     5   ⁢   a       +     IDLE_GAP   Y     +     SM_DELAY   Y       =       t   3     +     PHY_DELAY   Y       P   Y   ′     →     P   Y         +     packet_length     packet_speed   ·     BASERATE   Y         +     DATA_END   ⁢     _TIME   Y     P   Y         +     IDLE_GAP   Y     +     SM_DELAY   Y     -     transceiver_delay   Y     P   Y                   (   23   )                 t   6     =         t     5   ⁢   b       +     transceiver_delay   Y     P   Y         =       t   3     +     PHY_DELAY   Y       P   Y   ′     →     P   Y         +     packet_length     packet_speed   ·     BASERATE   Y         +     DATA_END   ⁢     _TIME   Y     P   Y         +     IDLE_GAP   Y     +     SM_DELAY   Y     +     transceiver_delay   Y     P   Y       -     transceiver_delay   Y     P   Y                   (   24   )                 
By definition,
 
                     RESPONSE_TIME   Y     P   Y   ′       =       t   6     -     t   5               (   25   )               
and through substitution:
 
                     RESPONSE_TIME   Y     P   Y   ′       =       PHY_DELAY   Y       P   Y   ′     →     P   Y         +     DE_delta     [       P   y     ,     P     y   -   1         ]       +     PPM_delta     [     Y   ,   Y     ]       +     IDLE_GAP   Y     +     SM_DELAY   Y     +     transceiver_delay   Y     P   Y       -     transceiver_delay   Y     P   Y                 (   26   )               
As such, the combination of RESPONSE_TIME, DE_delta, and PPM_delta from equation (18) can be represented as:
 
                     [             RESPONSE_TIME   Y     P   Y   ′       +                 DE_delta     [       P     y   -   1       ,     P   x       ]       +               PPM_delta     [       Y   -   1     ,   X     ]             ]     =             [           ⁢                   PHY_DELAY   Y       P   Y   ′     →     P   Y         +                 DE_delta     [       P   y     ,     P     y   -   1         ]       +     PPM_delta     [     Y   ,     Y   -   1       ]       +                       IDLE_GAP   Y     +     SM_DELAY   Y     +                 transceiver_delay   Y     P   Y       -     transceiver_delay   Y     P   Y       +                 DE_delta     [       P     y   -   1       ,     P   x       ]       +     PPM_delta     [       Y   -   1     ,   X     ]               ]               [                   PHY_DELAY   Y       P   Y   ′     →     P   Y         +                 DE_delta     [       P     y   ,       ,     P   x       ]       +     PPM_delta     [     Y   ,   X     ]       +                       IDLE_GAP   Y     +     SM_DELAY   Y     +                 transceiver_delay   Y     P   Y       -     transceiver_delay   Y     P   Y               ]           =             (   27   )               
Noting that if PHYs X and Y−1 both adhere to the same minimum timing requirement for DATA_END_TIME and maximum timing requirement for BASE_RATE, then
 
                                 DE_delta   max     [       P     Y   ,       ,     P   X       ]       =     DE_delta     [       P     Y   ,       ,     P     Y   -   1         ]                     PPM_delta   max     [     Y   ,   X     ]       =     PPM_delta   max     [     Y   ,     Y   -   1       ]                     (   28   )                 
The combined maximum can be rewritten as:
 
                       [             RESPONSE_TIME   Y     P   Y   ′       +                 DE_delta     [       P     Y   -   1       ,     P   X       ]       +               PPM_delta     [       Y   -   1     ,   X     ]             ]     max     =     
     ⁢     [           ⁢                   PHY_DELAY     Y   ,   max         P   Y   ′     →     P   Y         +                 DE_delta   max     [       P     Y   ,       ,     P     Y   -   1         ]       +     PPM_delta     [     Y   ,     Y   -   1       ]       +                       IDLE_GAP     Y   ,   max       +     SM_DELAY     Y   ,   max       +                 transceiver_delay     Y   ,   max       P   Y       -     transceiver_delay     Y   ,   min       P   Y               ]             (   29   )               
Comparing to equation (26) allows
 
                       [               RESPONSE   -     ⁢     TIME   Y     P   Y   ′         +                 DE_delta     [       P       Y   -   1     ,       ⁢     P   X       ]       +               PPM_delta     [       Y   -   1     ,   X     ]             ]       max   ⁢               =     RESPONSE_TIME     Y   ,   max       P   Y   ′               (   30   )               
Finally:
 
     
       
         
           
             
               
                 
                   gap_count 
                   &gt; 
                   
                     
                       
                         
                           BASERATE 
                           
                             X 
                             , 
                             max 
                           
                         
                         · 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     Round_Trip 
                                     ⁢ 
                                     
                                       _Delay 
                                       max 
                                       
                                         [ 
                                         
                                           
                                             P 
                                             X 
                                           
                                           ⁢ 
                                           O 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             P 
                                             Y 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     RESPONSE_TIME 
                                     
                                       Y 
                                       , 
                                       max 
                                     
                                     
                                       P 
                                       Y 
                                       ′ 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   PHY_DELAY 
                                   
                                     X 
                                     , 
                                     max 
                                   
                                   
                                     P 
                                     X 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       - 
                       27 
                     
                     16 
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
     For any given topology, the gap count must be set such that subaction gaps observed/generated at one PHY aren&#39;t falsely interpreted as arb_reset gaps by another PHY in the network. Subaction gaps are known to be at their largest nearest the PHY that originated the last packet. To ensure that the most recent originating PHY doesn&#39;t begin a new fairness interval before all PHYs exit the current one, its arb_reset_gap timeout must be greater than the largest subaction_gap which can legally occur.  FIG. 4  illustrates the case in which PHY X originated the most recent packet and PHY Y is responding after a subaction gap with arbitration for the current fairness interval. 
     For all topologies, the idle time observed at point P x  must not exceed the arbitration reset gap detection time: 
     
       
         
           
             
               
                 
                   
                     Idle 
                     max 
                     
                       P 
                       X 
                     
                   
                   &lt; 
                   
                     arb_reset 
                     ⁢ 
                     
                       _gap 
                       min 
                       
                         P 
                         X 
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     The analysis is identical to the case in which Ack and Iso gaps are preserved with the exception that PHY Y takes longer to respond to the trailing edge of DATA_END. Let PHY Y have a response time of subaction_response_time. Then, 
     
       
         
           
             
               
                 
                   
                     Idle 
                     
                       P 
                       
                         X 
                         ⁢ 
                         
                             
                         
                       
                     
                   
                   = 
                   
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         
                           [ 
                           
                             
                               P 
                               X 
                             
                             ⁢ 
                             O 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               Y 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       subaction_response 
                       ⁢ 
                       
                         _time 
                         Y 
                         
                           P 
                           Y 
                           ′ 
                         
                       
                     
                     + 
                     
                       DE_delta 
                       
                         [ 
                         
                           
                             P 
                             
                               
                                 Y 
                                 - 
                                 1 
                               
                               , 
                             
                           
                           ⁢ 
                           
                             P 
                             X 
                           
                         
                         ] 
                       
                     
                     + 
                     
                       PPM_delta 
                       
                         [ 
                         
                           
                             Y 
                             - 
                             1 
                           
                           , 
                           X 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     Substituting into Equation (32), subaction gaps are preserved network-wide if and only if: 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 Round_Trip 
                                 ⁢ 
                                 
                                   _Delay 
                                   
                                     [ 
                                     
                                       
                                         P 
                                         X 
                                       
                                       ⁢ 
                                       O 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         P 
                                         Y 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 subaction_response 
                                 ⁢ 
                                 
                                   _time 
                                   Y 
                                   
                                     P 
                                     Y 
                                     ′ 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 DE_delta 
                                 
                                   [ 
                                   
                                     
                                       P 
                                       
                                         
                                           Y 
                                           - 
                                           1 
                                         
                                         , 
                                       
                                     
                                     ⁢ 
                                     
                                       P 
                                       X 
                                     
                                   
                                   ] 
                                 
                               
                               + 
                               
                                 PPM_delta 
                                 
                                   [ 
                                   
                                     
                                       Y 
                                       - 
                                       1 
                                     
                                     , 
                                     X 
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     max 
                   
                   &lt; 
                   
                     arb_reset 
                     ⁢ 
                     
                       _gap 
                       min 
                       
                         P 
                         X 
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     The minimum arb_reset_gap at point P x  isn&#39;t well known. IEEE1394-1995, in Table 4-33, defines the minimum arb_reset_gap timeout used at a PHY&#39;s internal state machines, not at the external interface. It has been argued that the internal and external representations of time may differ by as much as ARB_RESPONSE_DELAY when a PHY is counting elapsed time between an internally generated event and an externally received event. However, the ARB_RESPONSE_DELAY value for a particular PHY isn&#39;t generally known externally. Fortunately, the ARB_RESPONSE_DELAY value for a PHY whose FIFO is known to be empty is bounded by the worst case PHY_DELAY reported within the PHY register map. This suggests a realistic bound for the minimum subaction_gap referenced at point P x : 
                     arb_reset   ⁢     _gap   min     P   x         ≥       arb_reset   ⁢     _gap   min     i   x         -     PHY_DELAY     X   ,   max         P   x     ⁢                         (   35   )               
where
 
     
       
         
           
             
               
                 
                   
                     arb_reset 
                     ⁢ 
                     
                       _gap 
                       min 
                       
                         i 
                         x 
                       
                     
                   
                   = 
                   
                     
                       51 
                       + 
                       
                         gap_count 
                         · 
                         32 
                       
                     
                     
                       BASERATE 
                       
                         X 
                         , 
                         max 
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     The maximum subaction_response_time for PHY Y parallels the earlier dissection of RESPONSE_TIME. The timing sequence for subaction_response_time is identical to that of RESPONSE_TIME except that PHY Y, after concluding stop_tx_Packet( ), must wait to detect a subaction gap and then wait an additional arb_delay before calling start_tx_packet( ). Said differently, the idle period timed internally is a subaction gap plus arb_delay rather than an IDLE_GAP. Consequently, t 5b  becomes:
 
 t   5b   =t   5a +subaction_gap ir   +arb _delay ir   +SM _DELAY Y   (37)
 
and
 
                     subaction_response   ⁢     _time   Y     P   Y   ′         =       RESPONSE_TIME   Y     P   Y   ′       -     IDLE_GAP   Y     +     subaction_gap     i   Y       +     arb_delay     i   Y                 (   38   )               
Substituting into Equation (34),
 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 Round_Trip 
                                 ⁢ 
                                 
                                   _Delay 
                                   
                                     [ 
                                     
                                       
                                         P 
                                         X 
                                       
                                       ⁢ 
                                       O 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         P 
                                         Y 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               + 
                               
                                 RESPONSE_TIME 
                                 Y 
                                 
                                   P 
                                   Y 
                                   ′ 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 DE_delta 
                                 
                                   [ 
                                   
                                     
                                       P 
                                       
                                         Y 
                                         - 
                                         1 
                                       
                                     
                                     , 
                                     
                                       P 
                                       X 
                                     
                                   
                                   ] 
                                 
                               
                               + 
                               
                                 PPM_delta 
                                 
                                   [ 
                                   
                                     
                                       Y 
                                       - 
                                       1 
                                     
                                     , 
                                     X 
                                   
                                   ] 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 subaction_gap 
                                 
                                   i 
                                   Y 
                                 
                               
                               + 
                               
                                 arb_delay 
                                 
                                   i 
                                   Y 
                                 
                               
                               - 
                               
                                 IDLE_GAP 
                                 Y 
                               
                             
                           
                         
                       
                       ] 
                     
                     max 
                   
                   &lt; 
                   
                     arb_reset 
                     ⁢ 
                     
                       _gap 
                       min 
                       
                         P 
                         x 
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     Again, RESPONSE_TIME, DE_delta, and PPM_delta are not independent parameters. As shown previously, if PHYs X and Y−1 adhere to the same timing constant limits, the explicit DE_Delta and PPM_delta terms can be subsumed within RESPONSE_TIME giving: 
                     [             Round_Trip   ⁢     _Delay   max     [       P   X     ⁢   O   ⁢           ⁢     P   Y       ]         +     RESPONSE_TIME     Y   ,   max       P   Y   ′       +                 subaction_gap   max     i   Y       +     arb_delay   max     i   Y       -     MIN_IDLE   ⁢     _TIME   Y               ]     &lt;     arb_reset   ⁢     _gap   min     P   x                 (   40   )               
where
 
                     subaction_gap   max     i   Y       =       29   +     gap_count   ·   16         BASERATE     Y   ,   min                 (   41   )                 arb_delay   max     i   Y       =       gap_count   ·   4       BASERATE     Y   ,   min                 (   42   )               
and
 IDLE_GAP Y,min =MIN_IDLE_TIME Y   (43) 
Combining Equations (35), (36), (40), (41), and (42):
 
                     [             Round_Trip   ⁢     _Delay   max     [       P   X     ⁢   O   ⁢           ⁢     P   Y       ]         +                 RESPONSE_TIME     Y   ,   max       P   Y   ′       -                 MIN_IDLE   ⁢     _TIME   Y       +                 29   +     gap_count   ·   20         BASERATE     Y   ,   min               ]     &lt;     [           ⁢         51   +     gap_count   ·   32         BASERATE     X   ,   max         -     PHY_DELAY     X   ,   max       P   x         ]             (   44   )               
Solving for gap_count:
 
     
       
         
           
             
               
                 
                   gap_count 
                   &gt; 
                   
                     
                       
                         
                           
                             
                               
                                 BASERATE 
                                 
                                   X 
                                   , 
                                   max 
                                 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           Round_Trip 
                                           ⁢ 
                                           
                                             _Delay 
                                             max 
                                             
                                               [ 
                                               
                                                 
                                                   P 
                                                   X 
                                                 
                                                 ⁢ 
                                                 O 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 
                                                   P 
                                                   Y 
                                                 
                                               
                                               ] 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           RESPONSE_TIME 
                                           
                                             Y 
                                             , 
                                             max 
                                           
                                           
                                             P 
                                             Y 
                                             ′ 
                                           
                                         
                                         - 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           MIN_IDLE 
                                           ⁢ 
                                           
                                             _TIME 
                                             y 
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         PHY_DELAY 
                                         
                                           X 
                                           , 
                                           max 
                                         
                                         
                                           P 
                                           X 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               29 
                               · 
                               
                                 
                                   BASERATE 
                                   
                                     X 
                                     , 
                                     max 
                                   
                                 
                                 
                                   BASERATE 
                                   
                                     Y 
                                     , 
                                     min 
                                   
                                 
                               
                             
                             - 
                             51 
                           
                         
                       
                     
                     
                       32 
                       - 
                       
                         20 
                         · 
                         
                           
                             BASERATE 
                             
                               X 
                               , 
                               max 
                             
                           
                           
                             BASERATE 
                             
                               Y 
                               , 
                               min 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     For any given topology, the gap count must be set such that if a subaction gap is observed following an isochronous packet at one PHY, it is observed at all PHYs. The danger occurs when a subsequent arbitration indication is transmitted in the same direction as the previous data packet. Given that arbitration indications may propagate through intervening PHYs faster than data bits, gaps may be shortened as they are repeated.  FIG. 5  illustrates the case in which PHY X originates an isochronous packet, observes a subaction_gap, and begins to drive an arbitration indication. 
     For all topologies, the minimum idle time observed at point P′ Y  must always exceed the maximum subaction gap detection time: 
     
       
         
           
             
               
                 
                   
                     Idle 
                     min 
                     
                       P 
                       Y 
                       ′ 
                     
                   
                   &gt; 
                   
                     subaction_gap 
                     max 
                     
                       P 
                       Y 
                       ′ 
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
           
         
       
     
     The time events t 0  through t 5  are identical to the previous analyses. In this scenario, t 6  follows t 2  by the time it takes PHY X to time subaction_gap and arb_delay: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           t 
                           6 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             t 
                             2 
                           
                           + 
                           
                             subaction_gap 
                             
                               P 
                               x 
                             
                           
                           + 
                           
                             arb_delay 
                             
                               P 
                               x 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             t 
                             0 
                           
                           + 
                           
                             packet_length 
                             
                               packet_speed 
                               · 
                               
                                 BASERATE 
                                 X 
                               
                             
                           
                           + 
                           
                             DATA_END 
                             ⁢ 
                             
                               _TIME 
                               X 
                               
                                 P 
                                 X 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             subaction_gap 
                             
                               P 
                               x 
                             
                           
                           + 
                           
                             arb_delay 
                             
                               P 
                               x 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
     The 1995 specification provides the timeouts used internally by the state machine. The externally observed timing requirements could differ (given possible mismatches in transceiver delay and state machines between the leading edge of IDLE and the leading edge of the subsequent arbitration indication). However, previous works have suggested any such delays could and should be well matched and that the external timing would follow the internal timing exactly. Consequently,
 
subaction_gap P     x     +arb _delay P     x   =subaction_gap i     x     +arb _delay i     x     (48)
 
T7 follows T6 by the time it takes the arbitration signal to propagate through the intervening PHYs and cables:
 
                           t   7     =       ⁢       t   6     +       ∑     n   =     X   +   1         Y   -   1       ⁢     (       cable_delay   n     +     ARB_RESPONSE   ⁢     _DELAY   n       P   n   ′     -&gt;     P   n             )       +                     ⁢     cable_delay   X                 =       ⁢       t   0     +     packet_length     packet_speed   ·     BASERATE   X         +     DATA_END   ⁢     _TIME   X     P   X         +                     ⁢       subtract_gap     i   x       +     arb_delay     i   x       +                     ⁢         ∑     n   =     X   +   1         Y   -   1       ⁢     (       cable_delay   n     +     ARB_RESPONSE   ⁢     _DELAY   n       P   n   ′     -&gt;     P   n             )       +                     ⁢     cable_delay   X                   (   49   )               
Given t 0  through t 7  above, the Idle time seen at point P′ Y  is given as:
 
     
       
         
           
               
             
               
                 
                   
                     
                       Idle 
                       
                         P 
                         y 
                         ′ 
                       
                     
                     = 
                     
                       
                         
                           t 
                           7 
                         
                         - 
                         
                           t 
                           5 
                         
                       
                       = 
                       
                         
                           subaction_gap 
                           
                             i 
                             x 
                           
                         
                         + 
                         
                           arb_delay 
                           
                             i 
                             x 
                           
                         
                         - 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               
                                 X 
                                 + 
                                 1 
                               
                             
                             
                               Y 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 PHY_DELAY 
                                 n 
                                 
                                   
                                     P 
                                     n 
                                     ′ 
                                   
                                   → 
                                   
                                     P 
                                     n 
                                   
                                 
                               
                               - 
                               
                                 ARB_RESPONSE 
                                 ⁢ 
                                 
                                   _DELAY 
                                   n 
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     → 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           DE_delta 
                           
                             [ 
                             
                               
                                 P 
                                 
                                   Y 
                                   - 
                                   1 
                                 
                               
                               , 
                               
                                 P 
                                 X 
                               
                             
                             ] 
                           
                         
                         - 
                         
                           PPM_delta 
                           
                             [ 
                             
                               
                                 Y 
                                 - 
                                 1 
                               
                               , 
                               X 
                             
                             ] 
                           
                         
                       
                     
                   
                 
                 
                   
                     ( 
                     50 
                     ) 
                   
                 
               
               
                 
                   Let 
                 
                 
                   
                       
                   
                 
               
               
                 
                   
                     
                       Data_Arb 
                       ⁢ 
                       
                         _Mismatch 
                         
                           [ 
                           
                             
                               P 
                               X 
                             
                             → 
                             
                               R 
                               Y 
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             PHY_DELAY 
                             n 
                             
                               
                                 P 
                                 n 
                                 ′ 
                               
                               → 
                               
                                 P 
                                 n 
                               
                             
                           
                           - 
                           
                             ARB_RESPONSE 
                             ⁢ 
                             
                               _DELAY 
                               n 
                               
                                 
                                   P 
                                   n 
                                   ′ 
                                 
                                 → 
                                 
                                   P 
                                   n 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     ( 
                     51 
                     ) 
                   
                 
               
               
                 
                   
                     Then 
                     , 
                   
                 
                 
                   
                       
                   
                 
               
               
                 
                   
                     
                       Idle 
                       
                         P 
                         y 
                         ′ 
                       
                     
                     = 
                     
                       
                         
                           t 
                           7 
                         
                         - 
                         
                           t 
                           5 
                         
                       
                       = 
                       
                         
                           subaction_gap 
                           
                             i 
                             x 
                           
                         
                         + 
                         
                           arb_delay 
                           
                             i 
                             x 
                           
                         
                         - 
                         
                           Data_Arb 
                           ⁢ 
                           
                             _Mismatch 
                             
                               [ 
                               
                                 
                                   P 
                                   X 
                                 
                                 → 
                                 
                                   R 
                                   Y 
                                 
                               
                               ] 
                             
                           
                         
                         - 
                         
                           DE_delta 
                           
                             [ 
                             
                               
                                 P 
                                 
                                   Y 
                                   - 
                                   1 
                                 
                               
                               , 
                               
                                 P 
                                 X 
                               
                             
                             ] 
                           
                         
                         - 
                         
                           PPM_delta 
                           
                             [ 
                             
                               
                                 Y 
                                 - 
                                 1 
                               
                               , 
                               X 
                             
                             ] 
                           
                         
                       
                     
                   
                 
                 
                   
                     ( 
                     52 
                     ) 
                   
                 
               
             
           
         
       
     
     For the maximum subaction_gap detection time at point P′ Y , the 1995 standard again only specifies the internal state machine timeout values.  FIG. 6  provides the timing reference for relating the external gap detection times to the internal ones. The elaborated timing sequence is identical to the case for RESPONSE_TIME through point t 5 ′. The remaining sequence is: 
                                    T 7     The arbitration indication launched by PHY X arrives at point P′ Y         T 7a     The arbitration indication launched by PHY X arrives at point iY. t 7a             lags t 7  by an unspecified arbitration detection time, herein termed           ARB_DETECTION_TIME                    
The externally seen gap at point P′ Y  is given as
 gap P′     Y     =t   7   −t   5   (53) 
The corresponding internal gap at point iY is
 gap i     y     =t   7a   −t   5a   (54) 
Given that
 
                                            t 7a  = t 7  + ARB_DETECTION_TIME Y   P′     Y                            
the external gap can be expressed as
 
                                 gap     P   Y   ′       =       ⁢       t   7     -     t   5                   =       ⁢       t     7   ⁢   a       -     t   5     -     ARB_DETECTION   ⁢     _TIME   Y     P   Y   ′                       =       ⁢       t     7   ⁢   a       -     t     5   ⁢   a       +     t     5   ⁢   a       -     t   5     -     ARB_DETECTION   ⁢     _TIME   Y     P   Y   ′                       =       ⁢       gap     i   Y       +     t     5   ⁢   a       -     t   5     -     ARB_DETECTION   ⁢     _TIME   Y     P   Y   ′                       =       ⁢       gap     i   Y       +     PHY_DELAY   n       P   n   ′     →     P   n         +     DE_delta     [       P   Y     ,     P     Y   -   1         ]       +                     ⁢       PPM_delta     [     Y   ,     Y   -   1       ]       -     transceiver_delay   Y     P   Y       -                     ⁢     ARB_DETECTION   ⁢     _TIME   Y     P   Y   ′                       (   56   )                 
Consequently,
 
                           subaction_gap     P   Y   ′       =       subaction_gap     i   Y       +     PHY_DELAY   Y       P   Y   ′     →     P   Y         +     DE_delta     [       P   Y     ,     P     Y   -   1         ]       +     PPM_delta     [     Y   ,     Y   -   1       ]       -     transceiver_delay   Y     P   Y       -     ARB_DETECTION   ⁢     _TIME   Y     P   Y   ′                   (   57   )                 
Substituting (52) and (57) into (46) yields
 
                           [             subaction_gap     i   x       +     arb_delay     i   x       -                 Data_Arb   ⁢     _Mismatch     [       P   x     →     P   Y       ]         -                 DE_delta     [       P     Y   -   1       ,     P   X       ]       -               PPM_delta     [       Y   -   1     ,   X     ]             ]     &gt;     
     ⁢     [             subaction_gap     i   x       +     PHY_DELAY   Y       P   Y   ′     →     P   Y         +                 DE_delta     [     P     Y   ,     P     Y   -   1           ]       +     PPM_delta     [       Y   -   1     ,   X     ]       -                 transceiver_delay   Y     P   Y       -     ARB_DETECTION   ⁢     _TIME   Y     P   Y   ′                 ]             (   58   )                 
The inequality holds generally if
 
                             [       subaction_gap     i   x       +     arb_delay     i   x         ]     min     &gt;     
     ⁢       [             subaction_gap     i   x       +     PHY_DELAY   Y       P   Y   ′     →     P   Y         +                 DE_delta     [     P     Y   ,     P     Y   -   1           ]       +     PPM_delta     [       Y   -   1     ,   X     ]       +                 DE_delta     [       P     Y   -   1       ,     P   X       ]       -     PPM_delta     [       Y   -   1     ,   X     ]       +                 Data_Arb   ⁢     _Mismatch     [       P   x     →     P   Y       ]         -                 transceiver_delay   Y     P   Y       -     ARB_DETECTION   ⁢     _TIME   Y     P   Y   ′                 ]     max             (   59   )                 
Combining the DE_Delta and PPM_delta terms gives:
 
                             [       subaction_gap     i   x       +     arb_delay     i   x         ]     min     &gt;     
     ⁢       [             subaction_gap     i   x       +     PHY_DELAY   Y       P   Y   ′     →     P   Y         +                 DE_delta     [       P   Y     ,     P   X       ]       +     PPM_delta     [     Y   ,   X     ]       +                 Data_Arb   ⁢     _Mismatch     [       P   x     →     P   Y       ]         -                 transceiver_delay   Y     P   Y       -     ARB_DETECTION   ⁢     _TIME   Y     P   Y   ′                 ]     max             (   60   )                 
By assuming
   DE _delta [P     Y     ,P     X     ]   +PPM _delta [Y,X] ≦transceiver_delay Y   P     Y     +ARB _DETECTION_TIME Y   P′     Y     (61) 
the constraining inequality can be further simplified to give
 
                             [       subaction_gap     i   x       +     arb_delay     i   x         ]     min     &gt;     
     ⁢     [             subaction_gap     i   x       +     PHY_DELAY     Y   ,   max         P   Y   ′     →     P   Y         +               Data_Arb   ⁢     _Mismatch   max     [       P   x     →     P   Y       ]               ]             (   62   )                 
where
 
                           subaction_gap   min     i   x       =       27   +     gap_count   ·   16         BASERATE     X   ,   max                 (   63   )                 arb_delay   min     i   x       =       gap_count   ·   4       BASERATE     X   ,   man                 (   64   )             and                           subaction_gap   max     i   y       =       29   +     gap_count   ·   16         BASERATE     Y   ,   min                 (   65   )                 
Solving for gap count,
 
     
       
         
           
             
               
                 
                   gap_count 
                   &gt; 
                   
                     
                       
                         
                           BASERATE 
                           
                             X 
                             , 
                             max 
                           
                         
                         · 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     PHY_DELAY 
                                     
                                       Y 
                                       , 
                                       max 
                                     
                                     
                                       
                                         P 
                                         Y 
                                         ′ 
                                       
                                       → 
                                       
                                         P 
                                         Y 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   Data_Arb 
                                   ⁢ 
                                   
                                     _Mismatch 
                                     max 
                                     
                                       [ 
                                       
                                         
                                           P 
                                           x 
                                         
                                         → 
                                         
                                           P 
                                           Y 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         29 
                         · 
                         
                           
                             BASERATE 
                             
                               X 
                               , 
                               max 
                             
                           
                           
                             BASERATE 
                             
                               Y 
                               , 
                               min 
                             
                           
                         
                       
                       - 
                       27 
                     
                     
                       20 
                       - 
                       
                         16 
                         · 
                         
                           
                             BASERATE 
                             
                               X 
                               , 
                               max 
                             
                           
                           
                             BASERATE 
                             
                               Y 
                               , 
                               min 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
           
         
       
     
     For any given topology, the gap count must be set such that if an arbitration reset gap is observed following an asynchronous packet at one PHY, it is observed at all PHYs. The danger occurs when a subsequent arbitration indication is transmitted in the same direction as the previous data packet. Given that arbitration indications may propagate through intervening PHYs faster than data bits, gaps may be shortened as they are repeated.  FIG. 7  illustrates the case in which PHY X originates an asynchronous packet, observes an arbitration reset gap, and begins to drive an arbitration indication. 
     For all topologies, the minimum idle time observed at point P′ Y  must always exceed the maximum arbitration reset gap detection time: 
     
       
         
           
             
               
                 
                   
                     Idle 
                     min 
                     
                       P 
                       Y 
                       ′ 
                     
                   
                   &gt; 
                   
                     arb_reset 
                     ⁢ 
                     
                       _gap 
                       max 
                       
                         
                           P 
                           Y 
                           
                             ′ 
                             ⁢ 
                             
                                 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   67 
                   ) 
                 
               
             
           
         
       
     
     The time events t 0  through t 5  are identical to the previous analyses. In this scenario, t 6  follows t 2  by the time it takes PHY X to time arb_reset_gap and arb_delay: 
     
       
         
           
             
               
                 
                   
                     t 
                     6 
                   
                   = 
                   
                     
                       
                         t 
                         2 
                       
                       + 
                       arb_reset 
                       - 
                       
                         gap 
                         
                           P 
                           X 
                         
                       
                       + 
                       
                         arb_delay 
                         
                           P 
                           X 
                         
                       
                     
                     = 
                     
                       
                         t 
                         0 
                       
                       + 
                       
                         packet_length 
                         
                           packet_speed 
                           · 
                           
                             BASERATE 
                             X 
                           
                         
                       
                       + 
                       
                         DATA_END 
                         ⁢ 
                         
                           _TIME 
                           X 
                           
                             P 
                             x 
                           
                         
                       
                       + 
                       
                         arb_reset 
                         ⁢ 
                         
                           _gap 
                           
                             P 
                             X 
                           
                         
                       
                       + 
                       
                         arb_delay 
                         
                           P 
                           X 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   68 
                   ) 
                 
               
             
           
         
       
     
     The 1995 IEEE 1394 standard provides the timeouts used internally by the state machine. The externally observed timing requirements could differ (given possible mismatches in transceiver delay and state machines between the leading edge of IDLE and the leading edge of the subsequent arbitration indication). However, previous works have suggested any such delays could and should be well matched and that the external timing would follow the internal timing exactly. Consequently,
 
 arb _reset_gap P     x     +arb _delay P     x     =arb _reset_gap i     x     +arb _delay i     x     (69)
 
     T7 follows T6 by the time it takes the arbitration signal to propagate through the intervening PHYs and cables: 
     
       
         
           
               
             
               
                 
                   
                     
                       
                         
                           
                             t 
                             7 
                           
                           = 
                             
                           ⁢ 
                           
                             
                               t 
                               6 
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   
                                     X 
                                     + 
                                     1 
                                   
                                 
                                 
                                   Y 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     cable_delay 
                                     n 
                                   
                                   + 
                                   
                                     ARB_RESPONSE 
                                     ⁢ 
                                     
                                       _DELAY 
                                       n 
                                       
                                         
                                           P 
                                           n 
                                           ′ 
                                         
                                         → 
                                         
                                           P 
                                           n 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             cable_delay 
                             - 
                             X 
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               t 
                               0 
                             
                             + 
                             
                               packet_length 
                               
                                 packet_speed 
                                 · 
                                 
                                   BASEREAT 
                                   X 
                                 
                               
                             
                             + 
                             
                               DATA_END 
                               ⁢ 
                               
                                 _TIME 
                                 X 
                                 
                                   P 
                                   x 
                                 
                               
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               arb_reset 
                               ⁢ 
                               
                                 _gap 
                                 
                                   i 
                                   X 
                                 
                               
                             
                             + 
                             
                               arb_delay 
                               
                                 i 
                                 X 
                               
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   
                                     X 
                                     + 
                                     1 
                                   
                                 
                                 
                                   Y 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     cable_delay 
                                     n 
                                   
                                   + 
                                   
                                     ARB_RESPONSE 
                                     ⁢ 
                                     
                                       _DELAY 
                                       n 
                                       
                                         
                                           P 
                                           n 
                                           ′ 
                                         
                                         → 
                                         
                                           P 
                                           n 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             cable_delay 
                             - 
                             X 
                           
                         
                       
                     
                   
                 
                 
                   
                     ( 
                     70 
                     ) 
                   
                 
               
             
           
         
       
     
     Given t 0  through t 7  above, the Idle time seen at point P′ Y  is given as: 
     
       
         
           
             
               
                 
                   
                     Idle 
                     
                       P 
                       Y 
                       ′ 
                     
                   
                   = 
                   
                     
                       
                         t 
                         7 
                       
                       - 
                       
                         t 
                         5 
                       
                     
                     = 
                     
                       
                         arb_reset 
                         ⁢ 
                         
                           _gap 
                           
                             i 
                             X 
                           
                         
                       
                       + 
                       
                         arb_delay 
                         
                           i 
                           X 
                         
                       
                       - 
                       
                         Data_Arb 
                         ⁢ 
                         
                           _Mismatch 
                           
                             [ 
                             
                               
                                 P 
                                 x 
                               
                               → 
                               
                                 P 
                                 Y 
                               
                             
                             ] 
                           
                         
                       
                       - 
                       
                         DE_delta 
                         
                           [ 
                           
                             
                               P 
                               
                                 Y 
                                 - 
                                 1 
                               
                             
                             , 
                             
                               P 
                               X 
                             
                           
                           ] 
                         
                       
                       - 
                       
                         PPM_delta 
                         
                           [ 
                           
                             
                               Y 
                               - 
                               1 
                             
                             , 
                             X 
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   71 
                   ) 
                 
               
             
           
         
       
     
     For the maximum arbitration_reset_gap detection time at point P′ Y , equation (56) gives: 
     
       
         
           
             
               
                 
                   
                     arb_reset 
                     ⁢ 
                     
                       _gap 
                       
                         P 
                         Y 
                         ′ 
                       
                     
                   
                   = 
                   
                     
                       arb_reset 
                       ⁢ 
                       
                         _gap 
                         
                           i 
                           Y 
                         
                       
                     
                     + 
                     
                       PHY_DELAY 
                       Y 
                       
                         
                           P 
                           Y 
                           ′ 
                         
                         → 
                         
                           P 
                           Y 
                         
                       
                     
                     + 
                     
                       DE_delta 
                       
                         [ 
                         
                           P 
                           
                             Y 
                             , 
                             
                               P 
                               
                                 Y 
                                 - 
                                 1 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       PPM_delta 
                       
                         [ 
                         
                           
                             Y 
                             - 
                             1 
                           
                           , 
                           X 
                         
                         ] 
                       
                     
                     - 
                     
                       transceiver_delay 
                       Y 
                       
                         P 
                         Y 
                       
                     
                     - 
                     
                       ARB_DETECTION 
                       ⁢ 
                       
                         _TIME 
                         Y 
                         
                           P 
                           Y 
                           ′ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   72 
                   ) 
                 
               
             
           
         
       
     
     Substituting (71) and (72) into (67) yields 
     
       
         
           
               
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 arb_reset 
                                 ⁢ 
                                 
                                   _gap 
                                   
                                     i 
                                     X 
                                   
                                 
                               
                               + 
                               
                                 arb_delay 
                                 
                                   i 
                                   x 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 Data_Arb 
                                 ⁢ 
                                 
                                   _Mismatch 
                                   
                                     [ 
                                     
                                       
                                         P 
                                         x 
                                       
                                       → 
                                       
                                         P 
                                         Y 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 DE_delta 
                                 
                                   [ 
                                   
                                     
                                       P 
                                       
                                         Y 
                                         - 
                                         1 
                                       
                                     
                                     , 
                                     
                                       P 
                                       X 
                                     
                                   
                                   ] 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               PPM_delta 
                               
                                 [ 
                                 
                                   
                                     Y 
                                     - 
                                     1 
                                   
                                   , 
                                   X 
                                 
                                 ] 
                               
                             
                           
                         
                       
                       ] 
                     
                     &gt; 
                     
                       
 
                     
                     ⁢ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 arb_reset 
                                 ⁢ 
                                 
                                   _gap 
                                   
                                     i 
                                     Y 
                                   
                                 
                               
                               + 
                               
                                 PHY_DELAY 
                                 Y 
                                 
                                   
                                     P 
                                     Y 
                                     ′ 
                                   
                                   → 
                                   
                                     P 
                                     Y 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 DE_delta 
                                 
                                   [ 
                                   
                                     P 
                                     
                                       Y 
                                       , 
                                       
                                         P 
                                         
                                           Y 
                                           - 
                                           1 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                               + 
                               
                                 PPM_delta 
                                 
                                   [ 
                                   
                                     
                                       Y 
                                       - 
                                       1 
                                     
                                     , 
                                     X 
                                   
                                   ] 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 transceiver_delay 
                                 Y 
                                 
                                   P 
                                   Y 
                                 
                               
                               - 
                               
                                 ARB_DETECTION 
                                 ⁢ 
                                 
                                   _TIME 
                                   Y 
                                   
                                     P 
                                     Y 
                                     ′ 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
                 
                   
                     ( 
                     73 
                     ) 
                   
                 
               
             
           
         
       
     
     The inequality holds generally if 
     
       
         
           
               
             
               
                 
                   
                     
                       
                         [ 
                         
                           
                             arb_reset 
                             ⁢ 
                             
                               _gap 
                               
                                 i 
                                 X 
                               
                             
                           
                           + 
                           
                             arb_delay 
                             
                               i 
                               x 
                             
                           
                         
                         ] 
                       
                       min 
                     
                     &gt; 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   arb_reset 
                                   ⁢ 
                                   
                                     _gap 
                                     
                                       i 
                                       Y 
                                     
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   Y 
                                   
                                     
                                       P 
                                       Y 
                                       ′ 
                                     
                                     → 
                                     
                                       P 
                                       Y 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   DE_delta 
                                   
                                     [ 
                                     
                                       
                                         P 
                                         Y 
                                       
                                       , 
                                       
                                         P 
                                         
                                           Y 
                                           - 
                                           1 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                 + 
                                 
                                   PPM_delta 
                                   
                                     [ 
                                     
                                       Y 
                                       , 
                                       
                                         Y 
                                         - 
                                         1 
                                       
                                     
                                     ] 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   DE_delta 
                                   
                                     [ 
                                     
                                       
                                         P 
                                         
                                           Y 
                                           - 
                                           1 
                                         
                                       
                                       , 
                                       
                                         P 
                                         X 
                                       
                                     
                                     ] 
                                   
                                 
                                 + 
                                 
                                   PPM_delta 
                                   
                                     [ 
                                     
                                       
                                         Y 
                                         - 
                                         1 
                                       
                                       , 
                                       X 
                                     
                                     ] 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   Data_Arb 
                                   ⁢ 
                                   
                                     _Mismatch 
                                     
                                       [ 
                                       
                                         
                                           P 
                                           x 
                                         
                                         → 
                                         
                                           P 
                                           Y 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   transceiver_delay 
                                   Y 
                                   
                                     P 
                                     Y 
                                   
                                 
                                 - 
                                 
                                   ARB_DETECTION 
                                   ⁢ 
                                   
                                     _TIME 
                                     Y 
                                     
                                       P 
                                       Y 
                                       ′ 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       max 
                     
                   
                 
                 
                   
                     ( 
                     74 
                     ) 
                   
                 
               
             
           
         
       
     
     Combining the DE_Delta and PPM_delta terms gives: 
                     ⁢               [       arb_reset   ⁢     _gap     i   x         +     arb_delay     i   x         ]     min     &gt;             [             arb_reset   ⁢     _gap     i   Y         +     PHY_DELAY   Y         P   ′     Y     -&gt;     P   Y         +                                   DE_delta     [       P   Y     ,     P   X       ]       +     PPM_delta     [     Y   ,   X     ]       +                 Data_Arb   ⁢     _Mismatch     [       P   X     -&gt;     P   Y       ]         -                 transceiver_delay   Y     P   Y       -     ARB_DETECTION   ⁢     _TIME   Y       P   ′     Y                                   ]     max               (   75   )                 
By requiring
   DE _delta [P     Y     ,P     X     ]   +PPM _delta [Y,X] ≦transceiver_delay Y   P     Y     +ARB _DETECTION_TIME Y   P′     Y     Y   (76) 
     the constraining inequality can be further simplified to give 
                     ⁢             [       arb_reset   ⁢     _gap   min     i   x         +     arb_delay   min     i   x         ]     &gt;           [             arb_reset   ⁢     _gap   max     i   Y         +     PHY_DELAY     Y   ,   max         P   Y   ′     -&gt;     P   Y         +                           Data_Arb   ⁢     _Mismatch   max     [       P   X     -&gt;     P   Y       ]                           ]               (   77   )                 
where
 
     
       
         
           
             
               
                 
                   
                     arb_reset 
                     ⁢ 
                     
                       _gap 
                       min 
                       
                         i 
                         x 
                       
                     
                   
                   = 
                   
                     
                       51 
                       + 
                       
                         gap_count 
                         · 
                         32 
                       
                     
                     
                       BASERATE 
                       
                         X 
                         , 
                         max 
                       
                     
                   
                 
               
               
                 
                   ( 
                   78 
                   ) 
                 
               
             
             
               
                 
                   
                     arb_delay 
                     min 
                     
                       i 
                       x 
                     
                   
                   = 
                   
                     
                       
                         gap_count 
                         · 
                         4 
                       
                       
                         BASERATE 
                         
                           X 
                           , 
                           max 
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     and 
                   
                 
               
               
                 
                   ( 
                   79 
                   ) 
                 
               
             
             
               
                 
                   
                     arb_reset 
                     ⁢ 
                     
                       _gap 
                       max 
                       
                         i 
                         Y 
                       
                     
                   
                   = 
                   
                     
                       53 
                       + 
                       
                         gap_count 
                         · 
                         32 
                       
                     
                     
                       BASERATE 
                       
                         Y 
                         , 
                         min 
                       
                     
                   
                 
               
               
                 
                   ( 
                   80 
                   ) 
                 
               
             
           
         
       
     
     Solving for gap count, 
     
       
         
           
             
               
                 
                   gap_count 
                   &gt; 
                   
                     
                       
                         
                           
                             
                               
                                 BASERATE 
                                 
                                   X 
                                   , 
                                   max 
                                 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           PHY_DELAY 
                                           
                                             Y 
                                             , 
                                             max 
                                           
                                           
                                             
                                               
                                                 P 
                                                 ′ 
                                               
                                               Y 
                                             
                                             -&gt; 
                                             
                                               P 
                                               Y 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         Data_Arb 
                                         ⁢ 
                                         
                                           _Mismatch 
                                           max 
                                           
                                             [ 
                                             
                                               
                                                 P 
                                                 X 
                                               
                                               -&gt; 
                                               
                                                 P 
                                                 Y 
                                               
                                             
                                             ] 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               53 
                               · 
                               
                                 
                                   BASERATE 
                                   
                                     X 
                                     , 
                                     max 
                                   
                                 
                                 
                                   BASERATE 
                                   
                                     Y 
                                     , 
                                     min 
                                   
                                 
                               
                             
                             - 
                             51 
                           
                         
                       
                     
                     
                       36 
                       - 
                       
                         32 
                         · 
                         
                           
                             BASERATE 
                             
                               X 
                               , 
                               max 
                             
                           
                           
                             BASERATE 
                             
                               Y 
                               , 
                               min 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   81 
                   ) 
                 
               
             
           
         
       
     
     Equations (31), (45), (66) and (81) place a lower bound on gap count. Let: 
     
       
         
           
             
               
                 
                   
                     gap_count 
                     A 
                   
                   = 
                   
                     
                       
                         
                           BASERATE 
                           
                             X 
                             , 
                             max 
                           
                         
                         · 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     Round_Trip 
                                     ⁢ 
                                     
                                       _Delay 
                                       max 
                                       
                                         [ 
                                         
                                           
                                             P 
                                             X 
                                           
                                           ⁢ 
                                           O 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             P 
                                             Y 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     RESPONSE_TIME 
                                     
                                       Y 
                                       , 
                                       max 
                                     
                                     
                                       P 
                                       Y 
                                       ′ 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   PHY_DELAY 
                                   
                                     X 
                                     , 
                                     max 
                                   
                                   
                                     P 
                                     X 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       - 
                       27 
                     
                     16 
                   
                 
               
               
                 
                   ( 
                   82 
                   ) 
                 
               
             
             
               
                 
                   
                     gap_count 
                     B 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 BASERATE 
                                 
                                   X 
                                   , 
                                   max 
                                 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           Round_Trip 
                                           ⁢ 
                                           
                                             _Delay 
                                             max 
                                             
                                               [ 
                                               
                                                 
                                                   P 
                                                   X 
                                                 
                                                 ⁢ 
                                                 O 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 
                                                   P 
                                                   Y 
                                                 
                                               
                                               ] 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           RESPONSE_TIME 
                                           
                                             Y 
                                             , 
                                             max 
                                           
                                           
                                             P 
                                             Y 
                                             ′ 
                                           
                                         
                                         - 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           MIN_IDLE 
                                           ⁢ 
                                           
                                             _TIME 
                                             Y 
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         PHY_DELAY 
                                         
                                           X 
                                           , 
                                           max 
                                         
                                         
                                           P 
                                           X 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               29 
                               · 
                               
                                 
                                   BASERATE 
                                   
                                     X 
                                     , 
                                     max 
                                   
                                 
                                 
                                   BASERATE 
                                   
                                     Y 
                                     , 
                                     min 
                                   
                                 
                               
                             
                             - 
                             51 
                           
                         
                       
                     
                     
                       32 
                       - 
                       
                         20 
                         · 
                         
                           
                             BASERATE 
                             
                               X 
                               , 
                               max 
                             
                           
                           
                             BASERATE 
                             
                               Y 
                               , 
                               min 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   83 
                   ) 
                 
               
             
             
               
                 
                   
                     gap_count 
                     C 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 BASERATE 
                                 
                                   X 
                                   , 
                                   max 
                                 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           Data_Arb 
                                           ⁢ 
                                           
                                             _Mismatch 
                                             max 
                                             
                                               [ 
                                               
                                                 
                                                   P 
                                                   X 
                                                 
                                                 -&gt; 
                                                 
                                                   P 
                                                   Y 
                                                 
                                               
                                               ] 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         PHY_DELAY 
                                         
                                           Y 
                                           , 
                                           max 
                                         
                                         
                                           
                                             P 
                                             Y 
                                             ′ 
                                           
                                           -&gt; 
                                           
                                             P 
                                             Y 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               29 
                               · 
                               
                                 
                                   BASERATE 
                                   
                                     X 
                                     , 
                                     max 
                                   
                                 
                                 
                                   BASERATE 
                                   
                                     Y 
                                     , 
                                     min 
                                   
                                 
                               
                             
                             - 
                             27 
                           
                         
                       
                     
                     
                       20 
                       - 
                       
                         16 
                         · 
                         
                           
                             BASERATE 
                             
                               X 
                               , 
                               max 
                             
                           
                           
                             BASERATE 
                             
                               Y 
                               , 
                               min 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   84 
                   ) 
                 
               
             
             
               
                 
                   
                     gap_count 
                     D 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 BASERATE 
                                 
                                   X 
                                   , 
                                   max 
                                 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           Data_Arb 
                                           ⁢ 
                                           
                                             _Mismatch 
                                             max 
                                             
                                               [ 
                                               
                                                 
                                                   P 
                                                   X 
                                                 
                                                 -&gt; 
                                                 
                                                   P 
                                                   Y 
                                                 
                                               
                                               ] 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         PHY_DELAY 
                                         
                                           Y 
                                           , 
                                           max 
                                         
                                         
                                           
                                             P 
                                             Y 
                                             ′ 
                                           
                                           -&gt; 
                                           
                                             P 
                                             Y 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               53 
                               · 
                               
                                 
                                   BASERATE 
                                   
                                     X 
                                     , 
                                     max 
                                   
                                 
                                 
                                   BASERATE 
                                   
                                     Y 
                                     , 
                                     min 
                                   
                                 
                               
                             
                             - 
                             51 
                           
                         
                       
                     
                     
                       36 
                       - 
                       
                         32 
                         · 
                         
                           
                             BASERATE 
                             
                               X 
                               , 
                               max 
                             
                           
                           
                             BASERATE 
                             
                               Y 
                               , 
                               min 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   85 
                   ) 
                 
               
             
           
         
       
     
     Given the ratio of maximum to minimum BASERATE is always &gt;1 and that MIN_IDLE_TIME is ˜40 ns, it is clear that:
 
gap_count B &gt;gap_count A   (86)
 
and
 
gap_count D &gt;gap_count C   (87)
 
     To select an appropriate gap count for a given topology, both gap_count B  and gap_count D  must be calculated, rounded up to the next integer, and the maximum of the two results selected. 
     For IEEE1394-1995 style topologies (assumed to be limited to 4.5 m cables and a worst case PHY_DELAY of 144 ns), a table can be constructed to provide the gap count setting as a function of hops. In constructing such a table, the constant values in Table 2 are assumed. 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 PHY Timing Constants 
               
            
           
           
               
               
               
            
               
                 Parameter 
                 Minimum 
                 Maximum 
               
               
                   
               
               
                 ARB_RESPONSE_DELAY 1   
                 PHY_DELAY(max) − 60 ns 
                 PHY_DELAY(max) 
               
               
                 BASERATE 
                 98.294 mbps 
                 98.314 mbps 
               
               
                 cable_delay 
                   
                 22.725 ns 
               
               
                 MIN_IDLE_TIME 
                 40 ns 
               
               
                 PHY_DELAY 
                   
                 144 ns 
               
               
                 RESPONSE_TIME 
                   
                 PHY_DELAY + 100 ns 
               
               
                   
               
            
           
         
       
     
     The resulting gap count versus Cable Hops can then be calculated: 
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 Gap Count as a function of hops 
               
            
           
           
               
               
               
            
               
                   
                 Hops 
                 Gap Count 
               
               
                   
                   
               
            
           
           
               
               
               
            
               
                   
                 1 
                 5 
               
               
                   
                 2 
                 7 
               
               
                   
                 3 
                 8 
               
               
                   
                 4 
                 10 
               
               
                   
                 5 
                 13 
               
               
                   
                 6 
                 16 
               
               
                   
                 7 
                 18 
               
               
                   
                 8 
                 21 
               
               
                   
                 9 
                 24 
               
               
                   
                 10 
                 26 
               
               
                   
                 11 
                 29 
               
               
                   
                 12 
                 32 
               
               
                   
                 13 
                 35 
               
               
                   
                 14 
                 37 
               
               
                   
                 15 
                 40 
               
               
                   
                 16 
                 43 
               
               
                   
                 17 
                 46 
               
               
                   
                 18 
                 48 
               
               
                   
                 19 
                 51 
               
               
                   
                 20 
                 54 
               
               
                   
                 21 
                 57 
               
               
                   
                 22 
                 59 
               
               
                   
                 23 
                 62 
               
               
                   
                   
               
            
           
         
       
     
     Pinging provides an effective way to set an optimal gap count for topologies with initially unspecified or unknown PHY or cable delays. Specifically, pinging allows determination of an instantaneous Round_Trip_Delay between two given points. Once the worst case Round_Trip Delay has been determined via pinging, gap_count b  and gap_count d  can be calculated and the appropriate gap count selected. 
     The Jitter value specified in the PHY register map was introduced to help relate instantaneous measurements of ROUND_TRIP_DELAY to the maximum possible ROUND_TRIP_DELAY between two points. Specifically, the outbound PHY_DELAY and return ARB_RESPONSE_DELAY measured between a given ordered pair of ports on a PHY (say P c  out to and back from P d ) can be related to the maximum outbound PHY_DELAY and return ARB_RESPONSE_DELAY between any and all ordered pairs of ports (referenced as P a &amp; P b ) on the same PHY: 
     
       
         
           
             
               
                 
                   0 
                   ≤ 
                   
                     [ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           
                             
                               
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     max 
                                   
                                   
                                     
                                       P 
                                       a 
                                     
                                     -&gt; 
                                     
                                       P 
                                       b 
                                     
                                   
                                 
                                 + 
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       max 
                                     
                                     
                                       
                                         P 
                                         b 
                                       
                                       -&gt; 
                                       
                                         P 
                                         a 
                                       
                                     
                                   
                                 
                               
                               2 
                             
                             - 
                           
                         
                       
                       
                         
                           
                             
                               
                                 PHY_DELAY 
                                 
                                   n 
                                   , 
                                   meas 
                                 
                                 
                                   
                                     P 
                                     c 
                                   
                                   -&gt; 
                                   
                                     P 
                                     d 
                                   
                                 
                               
                               + 
                               
                                 ARB_RESPONSE 
                                 ⁢ 
                                 
                                   _DELAY 
                                   
                                     n 
                                     , 
                                     meas 
                                   
                                   
                                     
                                       P 
                                       d 
                                     
                                     -&gt; 
                                     
                                       P 
                                       c 
                                     
                                   
                                 
                               
                             
                             2 
                           
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                   ≤ 
                   
                     Jitter 
                     n 
                   
                 
               
               
                 
                   ( 
                   88 
                   ) 
                 
               
             
           
         
       
     
     Noting that a measured value can never exceed a maximum value between order ports, the following corollary relating two independent measurements can be proven for any and all combination of ordered ports: 
     
       
         
           
             
               
                 
                   ❘ 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       j 
                                     
                                   
                                   
                                     
                                       P 
                                       a 
                                     
                                     -&gt; 
                                     
                                       P 
                                       
                                         b 
                                         ⁢ 
                                         
                                             
                                         
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       
                                         meas 
                                         j 
                                       
                                     
                                     
                                       
                                         P 
                                         b 
                                       
                                       -&gt; 
                                       
                                         P 
                                         a 
                                       
                                     
                                   
                                 
                               
                               2 
                             
                             - 
                           
                         
                       
                       
                         
                           
                             
                               
                                 PHY_DELAY 
                                 
                                   n 
                                   , 
                                   
                                     meas 
                                     2 
                                   
                                 
                                 
                                   
                                     P 
                                     c 
                                   
                                   -&gt; 
                                   
                                     P 
                                     d 
                                   
                                 
                               
                               + 
                               
                                 ARB_RESPONSE 
                                 ⁢ 
                                 
                                   _DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       2 
                                     
                                   
                                   
                                     
                                       P 
                                       d 
                                     
                                     -&gt; 
                                     
                                       P 
                                       c 
                                     
                                   
                                 
                               
                             
                             2 
                           
                         
                       
                     
                     ❘ 
                     
                       ≤ 
                       
                         Jitter 
                         n 
                       
                     
                   
                 
               
               
                 
                   ( 
                   89 
                   ) 
                 
               
             
           
         
       
     
     In order for a bus manager to calculate ordered leaf-to-leaf delays via a series of ping requests launched from the bus manager, a number of ROUND_TRIP_DELAY relationships will be required and are derived below. 
     
       
         
           
             Round_Trip 
             ⁢ 
             
               _Delay 
               max 
               
                 [ 
                 
                   
                     P 
                     X 
                   
                   ⁢ 
                   O 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     P 
                     Y 
                   
                 
                 ] 
               
             
           
         
       
     
     Using the definition of Round_Trip_Delay first provided in equation (12) as guidance, the roundtrip delay between Nodes X and Y from the perspective of Node X can be written as: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             X 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     max 
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       max 
                                     
                                     
                                       
                                         P 
                                         n 
                                       
                                       -&gt; 
                                       
                                         P 
                                         n 
                                         ′ 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       · 
                       
                         cable_delay 
                         X 
                       
                     
                   
                 
               
               
                 
                   ( 
                   90 
                   ) 
                 
               
             
           
         
       
     
     From equation (88), the maximum PHY_DELAY and ARB_RESPONSE_DELAY between an ordered pair of ports can be bounded by the measured delays plus the overall jitter sum yielding: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             X 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       j 
                                     
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       
                                         meas 
                                         j 
                                       
                                     
                                     
                                       
                                         P 
                                         n 
                                       
                                       -&gt; 
                                       
                                         P 
                                         n 
                                         ′ 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 2 
                                 · 
                                 
                                   Jitter 
                                   n 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       · 
                       
                         cable_delay 
                         X 
                       
                     
                   
                 
               
               
                 
                   ( 
                   91 
                   ) 
                 
               
             
           
         
       
     
     Comparison to the definition of Round_Trip_Delay then allows 
     
       
         
           
             
               
                 
                   
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         max 
                         
                           [ 
                           
                             
                               P 
                               X 
                             
                             ⁢ 
                             O 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               Y 
                             
                           
                           ] 
                         
                       
                     
                     ≤ 
                     
                       
                         Round_Trip 
                         ⁢ 
                         
                           _Delay 
                           
                             meas 
                             j 
                           
                           
                             [ 
                             
                               
                                 P 
                                 X 
                               
                               ⁢ 
                               O 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 P 
                                 Y 
                               
                             
                             ] 
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             
                               X 
                               + 
                               1 
                             
                           
                           
                             Y 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           2 
                           · 
                           
                             Jitter 
                             n 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             Y 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             X 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   92 
                   ) 
                 
               
             
           
         
       
     
     Using the definition of Round_Trip_Delay first provided in equation (12) as guidance, the roundtrip delay between Nodes X and Y from the perspective of Node Y can be written as: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             Y 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             X 
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     max 
                                   
                                   
                                     
                                       P 
                                       n 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 ARB_RESPONSE 
                                 ⁢ 
                                 
                                   _DELAY 
                                   
                                     n 
                                     , 
                                     max 
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       · 
                       
                         cable_delay 
                         X 
                       
                     
                   
                 
               
               
                 
                   ( 
                   93 
                   ) 
                 
               
             
           
         
       
     
     From equation (88), the maximum PHY_DELAY and ARB_RESPONSE_DELAY between an ordered pair of ports can be related to the measured delays observed in the reverse direction: 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             
                               PHY_DELAY 
                               
                                 n 
                                 , 
                                 max 
                               
                               
                                 
                                   P 
                                   n 
                                 
                                 -&gt; 
                                 
                                   P 
                                   n 
                                   ′ 
                                 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             ARB_RESPONSE 
                             ⁢ 
                             
                               _DELAY 
                               
                                 n 
                                 , 
                                 max 
                               
                               
                                 
                                   P 
                                   n 
                                   ′ 
                                 
                                 -&gt; 
                                 
                                   P 
                                   n 
                                 
                               
                             
                           
                         
                       
                     
                     ) 
                   
                   ≤ 
                   
                     
                       2 
                       · 
                       
                         Jitter 
                         n 
                       
                     
                     + 
                     
                       ( 
                       
                         
                           
                             
                               
                                 PHY_DELAY 
                                 
                                   n 
                                   , 
                                   
                                     meas 
                                     j 
                                   
                                 
                                 
                                   
                                     P 
                                     n 
                                     ′ 
                                   
                                   -&gt; 
                                   
                                     P 
                                     n 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               ARB_RESPONSE 
                               ⁢ 
                               
                                 _DELAY 
                                 
                                   n 
                                   , 
                                   
                                     meas 
                                     j 
                                   
                                 
                                 
                                   
                                     P 
                                     n 
                                   
                                   -&gt; 
                                   
                                     P 
                                     n 
                                     ′ 
                                   
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   94 
                   ) 
                 
               
             
           
         
       
     
     allowing the maximum round trip between Nodes X and Y to be rewritten as: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             Y 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             X 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       j 
                                     
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       
                                         meas 
                                         j 
                                       
                                     
                                     
                                       
                                         P 
                                         n 
                                       
                                       -&gt; 
                                       
                                         P 
                                         n 
                                         ′ 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 2 
                                 · 
                                 
                                   Jitter 
                                   n 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       · 
                       
                         cable_delay 
                         X 
                       
                     
                   
                 
               
               
                 
                   ( 
                   95 
                   ) 
                 
               
             
           
         
       
     
     Comparison to the definition of Round_Trip_Delay then allows 
     
       
         
           
             
               
                 
                   
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         max 
                         
                           [ 
                           
                             
                               P 
                               Y 
                             
                             ⁢ 
                             O 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               X 
                             
                           
                           ] 
                         
                       
                     
                     ≤ 
                     
                       
                         Round_Trip 
                         ⁢ 
                         
                           _Delay 
                           
                             meas 
                             j 
                           
                           
                             [ 
                             
                               
                                 P 
                                 X 
                               
                               ⁢ 
                               O 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 P 
                                 Y 
                               
                             
                             ] 
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             
                               X 
                               + 
                               1 
                             
                           
                           
                             Y 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           2 
                           · 
                           
                             Jitter 
                             n 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             N 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   96 
                   ) 
                 
               
             
           
         
       
     
     Using the definition of Round_Trip_Delay first provided in equation (12) as guidance, the roundtrip delay between Nodes N and Y from the perspective of Node N can be written as: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             N 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             N 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     max 
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 ARB_RESPONSE 
                                 ⁢ 
                                 
                                   _DELAY 
                                   
                                     n 
                                     , 
                                     max 
                                   
                                   
                                     
                                       P 
                                       n 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       · 
                       
                         cable_delay 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   97 
                   ) 
                 
               
             
           
         
       
     
     From equation (88), the maximum PHY_DELAY and ARB_RESPONSE_DELAY between an ordered pair of ports can be bounded by the measured delays plus the overall jitter sum yielding: 
     
       
         
           
             
               
                 
                   
                     Round 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Trip_Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             N 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             N 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       1 
                                     
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       
                                         meas 
                                         1 
                                       
                                     
                                     
                                       
                                         P 
                                         n 
                                       
                                       -&gt; 
                                       
                                         P 
                                         n 
                                         ′ 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 2 
                                 · 
                                 
                                   Jitter 
                                   n 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       · 
                       
                         cable_delay 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   98 
                   ) 
                 
               
             
           
         
       
     
     Introducing offsetting terms to the right side: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             N 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       1 
                                     
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       
                                         meas 
                                         1 
                                       
                                     
                                     
                                       
                                         P 
                                         n 
                                       
                                       -&gt; 
                                       
                                         P 
                                         n 
                                         ′ 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 2 
                                 · 
                                 
                                   Jitter 
                                   n 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       · 
                       
                         cable_delay 
                         X 
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       1 
                                     
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       
                                         meas 
                                         1 
                                       
                                     
                                     
                                       
                                         P 
                                         n 
                                       
                                       -&gt; 
                                       
                                         P 
                                         n 
                                         ′ 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 2 
                                 · 
                                 
                                   Jitter 
                                   n 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       2 
                       · 
                       
                         cable_delay 
                         X 
                       
                     
                     - 
                     
                       PHY_DELAY 
                       
                         N 
                         , 
                         
                           meas 
                           1 
                         
                       
                       
                         
                           P 
                           N 
                           ′ 
                         
                         -&gt; 
                         
                           P 
                           N 
                         
                       
                     
                     - 
                     
                       ARB_RESPONSE 
                       ⁢ 
                       
                         _DELAY 
                         
                           N 
                           , 
                           
                             meas 
                             1 
                           
                         
                         
                           
                             P 
                             N 
                           
                           -&gt; 
                           
                             P 
                             N 
                             ′ 
                           
                         
                       
                     
                     - 
                     
                       2 
                       · 
                       
                         Jitter 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   99 
                   ) 
                 
               
             
           
         
       
     
     Equations (89) and the fact that measured delays are at no smaller than minimum delays allow simplification to: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             N 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       1 
                                     
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       
                                         meas 
                                         1 
                                       
                                     
                                     
                                       
                                         P 
                                         n 
                                       
                                       -&gt; 
                                       
                                         P 
                                         n 
                                         ′ 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 2 
                                 · 
                                 
                                   Jitter 
                                   n 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       · 
                       
                         cable_delay 
                         X 
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       2 
                                     
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 ARB_RESPONSE 
                                 ⁢ 
                                 
                                   _DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       2 
                                     
                                   
                                   
                                     
                                       P 
                                       n 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       2 
                       · 
                       
                         cable_delay 
                         X 
                       
                     
                     - 
                     
                       PHY_DELAY 
                       
                         N 
                         , 
                         min 
                       
                       
                         
                           P 
                           N 
                           ′ 
                         
                         -&gt; 
                         
                           P 
                           N 
                         
                       
                     
                     - 
                     
                       ARB_RESPONSE 
                       ⁢ 
                       
                         _DELAY 
                         
                           N 
                           , 
                           min 
                         
                         
                           
                             P 
                             N 
                           
                           -&gt; 
                           
                             P 
                             N 
                             ′ 
                           
                         
                       
                     
                     - 
                     
                       2 
                       · 
                       
                         Jitter 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   100 
                   ) 
                 
               
             
           
         
       
     
     Comparison to the definition of Round_Trip_Delay then allows 
     
       
         
           
             
               
                 
                   
                     
                       
                         Round_Trip 
                         ⁢ 
                         
                           _Delay 
                           max 
                           
                             
                               [ 
                               
                                 
                                   P 
                                   N 
                                 
                                 ⁢ 
                                 O 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   P 
                                   Y 
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                                 
                             
                           
                         
                       
                       ≤ 
                       
                         
                           Round_Trip 
                           ⁢ 
                           
                             _Delay 
                             
                               meas 
                               1 
                             
                             
                               [ 
                               
                                 
                                   P 
                                   X 
                                 
                                 ⁢ 
                                 O 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   P 
                                   Y 
                                 
                               
                               ] 
                             
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               
                                 X 
                                 + 
                                 1 
                               
                             
                             
                               Y 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             2 
                             · 
                             
                               Jitter 
                               n 
                             
                           
                         
                         - 
                         
                           Round_Trip 
                           ⁢ 
                           
                             _Delay 
                             
                               meas 
                               2 
                             
                             
                               [ 
                               
                                 
                                   P 
                                   X 
                                 
                                 ⁢ 
                                 O 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   P 
                                   N 
                                 
                               
                               ] 
                             
                           
                         
                         - 
                         
                           PHY_DELAY 
                           
                             N 
                             , 
                             min 
                           
                           
                             
                               P 
                               N 
                               ′ 
                             
                             -&gt; 
                             
                               P 
                               N 
                             
                           
                         
                         - 
                         
                           ARB_RESPONSE 
                           ⁢ 
                           
                             _DELAY 
                             
                               N 
                               , 
                               min 
                             
                             
                               
                                 P 
                                 N 
                               
                               -&gt; 
                               
                                 P 
                                 N 
                                 ′ 
                               
                             
                           
                         
                         - 
                         
                           2 
                           · 
                           
                             Jitter 
                             N 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         max 
                         
                           [ 
                           
                             
                               P 
                               Y 
                             
                             ⁢ 
                             O 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               N 
                             
                           
                           ] 
                         
                       
                     
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                   ( 
                   101 
                   ) 
                 
               
             
           
         
       
     
     Using the definition of Round_Trip_Delay first provided in equation (12) as guidance, the roundtrip delay between Nodes N and Y from the perspective of Node Y can be written as: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             Y 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             N 
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             N 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     max 
                                   
                                   
                                     
                                       P 
                                       n 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 ARB_RESPONSE 
                                 ⁢ 
                                 
                                   _DELAY 
                                   
                                     n 
                                     , 
                                     max 
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       · 
                       
                         cable_delay 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   102 
                   ) 
                 
               
             
           
         
       
     
     From equation (88), the maximum PHY_DELAY and ARB_RESPONSE_DELAY between an ordered pair of ports can be related to the measured delays observed in the reverse direction: 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             
                               PHY_DELAY 
                               
                                 n 
                                 , 
                                 max 
                               
                               
                                 
                                   P 
                                   n 
                                 
                                 -&gt; 
                                 
                                   P 
                                   n 
                                   ′ 
                                 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             ARB_RESPONSE 
                             ⁢ 
                             
                               _DELAY 
                               
                                 n 
                                 , 
                                 max 
                               
                               
                                 
                                   P 
                                   n 
                                   ′ 
                                 
                                 -&gt; 
                                 
                                   P 
                                   n 
                                 
                               
                             
                           
                         
                       
                     
                     ) 
                   
                   ≤ 
                   
                     
                       2 
                       · 
                       
                         Jitter 
                         n 
                       
                     
                     + 
                     
                       ( 
                       
                         
                           
                             
                               
                                 PHY_DELAY 
                                 
                                   n 
                                   , 
                                   
                                     meas 
                                     1 
                                   
                                 
                                 
                                   
                                     P 
                                     n 
                                     ′ 
                                   
                                   -&gt; 
                                   
                                     P 
                                     n 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               ARB_RESPONSE 
                               ⁢ 
                               
                                 _DELAY 
                                 
                                   n 
                                   , 
                                   
                                     meas 
                                     1 
                                   
                                 
                                 
                                   
                                     P 
                                     n 
                                   
                                   -&gt; 
                                   
                                     P 
                                     n 
                                     ′ 
                                   
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   103 
                   ) 
                 
               
             
           
         
       
     
     allowing the maximum round trip between Nodes N and Y to be rewritten as: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             Y 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             N 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             N 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       1 
                                     
                                   
                                   
                                     
                                       P 
                                       n 
                                       
                                         ′ 
                                         ⁢ 
                                         
                                             
                                         
                                       
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       
                                         meas 
                                         1 
                                       
                                     
                                     
                                       
                                         P 
                                         n 
                                       
                                       -&gt; 
                                       
                                         P 
                                         n 
                                         ′ 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 2 
                                 · 
                                 
                                   Jitter 
                                   n 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       · 
                       
                         cable_delay 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   104 
                   ) 
                 
               
             
           
         
       
     
     Introducing offsetting terms to the right side: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             Y 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             N 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       1 
                                     
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       
                                         n 
                                         ⁢ 
                                         
                                             
                                         
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       
                                         meas 
                                         1 
                                       
                                     
                                     
                                       
                                         P 
                                         n 
                                       
                                       -&gt; 
                                       
                                         P 
                                         n 
                                         ′ 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 2 
                                 · 
                                 
                                   Jitter 
                                   n 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       · 
                       
                         cable_delay 
                         X 
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       1 
                                     
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       
                                         meas 
                                         1 
                                       
                                     
                                     
                                       
                                         P 
                                         n 
                                       
                                       -&gt; 
                                       
                                         P 
                                         n 
                                         ′ 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 2 
                                 · 
                                 
                                   Jitter 
                                   n 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       2 
                       · 
                       
                         cable_delay 
                         X 
                       
                     
                     - 
                     
                       PHY_DELAY 
                       
                         N 
                         , 
                         
                           meas 
                           1 
                         
                       
                       
                         
                           P 
                           N 
                           ′ 
                         
                         -&gt; 
                         
                           P 
                           N 
                         
                       
                     
                     - 
                     
                       ARB_RESPONSE 
                       ⁢ 
                       
                         _DELAY 
                         
                           N 
                           , 
                           
                             meas 
                             1 
                           
                         
                         
                           
                             P 
                             N 
                           
                           -&gt; 
                           
                             P 
                             N 
                             ′ 
                           
                         
                       
                     
                     - 
                     
                       2 
                       · 
                       
                         Jitter 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   105 
                   ) 
                 
               
             
           
         
       
     
     Equations (89) and the fact that measured delays are at no smaller than minimum delays allow simplification to: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             Y 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             N 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       1 
                                     
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       
                                         meas 
                                         1 
                                       
                                     
                                     
                                       
                                         P 
                                         n 
                                       
                                       -&gt; 
                                       
                                         P 
                                         n 
                                         ′ 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 2 
                                 · 
                                 
                                   Jitter 
                                   n 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       2 
                       · 
                       
                         cable_delay 
                         
                           X 
                           ⁢ 
                           
                               
                           
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     cable_delay 
                                     n 
                                   
                                 
                                 + 
                                 
                                   PHY_DELAY 
                                   
                                     n 
                                     , 
                                     
                                       meas 
                                       1 
                                     
                                   
                                   
                                     
                                       P 
                                       n 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       n 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ARB_RESPONSE 
                                   ⁢ 
                                   
                                     _DELAY 
                                     
                                       n 
                                       , 
                                       
                                         meas 
                                         2 
                                       
                                     
                                     
                                       
                                         P 
                                         n 
                                       
                                       -&gt; 
                                       
                                         P 
                                         n 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       2 
                       · 
                       
                         cable_delay 
                         X 
                       
                     
                     - 
                     
                       PHY_DELAY 
                       
                         N 
                         , 
                         min 
                       
                       
                         
                           P 
                           N 
                           ′ 
                         
                         -&gt; 
                         
                           P 
                           N 
                         
                       
                     
                     - 
                     
                       ARB_RESPONSE 
                       ⁢ 
                       
                         _DELAY 
                         
                           N 
                           , 
                           min 
                         
                         
                           
                             P 
                             N 
                           
                           -&gt; 
                           
                             P 
                             N 
                             ′ 
                           
                         
                       
                     
                     - 
                     
                       2 
                       · 
                       
                         Jitter 
                         
                           N 
                           ⁢ 
                           
                               
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   106 
                   ) 
                 
               
             
           
         
       
     
     Comparison to the definition of Round_Trip_Delay then allows 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             Y 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             N 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         
                           meas 
                           1 
                         
                         
                           [ 
                           
                             
                               P 
                               X 
                             
                             ⁢ 
                             O 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               Y 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             X 
                             + 
                             1 
                           
                         
                         
                           Y 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         2 
                         · 
                         
                           Jitter 
                           n 
                         
                       
                     
                     - 
                     
                       Round_Trip 
                       ⁢ 
                       
                         _delay 
                         
                           meas 
                           2 
                         
                         
                           [ 
                           
                             
                               P 
                               X 
                             
                             ⁢ 
                             O 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               N 
                             
                           
                           ] 
                         
                       
                     
                     - 
                     
                       PHY_DELAY 
                       
                         N 
                         , 
                         min 
                       
                       
                         
                           P 
                           N 
                           
                             ′ 
                             ⁢ 
                             
                                 
                             
                           
                         
                         -&gt; 
                         
                           P 
                           N 
                         
                       
                     
                     - 
                     
                       ARB_RESPONSE 
                       ⁢ 
                       
                         _DELAY 
                         
                           N 
                           , 
                           min 
                         
                         
                           
                             P 
                             N 
                           
                           -&gt; 
                           
                             P 
                             N 
                             ′ 
                           
                         
                       
                     
                     - 
                     
                       2 
                       · 
                       
                         Jitter 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   107 
                   ) 
                 
               
             
           
         
       
     
     PHY pinging provides a low level mechanism to directly measure round trip delays between two nodes by timing link initiated subactions. However, pinging does introduce some uncertainty in the measured delay. Any gap count algorithm which employs PHY pinging must compensate for such uncertainty.  FIG. 8  depicts a ping subaction issued by the link in Node X and directed to Node Y. 
     The timing reference points t 1  through t 7  are identical to those used in the previous gap count derivations. Additionally: 
     
       
         
           
               
               
             
               
                   
               
             
            
               
                 t 1 ′ 
                 Coincident with the rising SCLK edge in which the PHY first 
               
               
                   
                 samples IDLE after a link transmission. t 1 ′ leads t 1  by 
               
               
                   
                 LINK_TO_BUS_DELAY 
               
               
                 t 7 ′ 
                 Coincident with the rising SCLK edge in which the PHY is driving 
               
               
                   
                 the first RECEIVE indication to the link. (The PHY presumably 
               
               
                   
                 drove RECEIVE off of the previous clock transition.) t 7 ′ lags t 7  by 
               
               
                   
                 BUS_TO_LINK_DELAY 
               
               
                   
               
            
           
         
       
     
     The ping time measured by the link (in SCLK cycles) is then given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           Ping_Time 
                           meas 
                           
                             [ 
                             
                               
                                 P 
                                 X 
                               
                               ⁢ 
                               O 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 P 
                                 Y 
                               
                             
                             ] 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             t 
                             7 
                             ′ 
                           
                           - 
                           
                             t 
                             1 
                             ′ 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             BUS_TO 
                             ⁢ 
                             _LINK 
                             ⁢ 
                             
                               _DELAY 
                               
                                 X 
                                 , 
                                 meas 
                               
                             
                           
                           + 
                           
                             t 
                             7 
                           
                           - 
                           
                             t 
                             1 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           LINK_TO 
                           ⁢ 
                           _BUS 
                           ⁢ 
                           
                             _DELAY 
                             
                               X 
                               , 
                               meas 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             BUS_TO 
                             ⁢ 
                             _LINK 
                             ⁢ 
                             
                               _DELAY 
                               
                                 X 
                                 , 
                                 meas 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             LINK_TO 
                             ⁢ 
                             _BUS 
                             ⁢ 
                             
                               _DELAY 
                               
                                 X 
                                 , 
                                 meas 
                               
                             
                           
                           + 
                           
                             t 
                             0 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 
                                   X 
                                   + 
                                   1 
                                 
                               
                               
                                 Y 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         2 
                                         · 
                                         
                                           cable_delay 
                                           n 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         PHY_DELAY 
                                         
                                           n 
                                           , 
                                           meas 
                                         
                                         
                                           
                                             P 
                                             n 
                                             ′ 
                                           
                                           -&gt; 
                                           
                                             P 
                                             n 
                                           
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       ARB_RESPONSE 
                                       ⁢ 
                                       
                                         _DELAY 
                                         
                                           n 
                                           , 
                                           meas 
                                         
                                         
                                           
                                             P 
                                             n 
                                             ′ 
                                           
                                           -&gt; 
                                           
                                             P 
                                             n 
                                             ′ 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             2 
                             · 
                             
                               cable_delay 
                               X 
                             
                           
                           + 
                           
                             packet_length 
                             
                               
                                 
                                   
                                     packet_speed 
                                     · 
                                   
                                 
                               
                               
                                 
                                   
                                     BASERATE 
                                     
                                       Y 
                                       - 
                                       1 
                                     
                                   
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             DATA_END 
                             ⁢ 
                             
                               _TIME 
                               
                                 
                                   Y 
                                   - 
                                   1 
                                 
                                 , 
                                 meas 
                               
                               
                                 P 
                                 
                                   Y 
                                   - 
                                   1 
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             RESPONSE_TIME 
                             
                               Y 
                               , 
                               meas 
                             
                             
                               P 
                               Y 
                               ′ 
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             t 
                             0 
                           
                           - 
                           
                             packet_length 
                             
                               packet_speed 
                               · 
                               
                                 BASERATE 
                                 X 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             BUS_TO 
                             ⁢ 
                             _LINK 
                             ⁢ 
                             
                               _DELAY 
                               
                                 X 
                                 , 
                                 meas 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             LINK_TO 
                             ⁢ 
                             _BUS 
                             ⁢ 
                             
                               _DELAY 
                               
                                 X 
                                 , 
                                 meas 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             Round_Trip 
                             ⁢ 
                             
                               _Delay 
                               meas 
                               
                                 [ 
                                 
                                   
                                     P 
                                     X 
                                   
                                   ⁢ 
                                   O 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     P 
                                     Y 
                                   
                                 
                                 ] 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             PPM_delta 
                             
                               [ 
                               
                                 
                                   Y 
                                   - 
                                   1 
                                 
                                 , 
                                 X 
                               
                               ] 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             DATA_END 
                             ⁢ 
                             
                               _TIME 
                               
                                 
                                   Y 
                                   - 
                                   1 
                                 
                                 , 
                                 meas 
                               
                               
                                 P 
                                 
                                   Y 
                                   - 
                                   1 
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           RESPONSE_TIME 
                           
                             Y 
                             , 
                             meas 
                           
                           
                             P 
                             Y 
                             ′ 
                           
                         
                         ⁢ 
                         
                             
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   108 
                   ) 
                 
               
             
           
         
       
     
     Solving for the measured Round_Trip_Delay gives: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       meas 
                       
                         [ 
                         
                           
                             P 
                             X 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       Ping_Time 
                       meas 
                       
                         [ 
                         
                           
                             P 
                             X 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                     - 
                     
                       BUS_TO 
                       ⁢ 
                       _LINK 
                       ⁢ 
                       
                         _DELAY 
                         
                           X 
                           , 
                           meas 
                         
                       
                     
                     - 
                     
                       LINK_TO 
                       ⁢ 
                       _BUS 
                       ⁢ 
                       
                         _DELAY 
                         
                           X 
                           , 
                           meas 
                         
                       
                     
                     - 
                     
                       PPM_delta 
                       
                         [ 
                         
                           
                             Y 
                             - 
                             1 
                           
                           , 
                           X 
                         
                         ] 
                       
                     
                     - 
                     
                       DATA_END 
                       ⁢ 
                       
                         _TIME 
                         
                           
                             Y 
                             - 
                             1 
                           
                           , 
                           meas 
                         
                         
                           P 
                           
                             Y 
                             - 
                             1 
                           
                         
                       
                     
                     - 
                     
                       RESPONSE_TIME 
                       
                         Y 
                         , 
                         meas 
                       
                       
                         P 
                         Y 
                         ′ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   109 
                   ) 
                 
               
             
           
         
       
     
     Remembering that RESPONSE_TIME (min or max) absorbs PPM_delta, an upper and lower bound can be defined for Round_Trip_Delay: 
                     Round_Trip   ⁢     _Delay     Ping   ,   max       [       P   X     ⁢   O   ⁢           ⁢     P   Y       ]         =       Ping_Time   meas     [       P   X     ⁢   O   ⁢           ⁢     P   Y       ]       -     BUS_TO   ⁢   _LINK   ⁢     _DELAY     X   ,   min         -     LINK_TO   ⁢   _BUS   ⁢     _DELAY     X   ,   min         -     DATA_END   ⁢     _TIME       Y   -   1     ,   min       P     Y   -   1           -     RESPONSE_TIME     Y   ,   min       P   Y   ′                 (   110   )                 and   ⁢           ⁢   Round_Trip   ⁢     _Delay     Ping   ,   min       [       P   X     ⁢   O   ⁢           ⁢     P   Y       ]         =       Ping_Time   meas     [       P   X     ⁢   O   ⁢           ⁢     P   Y       ]       -     BUS_TO   ⁢   _LINK   ⁢     _DELAY     X   ,   max         -     LINK_TO   ⁢   _BUS   ⁢     _DELAY     X   ,   max         -     DATA_END   ⁢     _TIME       Y   -   1     ,   max       P     Y   -   1           -     RESPONSE_TIME     Y   ,   max       P   Y   ′                 (   111   )               
such that
 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       
                         Ping 
                         , 
                         min 
                       
                       
                         [ 
                         
                           
                             P 
                             X 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       meas 
                       
                         [ 
                         
                           
                             P 
                             X 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       
                         Ping 
                         , 
                         max 
                       
                       
                         [ 
                         
                           
                             P 
                             X 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   112 
                   ) 
                 
               
             
           
         
       
     
     Using the Round_Trip_Delay properties and the Ping_Time relationships, the maximum Round_Trip_Delay between two given leaf nodes can be bounded for any possible topology. 
     The simplest and most accurate Round_Trip_Delay determination is afforded when the Bus Manager is one of the leaf nodes in question as shown in  FIG. 9 . 
     From (92), 
                     Round_Trip   ⁢     _Delay   max     [       P   BM     ⁢   O   ⁢           ⁢     P   Y       ]         ≤       Round_Trip   ⁢     _Delay     meas   1       [       P   BM     ⁢   O   ⁢           ⁢     P   Y       ]         +       ∑   n     (     BM   ,   Y     )       ⁢           ⁢     2   ·     Jitter   n                   (   113   )               
And from (112),
 
                     Round_Trip   ⁢     _Delay   max     [       P   BM     ⁢   O   ⁢           ⁢     P   Y       ]         ≤       Round_Trip   ⁢     _Delay     Ping   ,   max       [       P   BM     ⁢   O   ⁢           ⁢     P   Y       ]         +       ∑   n     (     BM   ,   Y     )       ⁢           ⁢     2   ·     Jitter   n                   (   114   )               
Likewise, the reverse path is also bounded:
 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             Y 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             BM 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         
                           Ping 
                           , 
                           max 
                         
                         
                           [ 
                           
                             
                               P 
                               BM 
                             
                             ⁢ 
                             O 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               Y 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         n 
                         
                           ( 
                           
                             BM 
                             , 
                             Y 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         2 
                         · 
                         
                           Jitter 
                           n 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   115 
                   ) 
                 
               
             
           
         
       
     
     The second topology to consider is when the bus manager is not a leaf but is part of the connecting path between the two leaves as illustrated in  FIG. 10 . 
     Expressing the max delay piecewise, 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             X 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         max 
                         
                           [ 
                           
                             
                               P 
                               X 
                             
                             ⁢ 
                             O 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               BM 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         max 
                         
                           
                             [ 
                           
                           ⁢ 
                           
                             P 
                             BM 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                           ⁢ 
                           
                             ] 
                           
                         
                       
                     
                     + 
                     
                       PHY_DELAY 
                       
                         BM 
                         , 
                         max 
                       
                       
                         
                           P 
                           BM 
                           ′ 
                         
                         → 
                         
                           P 
                           BM 
                         
                       
                     
                     + 
                     
                       ARB_RESPONSE 
                       ⁢ 
                       
                         _DELAY 
                         
                           BM 
                           , 
                           max 
                         
                         
                           
                             P 
                             BM 
                           
                           → 
                           
                             P 
                             BM 
                             ′ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   116 
                   ) 
                 
               
             
           
         
       
     
     Equations (92) and (96) allow: 
                     Round_Trip   ⁢     _Delay   max     [       P   X     ⁢   O   ⁢           ⁢     P   Y       ]         ≤       Round_Trip   ⁢     _Delay   meas     [       P   BM     ⁢   O   ⁢           ⁢     P   X       ]         +       ∑   n     (     BM   ,   X     )       ⁢           ⁢     2   ·     Jitter   n         +     Round_Trip   ⁢     _Delay   meas     [       P   BM     ⁢   O   ⁢           ⁢     P   Y       ]         +       ∑   n     (     BM   ,   Y     )       ⁢           ⁢     2   ·     Jitter   n         +     PHY_DELAY     BM   ,   max         P   BM   ′     -&gt;     P   BM         +     ARB_RESPONSE   ⁢     _DELAY     BM   ,   max         P   BM     -&gt;     P   BM   ′                     (   117   )               
And from (112),
 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             X 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         
                           Ping 
                           , 
                           max 
                         
                         
                           [ 
                           
                             
                               P 
                               BM 
                             
                             ⁢ 
                             O 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               X 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         n 
                         
                           ( 
                           
                             BM 
                             , 
                             X 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         2 
                         · 
                         
                           Jitter 
                           n 
                         
                       
                     
                     + 
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         
                           Ping 
                           , 
                           max 
                         
                         
                           [ 
                           
                             
                               P 
                               BM 
                             
                             ⁢ 
                             O 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               Y 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         n 
                         
                           ( 
                           
                             BM 
                             , 
                             Y 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         2 
                         · 
                         
                           Jitter 
                           n 
                         
                       
                     
                     + 
                     
                       PHY_DELAY 
                       
                         BM 
                         , 
                         max 
                       
                       
                         
                           P 
                           BM 
                           ′ 
                         
                         -&gt; 
                         
                           P 
                           BM 
                         
                       
                     
                     + 
                     
                       ARB_RESPONSE 
                       ⁢ 
                       
                         _DELAY 
                         
                           BM 
                           , 
                           max 
                         
                         
                           
                             P 
                             BM 
                           
                           -&gt; 
                           
                             P 
                             BM 
                             ′ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   118 
                   ) 
                 
               
             
           
         
       
     
     Likewise, the reverse path is also bounded: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             Y 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             X 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         
                           Ping 
                           , 
                           max 
                         
                         
                           [ 
                           
                             
                               P 
                               BM 
                             
                             ⁢ 
                             O 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               X 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         n 
                         
                           ( 
                           
                             BM 
                             , 
                             X 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         2 
                         · 
                         
                           Jitter 
                           n 
                         
                       
                     
                     + 
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         
                           Ping 
                           , 
                           max 
                         
                         
                           [ 
                           
                             
                               P 
                               BM 
                             
                             ⁢ 
                             O 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               Y 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         n 
                         
                           ( 
                           
                             BM 
                             , 
                             Y 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         2 
                         · 
                         
                           Jitter 
                           n 
                         
                       
                     
                     + 
                     
                       PHY_DELAY 
                       
                         BM 
                         , 
                         max 
                       
                       
                         
                           P 
                           BM 
                         
                         -&gt; 
                         
                           P 
                           BM 
                           ′ 
                         
                       
                     
                     + 
                     
                       ARB_RESPONSE 
                       ⁢ 
                       
                         _DELAY 
                         
                           BM 
                           , 
                           max 
                         
                         
                           
                             P 
                             BM 
                             ′ 
                           
                           -&gt; 
                           
                             P 
                             BM 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   119 
                   ) 
                 
               
             
           
         
       
     
     The final topology to consider is when the bus manager is not a leaf but is not part of the connecting path between the two leaves as illustrated in  FIG. 11 . 
     Expressing the max delay piecewise, 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             X 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         max 
                         
                           
                             [ 
                           
                           ⁢ 
                           
                             P 
                             X 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             N 
                           
                           ⁢ 
                           
                             ] 
                           
                         
                       
                     
                     + 
                     
                       Round_Trip 
                       ⁢ 
                       
                         _Delay 
                         max 
                         
                           [ 
                           
                             
                               P 
                               N 
                             
                             ⁢ 
                             O 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               Y 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       PHY_DELAY 
                       
                         N 
                         , 
                         max 
                       
                       
                         
                           P 
                           N 
                           ′ 
                         
                         -&gt; 
                         
                           P 
                           N 
                         
                       
                     
                     + 
                     
                       ARB_RESPONSE 
                       ⁢ 
                       
                         _DELAY 
                         
                           N 
                           , 
                           max 
                         
                         
                           
                             P 
                             N 
                           
                           -&gt; 
                           
                             P 
                             N 
                             ′ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   120 
                   ) 
                 
               
             
           
         
       
     
     Equations (107) and (101) allow: 
     
       
         
           
             
               
                 
                   
                     Round_Trip 
                     ⁢ 
                     
                       _Delay 
                       max 
                       
                         [ 
                         
                           
                             P 
                             X 
                           
                           ⁢ 
                           O 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             P 
                             Y 
                           
                         
                         ] 
                       
                     
                   
                   ≤ 
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 Round_Trip 
                                 ⁢ 
                                 
                                   _Delay 
                                   
                                     meas 
                                     1 
                                   
                                   
                                     [ 
                                     
                                       
                                         P 
                                         BM 
                                       
                                       ⁢ 
                                       O 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         P 
                                         X 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   n 
                                   
                                     ( 
                                     
                                       BM 
                                       , 
                                       X 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   2 
                                   · 
                                   
                                     Jitter 
                                     n 
                                   
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 Round_Trip 
                                 ⁢ 
                                 
                                   _Delay 
                                   
                                     meas 
                                     2 
                                   
                                   
                                     [ 
                                     
                                       
                                         P 
                                         BM 
                                       
                                       ⁢ 
                                       O 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         P 
                                         N 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               - 
                               
                                 PHY_DELAY 
                                 
                                   N 
                                   , 
                                   min 
                                 
                                 
                                   
                                     P 
                                     N 
                                     BM 
                                   
                                   -&gt; 
                                   
                                     P 
                                     N 
                                     ′ 
                                   
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 ARB_RESPONSE 
                                 ⁢ 
                                 
                                   _DELAY 
                                   
                                     N 
                                     , 
                                     min 
                                   
                                   
                                     
                                       P 
                                       N 
                                       ′ 
                                     
                                     -&gt; 
                                     
                                       P 
                                       N 
                                       BM 
                                     
                                   
                                 
                               
                               - 
                               
                                 2 
                                 · 
                                 
                                   Jitter 
                                   N 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     + 
                     
                       
 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 Round_Trip 
                                 ⁢ 
                                 
                                   _Delay 
                                   
                                     meas 
                                     1 
                                   
                                   
                                     [ 
                                     
                                       
                                         P 
                                         BM 
                                       
                                       ⁢ 
                                       O 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         P 
                                         Y 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   n 
                                   
                                     ( 
                                     
                                       BM 
                                       , 
                                       Y 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   2 
                                   · 
                                   
                                     Jitter 
                                     n 
                                   
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 Round_Trip 
                                 ⁢ 
                                 
                                   _Delay 
                                   
                                     meas 
                                     2 
                                   
                                   
                                     [ 
                                     
                                       
                                         P 
                                         BM 
                                       
                                       ⁢ 
                                       O 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         P 
                                         N 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               - 
                               
                                 PHY_DELAY 
                                 
                                   N 
                                   , 
                                   min 
                                 
                                 
                                   
                                     P 
                                     N 
                                     BM 
                                   
                                   -&gt; 
                                   
                                     P 
                                     N 
                                   
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 ARB_RESPONSE 
                                 ⁢ 
                                 
                                   _DELAY 
                                   
                                     N 
                                     , 
                                     min 
                                   
                                   
                                     
                                       P 
                                       N 
                                     
                                     -&gt; 
                                     
                                       P 
                                       N 
                                       BM 
                                     
                                   
                                 
                               
                               - 
                               
                                 2 
                                 · 
                                 
                                   Jitter 
                                   N 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     + 
                     
                       PHY_DELAY 
                       
                         N 
                         , 
                         max 
                       
                       
                         
                           P 
                           N 
                           ′ 
                         
                         -&gt; 
                         
                           P 
                           N 
                         
                       
                     
                     + 
                     
                       ARB_RESPONSE 
                       ⁢ 
                       
                         _DELAY 
                         
                           N 
                           , 
                           max 
                         
                         
                           
                             P 
                             N 
                           
                           -&gt; 
                           
                             P 
                             N 
                             ′ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   121 
                   ) 
                 
               
             
           
         
       
     
     And from (112), 
                     Round_Trip   ⁢     _Delay   max     [       P   X     ⁢   O   ⁢           ⁢     P   Y       ]         ≤       (             Round_Trip   ⁢     _Delay     Ping   ,   max       [       P   BM     ⁢   O   ⁢           ⁢     P   X       ]         +       ∑   n     (     BM   ,   X     )       ⁢           ⁢     2   ·     Jitter   n         +                 Round_Trip   ⁢     _Delay     Ping   ,   max       [       P   BM     ⁢   O   ⁢           ⁢     P   Y       ]         +       ∑   n     (     BM   ,   Y     )       ⁢           ⁢     2   ·     Jitter   n         +                 PHY_DELAY     N   ,   max         P   N   ′     -&gt;     P   N         +     ARB_RESPONSE   ⁢     _DELAY     N   ,   max         P   N     -&gt;     P   N   ′                   )     -     
     ⁢     (               2   ·   Round_Trip     ⁢     _Delay     Ping   ,   min       [       P   BM     ⁢   O   ⁢           ⁢     P   N       ]         +     4   ·     Jitter   N       +                 PHY_DELAY     N   ,   min         P   N   BM     -&gt;     P   N   ′         +     ARB_RESPONSE   ⁢     _DELAY     N   ,   min         P   N   ′     -&gt;     P   N     BM   ⁢                     +                 PHY_DELAY     N   ,   min         P   N   BM     -&gt;     P   N         +     ARB_RESPONSE   ⁢     _DELAY     N   ,   min         P   N     -&gt;     P   N   BM                   )               (   122   )               
ARB_RESPONSE_DELAY is a difficult parameter to characterize. Proper PHY operation requires that arb signals propagate at least as fast as the data bits, otherwise the arbitration indications could shorten as they are repeated through a network. This fact places a bound on the maximum ARB_RESPONSE_DELAY: ARB_RESPONSE_DELAY between two ports at a particular instant must always be less than or equal to the data repeat delay at the very same instant. Although the distinction is subtle, this is not the same as saying the maximum ARB_RESPONSE_DELAY is PHY_DELAY. (PHY_DELAY only applies to the first bit of a packet and is known to have some jitter from one repeat operation to the next. Consequently, requiring ARB_RESPONSE_DELAY&lt;=PHY_DELAY doesn&#39;t force ARB_RESPONSE_DELAY to track the instantaneous PHY_DELAY nor does it allow ARB_RESPONSE_DELAY to track the data repeat time for the last bit of a packet which may actually exceed PHY_DELAY due to PPM drift.) Finally, the table approach to calculating gap_counta and gap_countb rely on ARB_RESPONSE_DELAY always being bounded by the maximum PHY_DELAY when determining the Round_Trip_Delay.
 
     The minimum ARB_RESPONSE_DELAY is only of significance when calculating Data_Arb_Mismatch as required by gap_countc and gap_countd. Ideally, Data_Arb_Mismatch should be a constant regardless of PHY_DELAY so that neither gap_countc nor gap_countd will begin to dominate the gap_count setting as PHY_DELAY increases. Consequently, the minimum ARB_RESPONSE_DELAY should track the instantaneous PHY_DELAY with some offset for margin. Simply specifying the min value as a function of PHY_DELAY is ambiguous, however, since PHY_DELAY can be easily confused with the max DELAY reported in the register map. (For example, with DELAY at 144 ns, it would be easy to assume a min of PHY_DELAY −60 ns would be equivalent to 84 ns. But if the worst case first bit repeat delay was only 100 ns, arb signals repeating with a delay of 40 ns ought to be considered within spec even though the delay is &lt;84 ns.) 
     Consequently, specifying an upper and a lower bound for ARB_RESPONSE_DELAY is best done in the standard with words rather than values. The minimum and maximum values for ARB_RESPONSE_DELAY include that between all ordered pairs of ports, the PHY shall repeat arbitration line states at least as fast as clocked data, but not more than 60 ns faster than clocked data. 
     A better approach is to replace ARB_RESPONSE_DELAY with the parameter DELAY_MISMATCH which is defined in the comment column as “Between all ordered pairs of ports, the instantaneous repeat delay for data less the instantaneous repeat delay for arbitration line states.” Then, the minimum would be given as 0 ns and the maximum would be 60 ns. 
     For a table based calculation of Round_Trip_Delay, either approach above allows the use of PHY_DELAY(max) for ARB_RESPONSE_DELAY. Since Round_Trip_Delay considers the arbitration repeat delay in the direction opposite to the original packet flow, the return arbitration indication of interest is known to arrive at the receive port when the PHY is idle (all caught up with nothing to repeat). At that point, the instantaneous PHY_DELAY is the same as the first data bit repeat delay which is bounded by PHY_DELAY(max). Since ARB_RESPONSE_DELAY is always bounded by the instantaneous PHY_DELAY, it to is bounded by PHY_DELAY(max) at the point the arbitration indication first arrives. 
     The minimum bound on PHY_DELAY is used by the bus manager when determining the round_trip_delay between leaf nodes that are not separated by the bus manager. The more precise the minimum bound, the more accurate the pinging calculation can be. Ideally then, the bound may want to scale with increasing PHY_DELAY. Alternatively, the lower bound could be calculated by examining the Delay field in the register map: if zero, the lower bound is assumed to be the fixed value specified (60 ns currently). If non-zero, the lower bound could then be determined by subtracting the jitter field (converted to ns) from the delay field (converted to ns). 
     The “Jitter” field was introduced to aid in selection of gap_count via pinging by describing the uncertainty found in any empirical measurement of Round_Trip_Delay. Since Round_Trip_Delay encompasses an “outbound” PHY_DELAY and a “return” ARB_RESPONSE_DELAY, the jitter term should capture uncertainty in both. The needs of pinging can be met with the following description for jitter: Upper bound of the mean average of the worst case data repeat jitter (max/min variance) and the worst case arbitration repeat jitter (max/min variance), expressed as 2*(jitter+1)/BASE_RATE. 
     Note that from the discussion on minimum PHY_DELAY, it may be desirable to require that if the delay field is non-zero, then the slowest first data bit repeat delay can be calculated by subtracting the jitter value from the delay value.

Metadata:
Filing Date: 20031229
Publication Date: 20071211
Grant Date: 20071211
Priority Date: 20031229
Inventors: HAUCK JERROLD VON
Assignee: APPLE INC
CPC Classifications: [{"code": "H04L12/40084", "inventive": true, "first": true, "tree": "[]"}, {"code": "H04L12/40084", "inventive": true, "first": true, "tree": "[]"}, {"code": "H04L12/40058", "inventive": true, "first": false, "tree": "[]"}, {"code": "H04L43/0858", "inventive": true, "first": false, "tree": "[]"}, {"code": "H04L43/0864", "inventive": true, "first": false, "tree": "[]"}, {"code": "H04L43/087", "inventive": true, "first": false, "tree": "[]"}, {"code": "H04L43/10", "inventive": true, "first": false, "tree": "[]"}, {"code": "H04L43/0864", "inventive": true, "first": false, "tree": "[]"}, {"code": "H04L43/0858", "inventive": true, "first": false, "tree": "[]"}, {"code": "H04L43/10", "inventive": true, "first": false, "tree": "[]"}, {"code": "H04L12/40058", "inventive": true, "first": false, "tree": "[]"}, {"code": "H04L43/087", "inventive": true, "first": false, "tree": "[]"}]
Family ID: 38792968