Abstract:
In an IPTV network, cache memory assigned to each DSLAM, Central Office (CO) or Intermediate Office (IO) can be determined by defining a network cost function having cache memory per DSLAM, cache memory per CO and cache memory per IO as decision variables. The network cost function can be minimized to determine the optimal cache memory size for the network. The network cost function can be subject to boundary constraints that the cache memory is between zero and a maximum cache memory size for each network node type, allowing the network cost function to be solved as a 3-ary tree.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims priority to U.S. provisional patent application Ser. No. 61/137,717, filed Aug. 1, 2008, the entire contents of which are herein incorporated by reference. 
    
    
     FIELD OF THE INVENTION 
     This invention relates to Internet Protocol Television (IPTV) networks and in particular to caching of video content within the network. 
     BACKGROUND OF THE INVENTION 
     IPTV networks in metro areas have a hierarchical structure. At the top of this hierarchy, shown in  FIG. 1 , is a Video Hub office (VHO)  19  where all video contents are stored in video servers. The VHO  19  is connected to several Intermediate Offices (IO)  18 , every IO  18  is connected to several Central Offices (CO)  16 , every CO  16  is connected to several Digital Subscriber Line Access Multiplexers (DSLAMs)  14 , and every DSLAM  14  is connected to a certain number of subscribers  12 . 
     In an IPTV network, Video on Demand and other video services generate large amounts of unicast traffic from VHO  19  to subscribers  12  and, therefore, require additional bandwidth and equipment resources in the network. To reduce this traffic (and overall network cost), part of video content (e.g. most popular titles) may be stored in caches  15  closer to subscribers, e.g., in DSLAM  14 , in service switches at COs  16 , or in service routers at IO  18 . 
     On one hand, cache memory has a cost, but on another hand, caching of video content allows decreasing amount of traffic from upstream levels of this hierarchy and, therefore decreasing of network cost. Cache effectiveness is defined as the hit rate, i.e. the ratio of traffic that is served from the cache to the total amount of requested traffic. Hit rate depends on size and throughput of the cache, statistical characteristics of the traffic (e.g., popularity distribution of different titles, their bit rate, size in memory) and on a caching algorithm. For an ideal caching algorithm, the cache size and statistical characteristics of the traffic define hit rate. 
     Another important factor is granularity of network elements. In case of cache that means that cache is allocated in units whose size does not exceed some limit; cost of cache includes cache unit cost and cost per unit of memory. In case of equipment (switches, routers), cost includes unit cost, I/O module (IOM) cost, Model Driven Architecture (MDA) cost and port cost. The number of ports, MDA, IOMs and equipment units are defined by traffic as a part of the dimensioning process. There is a limited number of IOMs per equipment, and these resources should be shared by the MDAs and cache units. 
     Overall, to find an optimal (in terms of network cost) location and amount of cache memory is a complex optimization problem. 
     What is required is a system and method for optimizing a cache arrangement in an IPTV network. 
     SUMMARY OF THE INVENTION 
     In one aspect of the disclosure there is provided a method for determining cache memories of one or more network nodes of an Internet Protocol Television (IPTV) network. A network cost function is defined comprising a plurality of cache memory decision variables that represent a cache memory size at one of the one or more network nodes and at least one cache hit rate function dependent on at least one of the cache memory decision variables. The network cost function is optimized to determine the plurality of cache memory decision variables. 
     In one aspect of the disclosure, there is provided an Internet Protocol Television network comprising a plurality of network nodes, a plurality of the network nodes comprising a cache, wherein a size of the memory of the cache is in accordance with a minimum solution of a network cost function for the network. 
     In one aspect of the disclosure, there is provided a computer-readable medium comprising computer-executable instructions for execution by a processor that, when executed cause the processor to receive a plurality of input parameters and calculate a cache memory size for a plurality of network nodes of an Internet Protocol TeleVision network that minimizes a network cost of the Internet Protocol TeleVision network. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Reference will now be made to specific embodiments, presented by way of example only, and to the accompanying drawings in which: 
         FIG. 1  illustrates an architecture of an IPTV network; 
         FIG. 2  illustrates a method for determining cache memory size; 
         FIG. 3  illustrates a hit rate function and its derivative; 
         FIG. 4  illustrates a minimum of a convex function on a segment; 
         FIG. 5  illustrates a logical 3-ary tree; 
         FIG. 6  illustrates an optimal cache configuration as a function of ZM power and traffic; 
         FIG. 7  illustrates an optimal cache configuration as a function of traffic; 
         FIG. 8  illustrates an equipment cost structure and its impact on cache memory; 
         FIG. 9  illustrates an optimal cache configuration as a function of equipment cost; 
         FIG. 10  illustrates an optimal cache configuration as a function of memory cost; 
         FIG. 11  illustrates a system processor and associated memory; 
         FIG. 12  illustrates an instruction set that may be executed on the processor of  FIG. 11 ; and 
         FIG. 13  illustrates a user processor and memory in association with the system processor of  FIG. 11 . 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In  FIG. 2 , there is shown a method  100  for determining cache memories of one or more nodes of the IPTV network. At step  101 , a network cost function is defined that includes a plurality of cache memory decision variables that represent a cache memory size the network nodes and at least one cache hit rate function dependent on the cache memory decision variables. The network cost function is optimized at step  102  to determine the plurality of cache memory decision variables. 
     The present disclosure thus describes an analytical model of hierarchical cache optimization. This model depends on basic parameters such as traffic, topology, and cost parameters. More simple cost structure and some reasonable assumptions allow an analytically optimal solution of the problem and an analysis of the factors that impact the solution. The optimal cache architecture may use caching at any combination of network hierarchy—DSLAM, CO and IO—depending on traffic, topology, and cost parameters, and thus hierarchical caching may be an important option for cost saving. 
     Content caching is one of the most effective solutions to improve performance and cost effectiveness of multimedia services. In the present Applicant&#39;s copending patent application, U.S. patent application Ser. No. 12/284,079, the entire contents of which are incorporated herein by reference, a simplified analytical model for cache optimization was presented. The total network cost in this model was the sum of transport cost and memory cost, where the transport cost was proportional to the traffic amount that traverses the equipment, and memory cost was proportional to the memory size. In the Applicant&#39;s earlier patent application, unlike the present disclosure, no boundary cases (i.e., case without cache or with maximum cache size) were considered and network cost minimum assumed to be at a stable point (i.e., where derivative of network cost by appropriate parameters is equal to zero). The present disclosure demonstrates that boundary cases can be important for understanding why in some cases optimal solution does not require caching at some levels, or requires using maximum allowed cache size on another levels. However, boundary cases introduce significant complexity to the problem because they generate a large number of cases to be considered. 
     The Applicant&#39;s copending patent application, PCT application Number PCT/US 08/10269, filed on Aug. 29, 2008 which is based on and claims priority from U.S. Provisional Patent Application No. 60/969,162 filed 30 Aug. 2007, the entire contents of which are herein incorporated by reference, describes the concept of content cacheability and a fast algorithm that uses cacheability to optimally partition a cache between several video services with different traffic characteristics and content sizes. The goal of the optimization was to serve maximum (in terms of bandwidth) amount of subscribers&#39; requests subject to constraints on cache memory and throughput. 
     In order to solve a cache optimization problem analytically in accordance with the present disclosure, some simplifying assumptions are made. 
     First, granularity factors mentioned above are ignored. In particular, it is assumed that cost of cache is proportional to cache size and equipment cost is proportional to amount of traffic that traverse the equipment. More specifically, the equipment cost is estimated based on amount of traffic received by this equipment from upstream levels of network hierarchy (upstream traffic) and on amount of traffic send by this equipment downstream (downstream traffic). In order to calculate a total network cost, a unit cost of cache memory is defined for each level of network hierarchy (DSLAM, CO, IO). A unit cost of upstream and downstream traffic is also defined, as will be described in greater detail below. 
     The next assumption regards hit rate, H(m), as a function of cache memory m. Hit rate, H(m)  31 , increases with memory m  32  as shown in  FIG. 3 . In addition to that, it is assumed that function H(m) has a continuous derivative H′(m)  33  and this derivative decreases (effect of diminishing return for hit rate). This condition means that function H(m) is strictly concave. To justify this assumption it is noted that function H′(m) in some ideal cases (when caching algorithm has complete information about statistical characteristics of the traffic) is equivalent to a popularity curve, which, indeed, decreases as a function of a title&#39;s rank. In case of multiple services, hit rate depends on popularity distribution and other characteristics of individual services, as described in the Applicant&#39;s copending patent application, PCT application Number PCT/US 08/10269, referenced above. 
     Because an IPTV network has a hierarchical (tree) structure and traffic is downstream-oriented, a concept of a “virtual” cache can be used. The “virtual” cache in any node of the tree is the actual cache at the node augmented by the cache sizes in a node deeper (towards the leafs) in the tree. A video content residing in the “virtual” cache of the node (i.e., in the node itself or in a node further down the tree) reduces the unicast traffic on the feeder link towards that node (and on all links towards the root of the tree). In the following, it is assumed that out of two cache resources—cache size and cache throughput—cache size is a limiting factor and the only resource to be considered. This assumption can be justified for a cache in DSLAM by setting a maximum cache size that guarantees that traffic from the cache does not exceed cache throughput. In this case, for a given traffic pattern of IPTV subscribers, if cache size per DSLAM is m 1 , cache size per service switch at CO is m 2 , and cache size per service router at IO is m 3 , then caching related traffic reduction (hit rate) at CO level is H(m 1 +m 2 ), and traffic reduction at IO level is H(m 1 +m 2 +m 3 ). 
     This model considers unicast traffic only. Multicast traffic is a relatively small portion of total traffic on CO and IO levels and, therefore, does not make a big impact on equipment cost on those levels. Multicast traffic is a significant part of the total traffic on DSLAM level. However, there is almost no dependency of DSLAM cost on amount of traffic that traverses DSLAMs. 
     NOTATION AND MATHEMATICAL FORMULATION 
     Notations 
     Decision Variables: 
     m 1 —cache memory size per DSLAM (GB); 
     m 2 —cache memory size per service switch at CO (GB); 
     m 3 —cache memory size per service router at IO (GB). 
     Traffic and Topology Parameters: 
     T—total amount of traffic requested by subscribers (Mbs); 
     H(m)—hit rate function; 
     N 1 —total number of DSLAMs in the network; 
     N 2 —total number of service switches in all COs in the network (it may be several service switches per CO); 
     N 3 —total number of service routers in all IOs in the network (it may be several service routers per IO); 
     M 1 —maximum cache size per DSLAM (GB); 
     M 2 —maximum cache size per service switch at CO (GB); 
     M 3 —maximum cache size per service router at IO (GB). 
     Cost Parameters: 
     c 1   m —unit cost of cache memory in DSLAM ($/GB); 
     c 1   m —unit cost of cache memory in CO ($/GB); 
     c 3   m —unit cost of cache memory in CO ($/GB); 
     c 11   t  and c 12   t —unit cost of downstream and upstream traffic for DSLAM ($/Mbs); 
     c 21   t  and c 22   t —unit cost of downstream and upstream traffic for CO ($/Mbs); 
     c 31   t  and c 32   t —unit cost of downstream and upstream traffic for IO ($/Mbs); 
     c 41   t  and c 42   t —unit cost of downstream and upstream traffic for VHO ($/Mbs); 
     Mathematical Formulation 
     Total network cost, NtwkCost, may be calculated as: 
     
       
         
           
             
               
                 
                   NtwkCost 
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         3 
                       
                       ⁢ 
                       
                         
                           N 
                           i 
                         
                         ⁢ 
                         
                           c 
                           i 
                           m 
                         
                         ⁢ 
                         
                           m 
                           i 
                         
                       
                     
                     + 
                     
                       Tc 
                       11 
                       t 
                     
                     + 
                     
                       
                         T 
                         ⁡ 
                         
                           ( 
                           
                             
                               c 
                               12 
                               t 
                             
                             + 
                             
                               c 
                               21 
                               t 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             H 
                             ⁡ 
                             
                               ( 
                               
                                 m 
                                 1 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         T 
                         ⁡ 
                         
                           ( 
                           
                             
                               c 
                               22 
                               t 
                             
                             + 
                             
                               c 
                               31 
                               t 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             H 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   m 
                                   1 
                                 
                                 + 
                                 
                                   m 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         T 
                         ⁡ 
                         
                           ( 
                           
                             
                               c 
                               32 
                               t 
                             
                             + 
                             
                               c 
                               41 
                               t 
                             
                             + 
                             
                               c 
                               42 
                               t 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             H 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   m 
                                   1 
                                 
                                 + 
                                 
                                   m 
                                   2 
                                 
                                 + 
                                 
                                   m 
                                   3 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     The goal is to minimize network cost subject to constraints on cache memory size:
 
 Ntwk Cost( m   1   ,m   2   ,m   3 )→min  (2)
 
Subject to 0 ≦m   i   ≦M   i   , i= 1, 2, 3  (3)
 
     Analytical Solution—Methodology 
     Network cost (1) is a function of three variables {right arrow over (m)}=(m 1 ,m 2 ,m 3 ) defined on polyhedral (3). Note, that this function is strictly convex (because hit rate function H is strictly concave according to the assumptions). The solution, uses the following result from calculus about minimum of a convex function: 
     Theorem 
     Let function f(x) be defined on an interval [a,b], convex, and has derivative f′(x). Then one of the following should take place ( FIG. 4 ):
         (a) f(x) attains its minimum 41 at x=a, i.e. f′(a)≧0;   (b) f(x) attains its minimum 42 at x=x min , a&lt;x min &lt;b, i.e. f′(x min )=0;   (c) f(x) attains its minimum 43 at x=b, i.e. f′(b)≦0.       

     Consider partial derivatives of the network cost function (1): 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   = 
                   
                     
                       
                         N 
                         1 
                       
                       ⁢ 
                       
                         c 
                         1 
                         m 
                       
                     
                     - 
                     
                       
                         Ts 
                         1 
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             m 
                             1 
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         Ts 
                         2 
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               m 
                               1 
                             
                             + 
                             
                               m 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         Ts 
                         3 
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               m 
                               1 
                             
                             + 
                             
                               m 
                               2 
                             
                             + 
                             
                               m 
                               3 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         2 
                       
                     
                   
                   = 
                   
                     
                       
                         N 
                         2 
                       
                       ⁢ 
                       
                         c 
                         2 
                         m 
                       
                     
                     - 
                     
                       
                         Ts 
                         2 
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               m 
                               1 
                             
                             + 
                             
                               m 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         Ts 
                         3 
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               m 
                               1 
                             
                             + 
                             
                               m 
                               2 
                             
                             + 
                             
                               m 
                               3 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ∂ 
                         
                           ( 
                           NtwkCost 
                           ) 
                         
                       
                       
                         ∂ 
                         
                           m 
                           3 
                         
                       
                     
                     = 
                     
                       
                         
                           N 
                           3 
                         
                         ⁢ 
                         
                           c 
                           3 
                           m 
                         
                       
                       - 
                       
                         
                           Ts 
                           3 
                         
                         ⁢ 
                         
                           
                             H 
                             ′ 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 m 
                                 1 
                               
                               + 
                               
                                 m 
                                 2 
                               
                               + 
                               
                                 m 
                                 3 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Where:
 
 s   1   =c   12   t   +c   21   t  
 
 s   2   =c   22   t   +c   31   t  
 
 s   3   =c   32   t   +c   41   t   +c   42   t   (7)
 
     Let {right arrow over (m)} o =(m 1   o , m 2   o , m 3   o ) denote an optimal solution of the problem. The process of network cost minimization for this problem has two phases—forward and backward, and each phase includes three steps. On each step, partial derivatives of network cost (1) are considered at {right arrow over (m)} o  by one variable and fixing two other variables. 
     Forward phase includes the following three steps: 
     1 st  step: minimize network cost (1) by m 3  for m 1 =m 1   o ,m 2 =m 2   o  using (6). 
     2 nd  step: minimize network cost (1) by m 2  for m 1 =m 1   o ,m 3 =m 3   o  using (5) and results of optimization on the previous step. 
     3 rd  step: minimize network cost (1) by m 1  for m 2 =m 2   o ,m 3 =m 3   o  using (4) and results of optimization on the previous step. 
     There are three alternatives to be considered on every step of the optimization process: zero cache size (no caching), some optimal non-zero cache size (moderate caching), or maximum cache size (maximum caching). 
     Overall, the optimization process may be presented as a movement along the logical 3-ary tree  50  with 3 3 =27 leaves, as shown in  FIG. 5 . The forward phase moves down to the leaves of the tree. The backward phase moves up from the leaves of the tree. 
     Description of Analytical Solution 
     Forward Phase 
     On the 1 st  step, network cost (1) is minimized by m 3  for m 1 =m 1   o ,m 2 =m 2   o  using (6). Applying the theorem described above, there are the following three alternatives: 
     
       
         
           
             
               
                 Case 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
                 ⁢ 
                 
                   : 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   m 
                   3 
                   o 
                 
               
               = 
               0 
             
             , 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       3 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       m 
                       → 
                     
                     o 
                   
                   ) 
                 
               
               ≥ 
               0 
             
             , 
             
                 
             
             ⁢ 
             or 
           
         
       
       
         
           
             
               
                 
                   Ts 
                   3 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         m 
                         2 
                         o 
                       
                     
                     ) 
                   
                 
               
               ≤ 
               
                 
                   N 
                   3 
                 
                 ⁢ 
                 
                   c 
                   3 
                   m 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 Case 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 2 
                 ⁢ 
                 
                   : 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 0 
               
               &lt; 
               
                 m 
                 3 
                 o 
               
               &lt; 
               
                 M 
                 3 
               
             
             , 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       3 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       m 
                       → 
                     
                     o 
                   
                   ) 
                 
               
               = 
               0 
             
             , 
             
                 
             
             ⁢ 
             or 
           
         
       
       
         
           
             
               
                 
                   Ts 
                   3 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         m 
                         2 
                         o 
                       
                       + 
                       
                         m 
                         3 
                         o 
                       
                     
                     ) 
                   
                 
               
               = 
               
                 
                   N 
                   3 
                 
                 ⁢ 
                 
                   c 
                   3 
                   m 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 Case 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 3 
                 ⁢ 
                 
                   : 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   m 
                   3 
                   o 
                 
               
               = 
               
                 M 
                 3 
               
             
             , 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       3 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       m 
                       → 
                     
                     o 
                   
                   ) 
                 
               
               ≤ 
               0 
             
             , 
             
                 
             
             ⁢ 
             or 
           
         
       
       
         
           
             
               
                 Ts 
                 3 
               
               ⁢ 
               
                 
                   H 
                   ′ 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       m 
                       1 
                       o 
                     
                     + 
                     
                       m 
                       2 
                       o 
                     
                     + 
                     
                       M 
                       3 
                     
                   
                   ) 
                 
               
             
             ≥ 
             
               
                 N 
                 3 
               
               ⁢ 
               
                 c 
                 . 
               
             
           
         
       
     
     On the 2 nd  step, for every case of the 1 st  step, three alternatives, m 2   o =0, 0&lt;m 2   o &lt;M 2 , and m 2   o =M 2  are considered, so there are a total of 9 cases on this step: 1.1, 1.2, 1.3, 2.1, 2.2, 2.3, 3.1, 3.2, and 3.3. 
     Finally, on the 3 rd  step, for every of the 9 cases of the 2 nd  step, three alternatives, m 1   o =0, 0&lt;m 1   0 &lt;M 1 , and m 1   o =M 1 , are considered, so there are a total of 27 cases on this step. By the end of this step, 9 values for m 1   o , an optimal value for m 1 , can be identified by analyzing 9 sets of cases: 1.1.x, 1.2.x, 1.3.x, 2.1.x, 2.2.x, 2.3.x, 3.1.x, 3.2.x and 3.3.x, where x=1, 2 and 3. For every of those alternative cases, m 1   o =0, if H′(0)≦P, m 1   o  is a solution of equation H′(m 1   o )=P, or m 1   o =M 1 , if H′(M 1 )≧P, where P depends on input parameters. These solutions will be termed “seeds” herein. 
     Here a detailed description of the 2 nd  and 3 rd  steps of the forward phase is provided. 
     On the 2 nd  step, the network cost (1) is minimized by m 2  for m 1 =m 1   o , m 3 =m 3   o  using (5). 
     For the Case 1, (5) becomes: 
     
       
         
           
             
               
                 
                   ∂ 
                   
                     ( 
                     NtwkCost 
                     ) 
                   
                 
                 
                   ∂ 
                   
                     m 
                     2 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     m 
                     → 
                   
                   o 
                 
                 ) 
               
             
             = 
             
               
                 
                   N 
                   2 
                 
                 ⁢ 
                 
                   c 
                   2 
                   m 
                 
               
               - 
               
                 
                   Ts 
                   2 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         m 
                         2 
                         o 
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   Ts 
                   3 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         m 
                         2 
                         o 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Applying the theorem, there are the following three alternatives: 
     
       
         
           
             
               
                 Case 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1.1 
                 ⁢ 
                 
                   : 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   m 
                   2 
                   o 
                 
               
               = 
               0 
             
             , 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       2 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       m 
                       → 
                     
                     o 
                   
                   ) 
                 
               
               ≥ 
               0 
             
             , 
             
                 
             
             ⁢ 
             or 
           
         
       
       
         
           
             
               
                 
                   T 
                   ⁡ 
                   
                     ( 
                     
                       
                         s 
                         2 
                       
                       + 
                       
                         s 
                         3 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       m 
                       1 
                       o 
                     
                     ) 
                   
                 
               
               ≤ 
               
                 
                   N 
                   2 
                 
                 ⁢ 
                 
                   c 
                   2 
                   m 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 Case 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1.2 
                 ⁢ 
                 
                   : 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 0 
               
               &lt; 
               
                 m 
                 2 
                 o 
               
               &lt; 
               
                 M 
                 2 
               
             
             , 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       2 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       m 
                       → 
                     
                     o 
                   
                   ) 
                 
               
               = 
               0 
             
             , 
             
                 
             
             ⁢ 
             or 
           
         
       
       
         
           
             
               
                 
                   T 
                   ⁡ 
                   
                     ( 
                     
                       
                         s 
                         2 
                       
                       + 
                       
                         s 
                         3 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         m 
                         2 
                         o 
                       
                     
                     ) 
                   
                 
               
               = 
               
                 
                   N 
                   2 
                 
                 ⁢ 
                 
                   c 
                   2 
                   m 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 Case 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1.3 
                 ⁢ 
                 
                   : 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   m 
                   2 
                   o 
                 
               
               = 
               
                 M 
                 2 
               
             
             , 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       2 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       m 
                       → 
                     
                     o 
                   
                   ) 
                 
               
               ≤ 
               0 
             
             , 
             
                 
             
             ⁢ 
             or 
           
         
       
       
         
           
             
               
                 T 
                 ⁡ 
                 
                   ( 
                   
                     
                       s 
                       2 
                     
                     + 
                     
                       s 
                       3 
                     
                   
                   ) 
                 
               
               ⁢ 
               
                 
                   H 
                   ′ 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       m 
                       1 
                       o 
                     
                     + 
                     
                       M 
                       2 
                     
                   
                   ) 
                 
               
             
             ≥ 
             
               
                 N 
                 2 
               
               ⁢ 
               
                 
                   c 
                   2 
                   m 
                 
                 . 
               
             
           
         
       
     
     For the Case 2, (5) becomes: 
     
       
         
           
             
               
                 
                   ∂ 
                   
                     ( 
                     NtwkCost 
                     ) 
                   
                 
                 
                   ∂ 
                   
                     m 
                     2 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     m 
                     → 
                   
                   o 
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     
                       N 
                       2 
                     
                     ⁢ 
                     
                       c 
                       2 
                       m 
                     
                   
                   - 
                   
                     
                       N 
                       3 
                     
                     ⁢ 
                     
                       c 
                       3 
                       m 
                     
                   
                 
                 ) 
               
               - 
               
                 
                   Ts 
                   2 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         m 
                         2 
                         o 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Applying the theorem, there are the following three alternatives: 
     
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2.1 
                   ⁢ 
                   
                     : 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     m 
                     2 
                     o 
                   
                 
                 = 
                 0 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         2 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≥ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
               
 
             
             ⁢ 
             
               
                 
                   
                     Ts 
                     2 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         m 
                         1 
                         o 
                       
                       ) 
                     
                   
                 
                 ≤ 
                 
                   
                     
                       N 
                       2 
                     
                     ⁢ 
                     
                       c 
                       2 
                       m 
                     
                   
                   - 
                   
                     
                       N 
                       3 
                     
                     ⁢ 
                     
                       c 
                       3 
                       m 
                     
                   
                 
               
               ; 
             
             ⁢ 
             
               
 
             
             ⁢ 
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2.2 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     : 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   0 
                 
                 &lt; 
                 
                   m 
                   2 
                   o 
                 
                 &lt; 
                 
                   M 
                   2 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         2 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 = 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
               
 
             
             ⁢ 
             
               
                 
                   
                     Ts 
                     2 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           m 
                           1 
                           o 
                         
                         + 
                         
                           m 
                           2 
                           o 
                         
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       N 
                       2 
                     
                     ⁢ 
                     
                       c 
                       2 
                       m 
                     
                   
                   - 
                   
                     
                       N 
                       3 
                     
                     ⁢ 
                     
                       c 
                       3 
                       m 
                     
                   
                 
               
               ; 
             
             ⁢ 
             
               
 
             
             ⁢ 
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2.3 
                   ⁢ 
                   
                     : 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     m 
                     2 
                     o 
                   
                 
                 = 
                 
                   M 
                   2 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         2 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≤ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
             ⁢ 
             
               
 
             
             ⁢ 
             
               
                 
                   Ts 
                   2 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         M 
                         2 
                       
                     
                     ) 
                   
                 
               
               ≥ 
               
                 
                   
                     N 
                     2 
                   
                   ⁢ 
                   
                     c 
                     2 
                     m 
                   
                 
                 - 
                 
                   
                     N 
                     3 
                   
                   ⁢ 
                   
                     
                       c 
                       3 
                       m 
                     
                     . 
                   
                 
               
             
           
         
       
     
     For the Case 3, (5) becomes: 
     
       
         
           
             
               
                 
                   ∂ 
                   
                     ( 
                     NtwkCost 
                     ) 
                   
                 
                 
                   ∂ 
                   
                     m 
                     2 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     m 
                     → 
                   
                   o 
                 
                 ) 
               
             
             = 
             
               
                 
                   N 
                   2 
                 
                 ⁢ 
                 
                   c 
                   2 
                   m 
                 
               
               - 
               
                 
                   Ts 
                   2 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         m 
                         2 
                         o 
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   Ts 
                   3 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         m 
                         2 
                         o 
                       
                       + 
                       
                         M 
                         3 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Applying the theorem, there are the following three alternatives: 
     
       
         
           
             
               
                 Case 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 3.1 
                 ⁢ 
                 
                   : 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   m 
                   2 
                   o 
                 
               
               = 
               0 
             
             , 
             
                 
             
             ⁢ 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       2 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       m 
                       → 
                     
                     o 
                   
                   ) 
                 
               
               ≥ 
               0 
             
             , 
             
                 
             
             ⁢ 
             or 
           
         
       
       
         
           
             
               
                 
                   
                     Ts 
                     2 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         m 
                         1 
                         o 
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     Ts 
                     3 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           m 
                           1 
                           o 
                         
                         + 
                         
                           M 
                           3 
                         
                       
                       ) 
                     
                   
                 
               
               ≤ 
               
                 
                   N 
                   2 
                 
                 ⁢ 
                 
                   c 
                   2 
                   m 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 Case 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 3.2 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   : 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 0 
               
               &lt; 
               
                 m 
                 2 
                 o 
               
               &lt; 
               
                 M 
                 2 
               
             
             , 
             
                 
             
             ⁢ 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       2 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       m 
                       → 
                     
                     o 
                   
                   ) 
                 
               
               = 
               0 
             
             , 
             
                 
             
             ⁢ 
             or 
           
         
       
       
         
           
             
               
                 
                   
                     Ts 
                     2 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           m 
                           1 
                           o 
                         
                         + 
                         
                           m 
                           2 
                           o 
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     Ts 
                     3 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           m 
                           1 
                           o 
                         
                         + 
                         
                           m 
                           2 
                           o 
                         
                         + 
                         
                           M 
                           3 
                         
                       
                       ) 
                     
                   
                 
               
               = 
               
                 
                   N 
                   2 
                 
                 ⁢ 
                 
                   c 
                   2 
                   m 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 Case 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 3.3 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   : 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   m 
                   2 
                   o 
                 
               
               = 
               
                 M 
                 2 
               
             
             , 
             
                 
             
             ⁢ 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       2 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       m 
                       → 
                     
                     o 
                   
                   ) 
                 
               
               ≤ 
               0 
             
             , 
             
                 
             
             ⁢ 
             or 
           
         
       
       
         
           
             
               
                 
                   Ts 
                   2 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         M 
                         2 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   Ts 
                   3 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         M 
                         2 
                       
                       + 
                       
                         M 
                         3 
                       
                     
                     ) 
                   
                 
               
             
             ≥ 
             
               
                 N 
                 2 
               
               ⁢ 
               
                 
                   c 
                   2 
                   m 
                 
                 . 
               
             
           
         
       
     
     Finally, network cost (1) is minimized by m 1  for m 2 =m 2   o , m 3 =m 3   o  using eq. 4. Here the following 9 cases from the 2 nd  level of logical tree ( FIG. 5 ) are considered. 
     For the Case 1.1, eq. 4 becomes: 
     
       
         
           
             
               
                 
                   ∂ 
                   
                     ( 
                     NtwkCost 
                     ) 
                   
                 
                 
                   ∂ 
                   
                     m 
                     1 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     m 
                     → 
                   
                   o 
                 
                 ) 
               
             
             = 
             
               
                 
                   N 
                   1 
                 
                 ⁢ 
                 
                   c 
                   1 
                   m 
                 
               
               - 
               
                 
                   T 
                   ⁡ 
                   
                     ( 
                     
                       
                         s 
                         1 
                       
                       + 
                       
                         s 
                         2 
                       
                       + 
                       
                         s 
                         3 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       m 
                       1 
                       o 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Case 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1.1 
                     ⁢ 
                     
                       .1 
                       . 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     m 
                     1 
                     o 
                   
                 
                 = 
                 0 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≥ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   T 
                   ⁡ 
                   
                     ( 
                     
                       
                         s 
                         1 
                       
                       + 
                       
                         s 
                         2 
                       
                       + 
                       
                         s 
                         3 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     0 
                     ) 
                   
                 
               
               ≤ 
               
                 
                   N 
                   1 
                 
                 ⁢ 
                 
                   c 
                   1 
                   m 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1.1 
                   ⁢ 
                   
                     .2 
                     . 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 &lt; 
                 
                   m 
                   1 
                   o 
                 
                 &lt; 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 = 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   T 
                   ⁡ 
                   
                     ( 
                     
                       
                         s 
                         1 
                       
                       + 
                       
                         s 
                         2 
                       
                       + 
                       
                         s 
                         3 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       m 
                       1 
                       o 
                     
                     ) 
                   
                 
               
               = 
               
                 
                   N 
                   1 
                 
                 ⁢ 
                 
                   c 
                   1 
                   m 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1.1 
                   ⁢ 
                   .3 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     m 
                     1 
                     o 
                   
                 
                 = 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≤ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 T 
                 ⁡ 
                 
                   ( 
                   
                     
                       s 
                       1 
                     
                     + 
                     
                       s 
                       2 
                     
                     + 
                     
                       s 
                       3 
                     
                   
                   ) 
                 
               
               ⁢ 
               
                 
                   H 
                   ′ 
                 
                 ⁡ 
                 
                   ( 
                   
                     M 
                     1 
                   
                   ) 
                 
               
             
             ≥ 
             
               
                 N 
                 1 
               
               ⁢ 
               
                 c 
                 1 
                 m 
               
             
           
         
       
     
     For the Case 1.2, eq. 4 becomes: 
     
       
         
           
             
               
                 
                   ∂ 
                   
                     ( 
                     NtwkCost 
                     ) 
                   
                 
                 
                   ∂ 
                   
                     m 
                     1 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     m 
                     → 
                   
                   o 
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     
                       N 
                       1 
                     
                     ⁢ 
                     
                       c 
                       1 
                       m 
                     
                   
                   - 
                   
                     
                       N 
                       2 
                     
                     ⁢ 
                     
                       c 
                       2 
                       m 
                     
                   
                 
                 ) 
               
               - 
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                   H 
                   ′ 
                 
                 ⁢ 
                 
                   ( 
                   
                     m 
                     1 
                     o 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1.2 
                   ⁢ 
                   
                     .1 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 0 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≥ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     0 
                     ) 
                   
                 
               
               ≤ 
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     c 
                     1 
                     m 
                   
                 
                 - 
                 
                   
                     N 
                     2 
                   
                   ⁢ 
                   
                     c 
                     2 
                     m 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1.2 
                   ⁢ 
                   
                     .2 
                     . 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 &lt; 
                 
                   m 
                   1 
                   o 
                 
                 &lt; 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 = 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       m 
                       1 
                       o 
                     
                     ) 
                   
                 
               
               = 
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     c 
                     1 
                     m 
                   
                 
                 - 
                 
                   
                     N 
                     2 
                   
                   ⁢ 
                   
                     c 
                     2 
                     m 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   
                     Case 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1.2 
                     ⁢ 
                     
                       .3 
                       . 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     m 
                     1 
                     o 
                   
                 
                 = 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≤ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       M 
                       1 
                     
                     ) 
                   
                 
               
               ≥ 
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     c 
                     1 
                     m 
                   
                 
                 - 
                 
                   
                     N 
                     2 
                   
                   ⁢ 
                   
                     
                       c 
                       2 
                       m 
                     
                     . 
                   
                 
               
             
             ⁢ 
             
                 
             
           
         
       
     
     For the Case 1.3, eq. 4 becomes: 
     
       
         
           
             
               
                 
                   ∂ 
                   
                     ( 
                     NtwkCost 
                     ) 
                   
                 
                 
                   ∂ 
                   
                     m 
                     1 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     m 
                     → 
                   
                   o 
                 
                 ) 
               
             
             = 
             
               
                 
                   N 
                   1 
                 
                 ⁢ 
                 
                   c 
                   1 
                   m 
                 
               
               - 
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       m 
                       1 
                       o 
                     
                     ) 
                   
                 
               
               - 
               
                 
                   Ts 
                   2 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         M 
                         2 
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   Ts 
                   3 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         M 
                         2 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1.3 
                   ⁢ 
                   
                     .1 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 0 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≥ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     Ts 
                     1 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                 
                 + 
                 
                   
                     Ts 
                     2 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         M 
                         2 
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     Ts 
                     3 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         M 
                         2 
                       
                       ) 
                     
                   
                 
               
               ≤ 
               
                 
                   N 
                   1 
                 
                 ⁢ 
                 
                   c 
                   1 
                   m 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1.3 
                   ⁢ 
                   
                     .2 
                     . 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 &lt; 
                 
                   m 
                   1 
                   o 
                 
                 &lt; 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 = 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       m 
                       1 
                       o 
                     
                     ) 
                   
                 
               
               + 
               
                 
                   Ts 
                   2 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         M 
                         2 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   Ts 
                   3 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         M 
                         2 
                       
                     
                     ) 
                   
                 
               
             
             = 
             
               
                 N 
                 1 
               
               ⁢ 
               
                 c 
                 1 
                 m 
               
             
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1.3 
                   ⁢ 
                   
                     .3 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≤ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       M 
                       1 
                     
                     ) 
                   
                 
               
               + 
               
                 
                   Ts 
                   2 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         M 
                         1 
                       
                       + 
                       
                         M 
                         2 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   Ts 
                   3 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         M 
                         1 
                       
                       + 
                       
                         M 
                         2 
                       
                     
                     ) 
                   
                 
               
             
             ≥ 
             
               
                 N 
                 1 
               
               ⁢ 
               
                 c 
                 1 
                 m 
               
             
           
         
       
     
     For the Case 2.1, eq. 4 becomes: 
     
       
         
           
             
               
                 
                   ∂ 
                   
                     ( 
                     NtwkCost 
                     ) 
                   
                 
                 
                   ∂ 
                   
                     m 
                     1 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     m 
                     → 
                   
                   o 
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     
                       N 
                       1 
                     
                     ⁢ 
                     
                       c 
                       1 
                       m 
                     
                   
                   - 
                   
                     
                       N 
                       3 
                     
                     ⁢ 
                     
                       c 
                       3 
                       m 
                     
                   
                 
                 ) 
               
               - 
               
                 
                   T 
                   ⁡ 
                   
                     ( 
                     
                       
                         s 
                         1 
                       
                       + 
                       
                         s 
                         2 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   H 
                   ′ 
                 
                 ⁢ 
                 
                   ( 
                   
                     m 
                     1 
                     o 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2.1 
                   ⁢ 
                   
                     .1 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 0 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≥ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   T 
                   ⁡ 
                   
                     ( 
                     
                       
                         s 
                         1 
                       
                       + 
                       
                         s 
                         2 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     0 
                     ) 
                   
                 
               
               ≤ 
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     c 
                     1 
                     m 
                   
                 
                 - 
                 
                   
                     N 
                     3 
                   
                   ⁢ 
                   
                     c 
                     3 
                     m 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2.1 
                   ⁢ 
                   
                     .2 
                     . 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 &lt; 
                 
                   m 
                   1 
                   o 
                 
                 &lt; 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 = 
                 
                   0 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   or 
                 
               
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   T 
                   ⁡ 
                   
                     ( 
                     
                       
                         s 
                         1 
                       
                       + 
                       
                         s 
                         2 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       m 
                       1 
                       o 
                     
                     ) 
                   
                 
               
               = 
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     c 
                     1 
                     m 
                   
                 
                 - 
                 
                   
                     N 
                     3 
                   
                   ⁢ 
                   
                     c 
                     3 
                     m 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2.1 
                   ⁢ 
                   
                     .3 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≤ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   T 
                   ⁡ 
                   
                     ( 
                     
                       
                         s 
                         1 
                       
                       + 
                       
                         s 
                         2 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       M 
                       1 
                     
                     ) 
                   
                 
               
               ≥ 
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     c 
                     1 
                     m 
                   
                 
                 - 
                 
                   
                     N 
                     3 
                   
                   ⁢ 
                   
                     
                       c 
                       3 
                       m 
                     
                     . 
                   
                 
               
             
             ⁢ 
             
                 
             
           
         
       
     
     For the Case 2.2, eq. 4 becomes: 
     
       
         
           
             
               
                 
                   ∂ 
                   
                     ( 
                     NtwkCost 
                     ) 
                   
                 
                 
                   ∂ 
                   
                     m 
                     1 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     m 
                     → 
                   
                   o 
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     
                       N 
                       1 
                     
                     ⁢ 
                     
                       c 
                       1 
                       m 
                     
                   
                   - 
                   
                     
                       N 
                       2 
                     
                     ⁢ 
                     
                       c 
                       2 
                       m 
                     
                   
                 
                 ) 
               
               - 
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       m 
                       1 
                       o 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2.2 
                   ⁢ 
                   
                     .1 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 0 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≥ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     0 
                     ) 
                   
                 
               
               ≤ 
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     c 
                     1 
                     m 
                   
                 
                 - 
                 
                   
                     N 
                     2 
                   
                   ⁢ 
                   
                     c 
                     2 
                     m 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2.2 
                   ⁢ 
                   
                     .2 
                     . 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 &lt; 
                 
                   m 
                   1 
                   o 
                 
                 &lt; 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 = 
                 0 
               
               ⁢ 
               
                   
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       m 
                       1 
                       o 
                     
                     ) 
                   
                 
               
               = 
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     c 
                     1 
                     m 
                   
                 
                 - 
                 
                   
                     N 
                     2 
                   
                   ⁢ 
                   
                     c 
                     2 
                     m 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2.2 
                   ⁢ 
                   
                     .3 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≤ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 Ts 
                 1 
               
               ⁢ 
               
                 
                   H 
                   ′ 
                 
                 ⁡ 
                 
                   ( 
                   
                     M 
                     1 
                   
                   ) 
                 
               
             
             ≥ 
             
               
                 
                   N 
                   1 
                 
                 ⁢ 
                 
                   c 
                   1 
                   m 
                 
               
               - 
               
                 
                   N 
                   2 
                 
                 ⁢ 
                 
                   
                     c 
                     2 
                     m 
                   
                   . 
                 
               
             
           
         
       
     
     For the Case 2.3, eq. 4 becomes: 
     
       
         
           
             
               
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       1 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       m 
                       → 
                     
                     o 
                   
                   ) 
                 
               
               = 
               
                   
               
               ⁢ 
               
                 
                   ( 
                   
                     
                       
                         N 
                         1 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         c 
                         1 
                         m 
                       
                     
                     - 
                     
                         
                     
                     ⁢ 
                     
                       
                         N 
                         3 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         c 
                         3 
                         m 
                       
                     
                   
                   ) 
                 
                 - 
                 
                     
                 
                 ⁢ 
                 
                   
                     Ts 
                     1 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ( 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                     ⁢ 
                     
                         
                     
                     ) 
                   
                 
                 ⁢ 
                 
                     
                 
                 - 
                 
                     
                 
                 ⁢ 
                 
                   
                     Ts 
                     2 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁢ 
                     
                         
                     
                     ( 
                     
                         
                     
                     ⁢ 
                     
                       
                         m 
                         1 
                         o 
                       
                       ⁢ 
                       
                           
                       
                       + 
                       
                           
                       
                       ⁢ 
                       
                         M 
                         2 
                       
                     
                     ⁢ 
                     
                         
                     
                     ) 
                   
                 
               
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2.3 
                   ⁢ 
                   
                     .1 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 0 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≥ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     Ts 
                     1 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                 
                 + 
                 
                   
                     Ts 
                     2 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         M 
                         2 
                       
                       ) 
                     
                   
                 
               
               ≤ 
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     c 
                     1 
                     m 
                   
                 
                 - 
                 
                   
                     N 
                     3 
                   
                   ⁢ 
                   
                     c 
                     3 
                     m 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2.3 
                   ⁢ 
                   
                     .2 
                     . 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 &lt; 
                 
                   m 
                   1 
                   o 
                 
                 &lt; 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 = 
                 0 
               
               ⁢ 
               
                   
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     Ts 
                     1 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         m 
                         1 
                         o 
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     Ts 
                     2 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           m 
                           1 
                           o 
                         
                         + 
                         
                           M 
                           2 
                         
                       
                       ) 
                     
                   
                 
               
               = 
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     c 
                     1 
                     m 
                   
                 
                 - 
                 
                   
                     N 
                     3 
                   
                   ⁢ 
                   
                     c 
                     3 
                     m 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2.3 
                   ⁢ 
                   
                     .3 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≤ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       M 
                       1 
                     
                     ) 
                   
                 
               
               + 
               
                 
                   Ts 
                   2 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         M 
                         1 
                       
                       + 
                       
                         M 
                         2 
                       
                     
                     ) 
                   
                 
               
             
             ≥ 
             
               
                 
                   N 
                   1 
                 
                 ⁢ 
                 
                   c 
                   1 
                   m 
                 
               
               - 
               
                 
                   N 
                   3 
                 
                 ⁢ 
                 
                   
                     c 
                     3 
                     m 
                   
                   . 
                 
               
             
           
         
       
     
     For the Case 3.1, eq. 4 becomes: 
     
       
         
           
             
               
                 
                   ∂ 
                   
                     ( 
                     NtwkCost 
                     ) 
                   
                 
                 
                   ∂ 
                   
                     m 
                     1 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   m 
                   
                     → 
                     o 
                   
                 
                 ) 
               
             
             = 
             
               
                 
                   N 
                   1 
                 
                 ⁢ 
                 
                   c 
                   1 
                   m 
                 
               
               - 
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       m 
                       1 
                       o 
                     
                     ) 
                   
                 
               
               - 
               
                 
                   Ts 
                   2 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       m 
                       1 
                       o 
                     
                     ) 
                   
                 
               
               - 
               
                 
                   Ts 
                   3 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         m 
                         1 
                         o 
                       
                       + 
                       
                         M 
                         3 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 Case 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 3.1 
                 ⁢ 
                 
                   .1 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     m 
                     1 
                     o 
                   
                 
               
               = 
               0 
             
             , 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       1 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     m 
                     
                       → 
                       o 
                     
                   
                   ) 
                 
               
               ≥ 
               0 
             
             , 
             
               
                 or 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         ⁡ 
                         
                           ( 
                           
                             
                               s 
                               1 
                             
                             + 
                             
                               s 
                               2 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         Ts 
                         3 
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             M 
                             3 
                           
                           ) 
                         
                       
                     
                   
                   ≤ 
                   
                     
                       N 
                       1 
                     
                     ⁢ 
                     
                       c 
                       1 
                       m 
                     
                   
                 
               
               ; 
               
                 
 
               
               ⁢ 
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3.1 
                   ⁢ 
                   
                     .2 
                     . 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 &lt; 
                 
                   m 
                   1 
                   o 
                 
                 &lt; 
                 
                   M 
                   1 
                 
               
             
             , 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       1 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     m 
                     
                       → 
                       o 
                     
                   
                   ) 
                 
               
               = 
               0 
             
             , 
             
               
                 or 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         ⁡ 
                         
                           ( 
                           
                             
                               s 
                               1 
                             
                             + 
                             
                               s 
                               2 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             m 
                             1 
                             o 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         Ts 
                         3 
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               m 
                               1 
                               o 
                             
                             + 
                             
                               M 
                               3 
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       N 
                       1 
                     
                     ⁢ 
                     
                       c 
                       1 
                       m 
                     
                   
                 
               
               ; 
               
                 
 
               
               ⁢ 
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3.1 
                   ⁢ 
                   
                     .3 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 
                   M 
                   1 
                 
               
             
             , 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       1 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     m 
                     
                       → 
                       o 
                     
                   
                   ) 
                 
               
               ≤ 
               0 
             
             , 
             
               
                 
                   or 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     T 
                     ⁡ 
                     
                       ( 
                       
                         
                           s 
                           1 
                         
                         + 
                         
                           s 
                           2 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         M 
                         1 
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     Ts 
                     3 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           M 
                           1 
                         
                         + 
                         
                           M 
                           3 
                         
                       
                       ) 
                     
                   
                 
               
               ≥ 
               
                 
                   N 
                   1 
                 
                 ⁢ 
                 
                   
                     c 
                     1 
                     m 
                   
                   . 
                 
               
             
           
         
       
     
     For the Case 3.2, eq. 4 becomes: 
     
       
         
           
             
               
                 
                   ∂ 
                   
                     ( 
                     NtwkCost 
                     ) 
                   
                 
                 
                   ∂ 
                   
                     m 
                     1 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     m 
                     → 
                   
                   o 
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     
                       N 
                       1 
                     
                     ⁢ 
                     
                       c 
                       1 
                       m 
                     
                   
                   - 
                   
                     
                       N 
                       2 
                     
                     ⁢ 
                     
                       c 
                       2 
                       m 
                     
                   
                 
                 ) 
               
               - 
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ( 
                   
                       
                   
                   ⁢ 
                   
                     m 
                     1 
                     o 
                   
                   ⁢ 
                   
                       
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3.2 
                   ⁢ 
                   
                     .1 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 0 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≥ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     0 
                     ) 
                   
                 
               
               ≤ 
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     c 
                     1 
                     m 
                   
                 
                 - 
                 
                   
                     N 
                     2 
                   
                   ⁢ 
                   
                     c 
                     2 
                     m 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3.2 
                   ⁢ 
                   
                     .2 
                     . 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 &lt; 
                 
                   m 
                   1 
                   o 
                 
                 &lt; 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 = 
                 0 
               
               ⁢ 
               
                   
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   Ts 
                   1 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     H 
                     ′ 
                   
                   ( 
                   
                       
                   
                   ⁢ 
                   
                     m 
                     1 
                     o 
                   
                   ⁢ 
                   
                       
                   
                   ) 
                 
               
               ⁢ 
               
                   
               
               = 
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     c 
                     1 
                     m 
                   
                 
                 - 
                 
                   
                     N 
                     2 
                   
                   ⁢ 
                   
                     c 
                     2 
                     m 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3.2 
                   ⁢ 
                   
                     .3 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≤ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 Ts 
                 1 
               
               ⁢ 
               
                 
                   H 
                   ′ 
                 
                 ⁡ 
                 
                   ( 
                   
                     M 
                     1 
                   
                   ) 
                 
               
             
             ≥ 
             
               
                 
                   N 
                   1 
                 
                 ⁢ 
                 
                   c 
                   1 
                   m 
                 
               
               - 
               
                 
                   N 
                   2 
                 
                 ⁢ 
                 
                   
                     c 
                     2 
                     m 
                   
                   . 
                 
               
             
           
         
       
     
     For the Case 3.3, eq. 4 becomes as: 
     
       
         
           
             
               
                 
                   
                     ∂ 
                     
                       ( 
                       NtwkCost 
                       ) 
                     
                   
                   
                     ∂ 
                     
                       m 
                       1 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       m 
                       → 
                     
                     o 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     c 
                     1 
                     m 
                   
                 
                 - 
                 
                   
                     Ts 
                     1 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ( 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                     ⁢ 
                     
                         
                     
                     ) 
                   
                 
                 ⁢ 
                 
                     
                 
                 - 
                 
                   
                     Ts 
                     2 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ( 
                     
                         
                     
                     ⁢ 
                     
                       
                         m 
                         1 
                         o 
                       
                       ⁢ 
                       
                           
                       
                       + 
                       
                         M 
                         2 
                       
                     
                     ) 
                   
                 
                 - 
                 
                   
                     Ts 
                     3 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ( 
                     
                         
                     
                     ⁢ 
                     
                       
                         m 
                         1 
                         o 
                       
                       ⁢ 
                       
                           
                       
                       + 
                       
                         M 
                         2 
                       
                       + 
                       
                         M 
                         3 
                       
                     
                     ) 
                   
                 
               
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3.3 
                   ⁢ 
                   
                     .1 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 0 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≥ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     Ts 
                     1 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                 
                 + 
                 
                   
                     Ts 
                     2 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         M 
                         2 
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     Ts 
                     3 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           M 
                           2 
                         
                         + 
                         
                           M 
                           3 
                         
                       
                       ) 
                     
                   
                 
               
               ≤ 
               
                 
                   N 
                   1 
                 
                 ⁢ 
                 
                   c 
                   1 
                   m 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3.3 
                   ⁢ 
                   
                     .2 
                     . 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 &lt; 
                 
                   m 
                   1 
                   o 
                 
                 &lt; 
                 
                   M 
                   1 
                 
               
               , 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 = 
                 0 
               
               ⁢ 
               
                   
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     Ts 
                     1 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         m 
                         1 
                         o 
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     Ts 
                     2 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           m 
                           1 
                           o 
                         
                         + 
                         
                           M 
                           2 
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     Ts 
                     3 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           m 
                           1 
                           o 
                         
                         + 
                         
                           M 
                           2 
                         
                         + 
                         
                           M 
                           3 
                         
                       
                       ) 
                     
                   
                 
               
               = 
               
                 
                   N 
                   1 
                 
                 ⁢ 
                 
                   c 
                   1 
                   m 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   Case 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3.3 
                   ⁢ 
                   
                     .3 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       m 
                       1 
                       o 
                     
                   
                 
                 = 
                 
                   M 
                   1 
                 
               
               , 
               
                   
               
               ⁢ 
               
                 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         ( 
                         NtwkCost 
                         ) 
                       
                     
                     
                       ∂ 
                       
                         m 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         → 
                       
                       o 
                     
                     ) 
                   
                 
                 ≤ 
                 0 
               
               , 
               
                   
               
               ⁢ 
               or 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     Ts 
                     1 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         M 
                         1 
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     Ts 
                     2 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           M 
                           1 
                         
                         + 
                         
                           M 
                           2 
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     Ts 
                     3 
                   
                   ⁢ 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           M 
                           1 
                         
                         + 
                         
                           M 
                           2 
                         
                         + 
                         
                           M 
                           3 
                         
                       
                       ) 
                     
                   
                 
               
               ≥ 
               
                 
                   N 
                   1 
                 
                 ⁢ 
                 
                   c 
                   1 
                   m 
                 
               
             
             ; 
           
         
       
     
     At this point, up to 9 candidates for the optimal solution can be identified though the actual number of candidates for the optimal solution may be less than 9, because of possible infeasibility. In order to select the most optimal solution, a “backward” process can be used in the tree hierarchy. 
     Backward Phase 
     Every “seed” m 1   o  found on the 3 rd  step of the forward phase, can be checked to see if a feasible solution m 2   o  exists for a “parent” case. For example, if “seed” m 1   o  was found as a solution of case 2.3.x of the 3 rd  step of forwarding phase, then a check is made of the inequality in case 2.3: Ts 2 H′(m 1   o +M 2 )≧N 2 c 2   m −N 3 c 3   m . If this inequality is true, then corresponding value for m 2   o  is M 2 , which may be defined as a “seed pair” (m 1   o ,m 2   o ). Otherwise, there is no feasible solution for this “seed”. 
     For every “seed pair” (m 1   0 ,m 2   0 ) found on the previous step, a check can be made to see if a feasible solution m 3   o  exists for a “parent” case. For example, if “seed pair” (m 1   0 ,m 2   0 ) was found as a solution of case 2.3 on the previous step, then a check may be made of case 2. If equation Ts 3 H′(m 3   o +m 2   o +m 3   o )=N 3 c 3   m  from case 2 has a feasible solution m 3   o , 0&lt;m 3   o &lt;M 3 , then a “seed triple” (m 1   0 ,m 2   0 ,m 3   0 ) may be defined. Otherwise, there is no feasible solution for this “seed pair”. 
     Finally, the optimal solution can be selected out of the “seed triples” found above. 
     Analysis and Observations 
     The analytical solution presented above allows identifying conditions when the optimal solution requires moderate, maximum caching or does not require caching on different levels (DSLAM, CO, IO). Consider several examples. 
     Example 1 
     Optimal solution that does not require caching at all corresponds to the case 1.1.1 (and “parent” cases 1.1 and 1 in logical hierarchy). The corresponding equations are: 
     
       
         
           
             
               
                 
                   
                     
                       
                         H 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         0 
                         ) 
                       
                     
                     ≤ 
                     
                       
                         
                           
                             N 
                             1 
                           
                           ⁢ 
                           
                             c 
                             1 
                             m 
                           
                         
                         
                           T 
                           ⁡ 
                           
                             ( 
                             
                               
                                 s 
                                 1 
                               
                               + 
                               
                                 s 
                                 2 
                               
                               + 
                               
                                 s 
                                 3 
                               
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         m 
                         1 
                         o 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         H 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         
                           m 
                           1 
                           o 
                         
                         ) 
                       
                     
                     ≤ 
                     
                       
                         
                           
                             N 
                             2 
                           
                           ⁢ 
                           
                             c 
                             2 
                             m 
                           
                         
                         
                           T 
                           ⁡ 
                           
                             ( 
                             
                               
                                 s 
                                 2 
                               
                               + 
                               
                                 s 
                                 3 
                               
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         m 
                         2 
                         o 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         H 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             m 
                             1 
                             o 
                           
                           + 
                           
                             m 
                             2 
                             o 
                           
                         
                         ) 
                       
                     
                     ≤ 
                     
                       
                         
                           
                             N 
                             3 
                           
                           ⁢ 
                           
                             c 
                             3 
                             m 
                           
                         
                         
                           Ts 
                           3 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         m 
                         3 
                         o 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     In order that equations (8)-(10) have a solution, the following inequality should take place: 
     
       
         
           
             
               
                 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   ≤ 
                   
                     min 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             
                               N 
                               1 
                             
                             ⁢ 
                             
                               c 
                               1 
                               m 
                             
                           
                           
                             T 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   s 
                                   1 
                                 
                                 + 
                                 
                                   s 
                                   2 
                                 
                                 + 
                                 
                                   s 
                                   3 
                                 
                               
                               ) 
                             
                           
                         
                         , 
                         
                           
                             
                               N 
                               2 
                             
                             ⁢ 
                             
                               c 
                               2 
                               m 
                             
                           
                           
                             T 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   s 
                                   2 
                                 
                                 + 
                                 
                                   s 
                                   3 
                                 
                               
                               ) 
                             
                           
                         
                         , 
                         
                           
                             
                               N 
                               3 
                             
                             ⁢ 
                             
                               c 
                               3 
                               m 
                             
                           
                           
                             Ts 
                             3 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Equation (11) indicates that if hit rate growth at 0 is below some threshold (this threshold depends on topology, traffic and cost parameters), no caching is required for the optimal solution. 
     Example 2 
     Optimal solution that requires moderate (i.e., below maximum cache size) caching on all three levels corresponds to the case 2.2.2 (and “parent” cases 2.2 and 2 in logical hierarchy). The corresponding equations are: 
     
       
         
           
             
               
                 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         m 
                         1 
                         o 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               N 
                               1 
                             
                             ⁢ 
                             
                               c 
                               1 
                               m 
                             
                           
                           - 
                           
                             
                               N 
                               2 
                             
                             ⁢ 
                             
                               c 
                               2 
                               m 
                             
                           
                         
                         
                           Ts 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                     &lt; 
                     
                       m 
                       1 
                       o 
                     
                     &lt; 
                     
                       M 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           m 
                           1 
                           o 
                         
                         + 
                         
                           m 
                           2 
                           o 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               N 
                               2 
                             
                             ⁢ 
                             
                               c 
                               2 
                               m 
                             
                           
                           - 
                           
                             
                               N 
                               3 
                             
                             ⁢ 
                             
                               c 
                               3 
                               m 
                             
                           
                         
                         
                           Ts 
                           2 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                     &lt; 
                     
                       m 
                       2 
                       o 
                     
                     &lt; 
                     
                       M 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           m 
                           1 
                           o 
                         
                         + 
                         
                           m 
                           2 
                           o 
                         
                         + 
                         
                           m 
                           3 
                           o 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             N 
                             3 
                           
                           ⁢ 
                           
                             c 
                             3 
                             m 
                           
                         
                         
                           Ts 
                           3 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                     &lt; 
                     
                       m 
                       3 
                       o 
                     
                     &lt; 
                     
                       M 
                       3 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     If (12) has a solution m 1   o , its value may be substituted in (13). If (13) has a solution m 2   o , its value may be substituted in (14). If (14) has a solution m 3   o , that gives an optimal solution (m 1   o , m 2   o , m 3   o ) that uses caching in DSLAM, CO and IO levels. 
     Example 3 
     Optimal solution that requires maximum caching (i.e., m i   o =M i , i=1, 2, 3) on all three levels corresponds to the case 3.3.3 (and “parent” cases 3.3 and 3 in logical hierarchy). The corresponding equations are: 
     
       
         
           
             
               
                 
                   
                     
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             M 
                             1 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         s 
                         1 
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               M 
                               1 
                             
                             + 
                             
                               M 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         s 
                         3 
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               M 
                               1 
                             
                             + 
                             
                               M 
                               2 
                             
                             + 
                             
                               M 
                               3 
                             
                           
                           ) 
                         
                       
                     
                   
                   ≥ 
                   
                     
                       
                         N 
                         1 
                       
                       ⁢ 
                       
                         c 
                         1 
                         m 
                       
                     
                     T 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         s 
                         2 
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               M 
                               1 
                             
                             + 
                             
                               M 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         s 
                         3 
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               M 
                               1 
                             
                             + 
                             
                               M 
                               2 
                             
                             + 
                             
                               M 
                               3 
                             
                           
                           ) 
                         
                       
                     
                   
                   ≥ 
                   
                     
                       
                         N 
                         2 
                       
                       ⁢ 
                       
                         c 
                         2 
                         m 
                       
                     
                     T 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       s 
                       3 
                     
                     ⁢ 
                     
                       
                         H 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             M 
                             1 
                           
                           + 
                           
                             M 
                             2 
                           
                           + 
                           
                             M 
                             3 
                           
                         
                         ) 
                       
                     
                   
                   ≥ 
                   
                     
                       
                         N 
                         3 
                       
                       ⁢ 
                       
                         c 
                         3 
                         m 
                       
                     
                     T 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Consider now the case when cache memory cost and equipment cost (in terms of cost per unit of traffic) does not depend on level of hierarchy (homogeneous cost):
 
 c   1   m   =c   2   m   =c   3   m   =c   m   (18)
 
 c   11   t   =c   12   t   =c   21   t   =c   22   t   =c   31   t   =c   32   t   =c   41   t   =c   42   t   =c   t   (19)
 
     Therefore, from eq. 7:
 
 s   1   =s   2 =2 c   t   (20)
 
 s   3 =3 c   t   (21)
 
     Note, that if cost is homogeneous, optimal cache configuration for a given topology (N 1 , N 2 , N 3 ), hit rate function H and maximum cache sizes (M 1 , M 2 , M 3 ) depends on the ratio 
     
       
         
           
             
               
                 c 
                 m 
               
               
                 Tc 
                 t 
               
             
             . 
           
         
       
     
     Substituting (18)-(21) into (11), gives rise to the following condition of optimal solution without cache: 
     
       
         
           
             
               
                 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   ≤ 
                   
                     
                       
                         c 
                         m 
                       
                       
                         Tc 
                         t 
                       
                     
                     ⁢ 
                     
                       min 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               N 
                               1 
                             
                             7 
                           
                           , 
                           
                             
                               N 
                               2 
                             
                             5 
                           
                           , 
                           
                             
                               N 
                               3 
                             
                             3 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     11 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     Equation (11a) demonstrates an impact of hit rate, cost, traffic and topology parameters on optimal cache architecture. For example, the chances that the optimal solution does not require caching increases when:
         Ratio       

               c   m       Tc   t           
of memory cost c m  over equipment cost c t  and traffic T is high;
         Numbers of DSLAMs (N 1 ), COs (N 2 ) and IOs (N 3 ), but especially DSLAMs are large;   Hit rate function H has a small level of growth at 0.       

     Substituting (18)-(21) into (12)-(14), the following equations define a condition of optimal solution with moderate caching on all levels: 
     
       
         
           
             
               
                 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         m 
                         1 
                         o 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 N 
                                 1 
                               
                               - 
                               
                                 N 
                                 2 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             c 
                             m 
                           
                         
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Tc 
                             1 
                             t 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                     &lt; 
                     
                       m 
                       1 
                       o 
                     
                     &lt; 
                     
                       M 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   
                     12 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           m 
                           1 
                           o 
                         
                         + 
                         
                           m 
                           2 
                           o 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 N 
                                 2 
                               
                               - 
                               
                                 N 
                                 3 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             c 
                             m 
                           
                         
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Tc 
                             t 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                     &lt; 
                     
                       m 
                       2 
                       o 
                     
                     &lt; 
                     
                       M 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   
                     13 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       H 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           m 
                           1 
                           o 
                         
                         + 
                         
                           m 
                           2 
                           o 
                         
                         + 
                         
                           m 
                           3 
                           o 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             N 
                             3 
                           
                           ⁢ 
                           
                             c 
                             m 
                           
                         
                         
                           3 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Tc 
                             t 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                     &lt; 
                     
                       m 
                       3 
                       o 
                     
                     &lt; 
                     
                       M 
                       3 
                     
                   
                 
               
               
                 
                   ( 
                   
                     14 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     Substituting (18)-(21) into (15)-(17), the following equations define a condition of optimal solution with maximum caching on all levels: 
     
       
         
           
             
               
                 
                   
                     
                       2 
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             M 
                             1 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               M 
                               1 
                             
                             + 
                             
                               M 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               M 
                               1 
                             
                             + 
                             
                               M 
                               2 
                             
                             + 
                             
                               M 
                               3 
                             
                           
                           ) 
                         
                       
                     
                   
                   ≥ 
                   
                     
                       
                         N 
                         1 
                       
                       ⁢ 
                       
                         c 
                         m 
                       
                     
                     
                       Tc 
                       t 
                     
                   
                 
               
               
                 
                   ( 
                   
                     15 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       2 
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               M 
                               1 
                             
                             + 
                             
                               M 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           H 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               M 
                               1 
                             
                             + 
                             
                               M 
                               2 
                             
                             + 
                             
                               M 
                               3 
                             
                           
                           ) 
                         
                       
                     
                   
                   ≥ 
                   
                     
                       
                         N 
                         2 
                       
                       ⁢ 
                       
                         c 
                         m 
                       
                     
                     
                       Tc 
                       t 
                     
                   
                 
               
               
                 
                   ( 
                   
                     16 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     3 
                     ⁢ 
                     
                       
                         H 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             M 
                             1 
                           
                           + 
                           
                             M 
                             2 
                           
                           + 
                           
                             M 
                             3 
                           
                         
                         ) 
                       
                     
                   
                   ≥ 
                   
                     
                       
                         N 
                         3 
                       
                       ⁢ 
                       
                         c 
                         m 
                       
                     
                     
                       Tc 
                       t 
                     
                   
                 
               
               
                 
                   ( 
                   
                     17 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     Equations (15a)-(17a) demonstrate that the chances that the optimal solution requires maximum caching on all levels increases when:
         Ratio       

               c   m       Tc   t           
of memory cost c m  over equipment cost c t  and traffic T is low;
         Numbers of DSLAMs (N 1 ), COs (N 2 ) and IOs (N 3 ) are small;   Hit rate function H has a high level of growth at points M 1 , M 1 +M 2  and M 1 +M 2 +M 3 .       

     Sensitivity Study 
     Objective 
     The impact of various parameters on optimal cache configuration will now be described using the following two metrics of optimal cache configuration:
         Size of cache memory in DSLAM, CO and IO nodes, that is {right arrow over (m)} o =(m 1   o ,m 2   o ,m 3   o );   Cost gain characterizes a potential cost savings from caching and defined as 1−NtwkCost opt /NtwkCoSt NoCache , where NtwkCost opt  is a cost (1) of network with optimal cache configuration, and NtwkCost NoCache  is a cost (1) of network without cache.       

     Reference Scenario 
     As a reference scenario, there is considered a network defined by the following parameters:
         N 1 , a total number of DSLAMs in the network, is 9,600;   N 2 , a total number of service switches in all COs, is 100;   N 3 , a total number of service routers in all IOs, is 16;   Total amount of traffic, T, in reference network is 9,600,000 Mbs (i.e., 1 Gbs per DSLAM);   M 1 , a maximum cache size per DSLAM, is 100 GB;   M 2 , a maximum cache size per service switch at CO, is 12,000 GB;   M 3 , a maximum cache size per service router at IO, is 24,000 GB.       

     It is assumed that the unit cost of cache memory in DSLAM, CO and IO is $22/GB. As for equipment cost (i.e., cost per unit of traffic), it is assumed that the unit cost of upstream DSLAM traffic is $1.5/Mbs, unit cost of upstream and downstream CO traffic is $2.5/Mbs, and unit cost of upstream and downstream IO and VHO traffic is $4/Mbs. That corresponds to S1=$4/Mbs, S2=$6.5/Mbs, and S3=$12/Mbs. 
     Hit rate function H(m) may be modeled in such a way that its derivative, H′(m), represents continuous version of Zipf-Mandelbrot distribution: 
                   H   ′     ⁡     (   m   )       =     K       (     m   +   q     )     α         ,         
truncated on interval [ 0 ,M], where 0&lt;M≦∞. Here, ZM power parameter α≧0 characterizes hit rate steepness at 0, shift parameter q&gt;0, and K is normalization coefficient:
 
     
       
         
           
             
               
                 ∫ 
                 0 
                 M 
               
               ⁢ 
               
                 
                   
                     H 
                     ′ 
                   
                   ⁡ 
                   
                     ( 
                     m 
                     ) 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   m 
                 
               
             
             = 
             1 
           
         
       
     
     For the reference case, chosen parameters may be a α=1, q=1, and M=100,000. 
     In the following, there is shown the sensitivity of optimal cache sizes to various parameters. 
     Sensitivity to Power Parameter of ZM Distribution 
     Sensitivity to power parameter of ZM distribution is shown for several levels of traffic per DSLAM: 0.1 Gbs, 0.5 Gbs, 1 Gbs and 10 Gbs (see  FIG. 6 ). 
     Note, that for a small amount of traffic (e.g., 0.1 Gbs per DSLAM), the optimal amount of cache per node increases with ZM-power parameter up to some point, and then starts to decrease. When ZM-power parameter is small (i.e., hit rate is relatively flat) cache effectiveness is also small, so the optimal cache configuration uses a small amount of cache or no cache at all. With an increase of the ZM-power parameter, the cache effectiveness as well as cache size in the optimal configuration increases. After some point, further increases of ZM-power parameter has an inverse effect: i.e. the hit rate becomes so steep and cache effectiveness so high that the optimal result can be achieved with smaller and smaller amounts of cache size per node. Therefore, optimal cache size as a function of ZM-power parameter obtains its maximum at some point and declines after that. The maximum point of optimal cache size depends on total amount of traffic T and increases with increase of traffic. For a larger amount of traffic, when optimal cache size attains its limit, some boundary effect starts to play its role at first for cache at IO level, and then for cache at all three levels (IO, CO and DSLAM). In all considered scenarios, cost gain increases with power parameter. 
     Sensitivity to Traffic 
     The optimal amount of cache per DSLAM, CO and IO nodes, as well as cost gain, increases with traffic (see  FIG. 7 ). To justify cache at DSLAM, traffic should exceed some threshold. This threshold decreases with increase of ZM-power parameter as we demonstrated earlier. 
     Sensitivity to Equipment Cost 
     The optimal cache configuration depends on cost parameters S 1 , S 2 , and S 3 , defined by (7). There is a strong impact of value S 1  on size of DSLAM cache, value of S 2  on size of CO cache and value S 3  on size of IO cache (see  FIG. 8 ). 
     In the same time, there may be an “isolation effect”: i.e. there is a relatively small impact of S 1  on size of CO and IO cache, S 2  on size of DSLAM and IO cache and S 3  on size of DSLAM and CO cache (see  FIG. 9 ). 
     For example, when S 1  tends to 0, DSLAM cache size also tends to 0 (similar effect for S 2  and S 3 ). Caching in DSLAM impacts all network cost elements upstream from DSLAM by reducing traffic (a) from CO to DSLAM, (b) from IO to CO and (c) from VHO to IO. However, while (a) may be reduced by DSLAM caching (but not by CO and IO caching), for reduction of (b) caching at DSLAM and/or at CO can be used, and for reduction of (c) caching at DSLAM, at CO, and/or at IO can be used. Now, assume that S 1 =0 (or very small). In this case, caching at DSLAM makes no or small difference on network cost related to S 1 , because for upstream network cost (related to S 2  and S 3  elements), it is preferred to use the same amount X GB of cache memory at CO than at DSLAM to achieve the same network cost reduction effect, but with smaller cost there is X GB of cache memory at each CO instead of X GB of memory at each DSLAM. Cost gain decreases with increase of equipment cost. 
     Sensitivity to Memory Cost 
     When memory cost drops, more cache is used in an optimal configuration ( FIG. 10 ). The DSLAM cache as a part of optimal hierarchical configuration is used when cache cost is below some threshold. 
     As described herein, the analytical model allows a calculation of optimal cache configuration and to investigate how sensitive the solution is to various parameters of the system (traffic, topology, cost parameters). 
     Once an optimal solution of the network cost function has been determined and optimal values for the cache memory sizes m 1 , m 2 , and m 3  have been selected, the nodes of the network, i.e. the IOs  18 , COs  16  and DSLAMs  14  can be provided with cache memory in accordance with the optimized solution. 
     In one embodiment, the analytical model of the network cost function may embodied in a computer system including one or more processors  1101  as shown in  FIG. 11 . The processor  1101  may be a system processor operatively associated with a system memory  1102  that stores an instruction set such as software for calculating a network cost function. The process may cause a user interface (not shown) to display user prompts that allow a user to provide input parameters such as the traffic, topology and costing parameters as described above. An example instruction set is shown in the flowchart  200  of  FIG. 12 . At step  201 , the processor receives the input parameters, e.g. through the user interface, and then at step  202 , the processor calculates a cache memory size for a plurality of network nodes of an Internet Protocol TeleVision network that minimizes a network cost of the Internet Protocol TeleVision network. 
     The user interface may be displayed on a separate computer terminal such that the system processor  1101  may receive parameter information from a second processor  1107 , such as a user processor which is also operatively associated with a memory  1108 . The communication between the processors  1101 ,  1107  may be provided by a suitable communications link  1105  which may be hard wired, wireless, internet or FTP based, or any other suitable communications link. 
     Although embodiments of the present invention have been illustrated in the accompanied drawings and described in the foregoing description, it will be understood that the invention is not limited to the embodiments disclosed, but is capable of numerous rearrangements, modifications, and substitutions without departing from the spirit of the invention as set forth and defined by the following claims. For example, the capabilities of the invention can be performed fully and/or partially by one or more of the blocks, modules, processors or memories. Also, these capabilities may be performed in the current manner or in a distributed manner and on, or via, any device able to provide and/or receive information. Further, although depicted in a particular manner, various modules or blocks may be repositioned without departing from the scope of the current invention. Still further, although depicted in a particular manner, a greater or lesser number of modules and connections can be utilized with the present invention in order to accomplish the present invention, to provide additional known features to the present invention, and/or to make the present invention more efficient. Also, the information sent between various modules can be sent between the modules via at least one of a data network, the Internet, an Internet Protocol network, a wireless source, and a wired source and via plurality of protocols.