Abstract:
Self-similar data communication in network traffic is modeled real time and is analyzed using a Markov modified Poissen process (MMPP) to characterize the traffic flow and to accommodate high variability in traffic flow from one time period to the other. The analysis is performed at multiple time levels using a bottom-up approach. The parameters of the model are adjustable at each level according to the traffic parameters at that level. Each model consists of 2 states of network traffic behavior comprising a bursty state representing heavy traffic conditions and an idle state representing light traffic conditions. A transition window defines the upper time interval for the receipt of packets in the bursty state and the lower time interval for the receipt of packets in the idle state. If the inter-rival times for the bursty state and the idle state become approximately equal, the model defaults to a single state model.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS  
       [0001]    The present application is related to the following co-pending U.S. patent applications: U.S. Ser. No. 09/607,103, filed Jun. 29, 2000, entitled “Method and System for Reducing Latency in Message Passing Systems” (Docket No. RPS920000014US1); U.S. Ser. No. 09/607,113, filed Jun. 29, 2000, for “Method and System for Predicting Inter-Packet Delays” (Docket No RPS920000017US1); and U.S. Ser. No. ______, filed ______, for “MMPP Analysis of Network Traffic Using a Transition Window” (Docket No. RPS920030018US1). These patent applications all are assigned to the assignee of the present invention. The content of these cross-referenced co-pending applications is hereby incorporated herein by reference. 
     
    
     
       FIELD OF THE INVENTION  
         [0002]    This invention relates in general to the field of computer technology, and particularly to systems for the transfer of data. More specifically, the invention relates to the real-time modeling and analysis of data communication of self-similar network traffic at multiple levels.  
         BACKGROUND OF THE INVENTION  
         [0003]    The flow of information in a network is often called ‘traffic’. Units of information used in network communication are referred to as ‘packet’. Packets generally arrive at a point in the network at random intervals resulting in ‘bursts’ of traffic resulting in congestion and ‘idle’ periods in which traffic is somewhat more sparse.  
           [0004]    Systems that use a network to communicate messages can derive significant benefits from analysis that provides to the system a characterization of the network traffic. The Poisson Process is widely utilized to model aggregate traffic from voice sources. A Markov Modulated Poisson Process (MMPP) is often utilized to model aggregate traffic from data sources. Network traffic has been shown to be self-similar, therefore, a method used to analyze network traffic should be able to display behavior that is bursty and self-similar.  
           [0005]    The present method uses a multilevel model that utilizes the model claimed and described in co-pending patent application Docket Number RPS920030018US1 filed concurrently herewith and entitled MMPP ANALYSIS OF NETWORK TRAFFIC USING A TRANSITION WINDOW as the base and replicating the model once for each time-scale displayed by the self-similar traffic. A single level, 2 state MMPP model is shown in FIG. 1.  
         SUMMARY OF THE INVENTION  
         [0006]    The present method and system serve to model and analyze asynchronous network traffic that is bursty and self-similar using a Markov modulated Poisson process (MMPP) and self-similar traffic by making the MMPP model multilevel, where each level in the model represents a different time scale. By ‘self-similar’ is meant that the traffic displays the same characteristics of behavior (e.g. bursty or idle) at different time scales. This permits the same principles such as an MMPP model to be applied at each different scale. The model employs a transition window to determine the transition between states. This transition window is represented as [λ B   min , λ I   min ] wherein λ B   max  is the upper boundary for heavy traffic arrival in the bursty state and λ I   min  is the lower boundary for light traffic arrival in the idle state.  
           [0007]    The complexity of the model grows as the number of levels in the model increases. This is not a problem because a model with something in the order of four levels has been deemed to be adequate. For example, others have confirmed that TCP traffic has been is described by at most four time scales. The present invention describes an example of a three-level model, although the model is general enough to represent any number of time scales. This model is an effective means to provide for network traffic analysis either in batch mode or in real time.  
           [0008]    The invention relates to an article comprising a computer-readable medium which stores computer-executable instructions for processing traffic flow patterns associated with network data transmission. The instructions cause a machine to: a) receive traffic pattern data associated with the network transmission of data packets relating to the times of arrival of network data packets; b) apply an MMPP algorithm to the received pattern to define the traffic as being in the bursty state or the idle state; and c) repeat the steps a) and b) one or more additional times at different time-scale levels. The different time levels are based on a bottom-up analysis and rely on the generation of a trace of a traffic pattern for a given time scale and the analysis of the trace to generate a trace of the next scale pattern. The algorithm utilizes a transition window to determine the transition between states. This transition window is represented as [λ B   max , λ I   min ] wherein λ B   max  is the upper boundary for heavy traffic arrival in the bursty state and λ I   min  is the lower boundary for light traffic arrival in the idle state.  
           [0009]    The system analyzes network traffic that is bursty and self similar. It employs an MMPP to model network traffic (in real time or in a batch model) at a first level representing a given time scale. It then repeats the process to model the network traffic at one or more additional levels representing time scales that differ from the time scale in the first step. Each level typically includes 2 states of network traffic behavior comprising a bursty state representing heavy traffic conditions and an idle state representing light traffic conditions. The system employs a transition window to determine the transition between states. This transition window is represented as [λ B   max , λ I   min ] wherein λ B   max  is the upper boundary for heavy traffic arrival in the bursty state and λ I   min  is the lower boundary for light traffic arrival in the idle state. However, when the inter-arrival times for the bursty state and the idle state become approximately equal, the system defaults to a single state model. The analysis comprises the generation of a trace of a traffic pattern for a given time scale. This generated trace is then used to generate a trace of the next time scale pattern. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0010]    [0010]FIG. 1 depicts an MMPP for modeling burst network traffic;  
         [0011]    [0011]FIG. 2 is an illustration of a 3 level MMPP model;  
         [0012]    [0012]FIG. 3 is a diagram of bursty, self similar network traffic;  
         [0013]    [0013]FIG. 4 is a flow chart showing the generation of algorithm τ A ;  
         [0014]    [0014]FIG. 5 is a flow chart showing the generation of algorithm τ B ;  
         [0015]    [0015]FIG. 6 is a diagram of a 3-level, 6-state heavy, bursty, self similar network traffic;  
         [0016]    [0016]FIG. 7 is a diagram of a 2-level, 4 state MMPP model;  
         [0017]    [0017]FIG. 8 is a diagram of 1 level, 2 state bursty, self similar network traffic;  
         [0018]    [0018]FIG. 9 is a diagram of 1 level, 1 state bursty model with light traffic, not bursty;  
         [0019]    [0019]FIG. 10 shows an MMPP model having K-levels, and N-states for handling variable traffic; and  
         [0020]    [0020]FIG. 11 represents one medium for the execution of the program. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0021]    An MMPP model  110 , also sometimes referred to as a bimodal sequencer, is shown in FIG. 1. The model serves to predict inter-message arrival delays. The bursty state  112  describes the network traffic behavior when a burst of packets  114  occurs during heavy traffic conditions. During these bursts, the inter-arrival time between packets is Poisson distributed having a mean value for the time of λ B   mean . The idle state  118  describes the network traffic between bursts, when the traffic characteristic  120  is light traffic with a Poisson distribution having a mean value of λ I   mean  for the inter-arrival time. For the 2-state MMPP model to be a valid representation of the network traffic, the characteristics of the traffic are such that the mean time intervals during heavy (bursty) traffic are substantially shorter than the corresponding time intervals during light (idle) traffic, i.e. λ B   mean &lt;λ I   mean . In the model, the traffic inter-arrival times for the bursty and idle states are represented by the boundary values λ B   max  and λ I   min  respectively. For the special case where λ I   mean  approximately equal to λ B   mean , the model defaults to a single state model. These values are used as the transition criteria between states, as shown in FIG. 1 (λ represents the inter-arrival time of the last packet received). When the inter-arrival time slows down so that λ&gt;λ I   min  and λ&gt;λ B   max , the model transitions at  116  from the bursty state to the idle state. Conversely, when the arrival time becomes faster and λ&lt;λ I   min  and λ&lt; B   max , the model returns along  122  to the bursty state.  
         [0022]    Since the representation of the network traffic in a model is an approximation, the length of the burst during state P B  is an approximation with burst edges that are defined somewhat arbitrarily. In practice, the burst length is defined to satisfy the requirements of the user process. As previously mentioned, these values are used as the transition criteria between states. These boundary values define a transition window [λ B   max , λ I   min ] that has as the left side the parameter λ B   max  and as the right hand side the parameter λ I   min . The first parameter λ B   max  determines an upper bound for the packet inter-arrival time for the bursty state and the parameter λ I   min  determines a lower bound for the packet inter-arrival time for the idle state. For the bursty state, λ B   max  defines the probability ρ B  that a packet with inter-arrival time lower than λ B   max  belongs to the bursty state. Similarly, for the idle state, λ I   min  defines the probability ρ I  that a packet with inter-arrival time higher than λ I   min  belongs to the idle state. Based on these probabilities, a decision can be made for each arriving packet of the particular state transition induced by the arrival.  
         [0023]    Algorithms are described that allow the model to track changes in the network traffic dynamically. As the network traffic characteristics change over time, the mean inter-arrival times for the bursty state (λ B   mean ) and for the idle state (λ I   mean ) also change over time. For the model to track these changes over time, the values λ B   max  and λ I   min  change in proportion to the changes in the traffic. The values λ B   max  and λ I   min  define the sides of a transition window of length k=λ I   min −λ B   max . The size of the transition window [λ B   max ,λ I   min ] can be changed dynamically to be used in adaptive algorithms that control the process transition between states. For implementation in an algorithm used in that fashion, the transition window [λ B   max , λ I   min ] can grow larger and smaller by changing the value of λ I   min  and λ B   max  accordingly. The specific value of the parameters used depends on the specific application of the algorithm.  
         [0024]    Referring now to FIG. 2, the model consists of multiple 2-state MMPP models, with each level specialized to a given scale of the self-similar network traffic. Each 2-state MMPP model shown in FIG. 2 consists of a “bursty” state P B , and an “idle” state P I . The multilevel model involves the analysis needed to determine three different time scales, it being understood that this number of levels is illustrative only. The present invention uses a constructive/bottom-up approach, commencing with the shortest intervals between packet arrivals, and moving progressively to the longest inter-arrival times. Each level of self-similarity is analyzed and a trace representation of traffic at each corresponding level is generated and analyzed. Starting with the low-level traffic pattern  202  where the inter-arrival times are, e.g. 1 ms or less, λ is in the idle state if it is more than λ 1   max  and more than λ 2   min  and in the bursty state if less than λ 1   max  and λ 2   min . A trace of that traffic pattern is generated, stored and then analyzed to generate a trace of the next time scale pattern  204  having inter-arrival times are e.g., 1000 ms or less. That trace and the time scale parameters are analyzed and stored. If λ is more than λ 3   max , it is in the bursty state and more than λ 4   min , it is in the idle state. Each trace is then analyzed in similar fashion until the highest level time scale  206  (inter-arrival times of, for instance, 1 second or less) is reached and λ is compared with λ 5   max  and λ 6  min. The stored traces are used to maintain historical data for the analysis algorithms. These algorithms are described here.  
         [0025]    The analysis generates a sequence of traces of the form τ 1 .τ 2 , . . . , τ k , where τ k ={(Λ k   1 ,P k   1 ), (Λ k   2 ,P k   2 ), . . . , (Λ k   i ,P k   i )}. Each trace τ k  represents a different characteristic scale of the self-similar network traffic at a different scale and where, Λ k   i =the time stamp of leading of the packet and P k   i =the packet/burst size.  
         [0026]    Packet arrivals and inter-burst transitions are detected in the following manner. Assume that packet P i−1  presently belongs to burst state P 1 . Then, the task is to detect whether packet P i  belongs still to burst state P 1  or to the idle state P 2 . The detection logic compares the incoming packet inter-arrival time λ i  with λ B   max  and λ I   min . Four cases are possible:  
         [0027]    Case 1. λ i &lt;λ B   max  and λ i &lt;λ I   min : P i  is detected to belong to burst state P B .  
         [0028]    Case 2. λ i &gt;λ B   max  and λ i &gt;λ I   min : P i  is detected to belong to idle state P I .  
         [0029]    Case 3. λ i &gt;λ B   max  and λ i &lt;λ I   min : P i  is detected to be inside of the transition window [λ B   max , λ I   min ] 
         [0030]    In case 3, the next state transition selected is dependent on the user process requirements. This method can be applied to improve the performance of the network attached devices as will be described hereinafter. In particular, the application of the transition window approach will be described for managing the synchronization process in low-latency, high-bandwidth networks.  
         [0031]    Case 4. λ i &lt;λ B   max  and λ i &gt;λ I   min : This is not a valid combination because both events can not occur at the same time.  
         [0032]    These four cases ( 304 ,  306 ,  308 , and  310 ) are illustrated in FIG. 3. The transition window [λ B   max , λ I   min ] is illustrated as a rectangle  312 , the packet arrivals as vertical arrows  302 , and the incoming packet P i    302  illustrated as an X.  
         [0033]    The traffic analysis algorithm has to accomplish several things:  
         [0034]    1. Analyze the traffic to determine the type of traffic at the time. Then it has to adjust the parameters of the model to the traffic parameters.  
         [0035]    2. For bursty self-similar traffic, but for other traffic as well, the algorithm analyzes the traffic to determine the parameters for each different time scale. Then it adjusts the parameters of the model to the traffic parameters.  
         [0036]    3. It detects the changes to the traffic from time to time, adjusting the model parameters as needed.  
         [0037]    Of course, these are all different aspects of the same problem, that of having the capability to represent highly variable network traffic.  
         [0038]    Although typically the network traffic can be characterized as being bursty and self-similar, as described by the multi-scale MMPP model described above, the conditions of the traffic are such that a high variability can be expected from one time period to the other (minute to minute, hour to hour, day to day, week to week, etc.). It is important for the model used by the methodology to be able to capture this variability in a dynamic way, real-time. The model of the invention assumes bursty self-similar traffic, but it adapts to changes in the traffic from very light traffic, when traffic is not bursty, to more heavy bursty traffic, on to very heavy peaks where the traffic acquires self-similar characteristics. The model assumes the structure of a 2-state multilevel model, or a simpler 2-state single-level model or a single-state model, depending on the traffic. Between the two extremes, there is a continuum of conditions that are represented by the model. The model restructures itself adaptively to the changing conditions of the network traffic. For example, under light traffic conditions, the network traffic can be characterized by a simple Poisson distribution. As traffic intensity goes down, the mean inter-arrival time λ B   mean  approaches that of the idle state inter-arrival time λ I   mean  to the point where the two are no longer distinguished by the model. When the characteristics of the traffic are such that λ I   mean  approximately equals λ B   mean , a single state in the MMPP model represents the network traffic.  
         [0039]    For the purposes of the model, the following states are considered, along with the related parameters in FIGS. 6-9.  
         [0040]    a) 3-level, 6-state MMPP model: heavy, bursty, self-similar traffic. (FIG. 6)  
         [0041]    b) 2-level, 4-state MMPP model: bursty, self-similar traffic. (FIG. 7)  
         [0042]    c) 1-level, 2-state MMPP model: bursty traffic. (FIG. 8)  
         [0043]    d) 1-level, 1-state MMPP model: light traffic, not bursty. (FIG. 9)  
         [0044]    The theoretical maximum number of levels in the model depends on the number of different time scales displayed by the network traffic. That number is believed to be in the range of four levels. In practice, the scale will also be determined by one of the following conditions:  
         [0045]    1. The length of the trace is too short to capture the information at scales beyond some number of levels.  
         [0046]    2. The higher the time scale, both the storage requirements needed to handle a longer trace and the required computational times will be higher. Because of the storage and computational requirements for each level, some practical limit needs to be defined to the system.  
         [0047]    3. The number of levels is defined as a system parameter. The time scales will not reflect self-similar traffic beyond some specified value. Therefore, the value should be established experimentally by the designer. Once an upper limit has been established for the different time scales, the procedure will track changes to the workload automatically and will adjust to the instantaneous burstiness of the workload. The method must then track traffic changes from one end of the spectrum to the other (a to d above).  
         [0048]    Other, more complicated modeling schemes are possible. However, this example represents an adequate implementation of the invention.  
         [0049]    As previously noted, the traffic burst analysis and trace generation consists of the sequential generation of traces of the form τ 1 , τ 2 , . . . , τ k , where τ k ={(Λ k   1 ,P k   1 ), (Λ k   2 ,P k   2 ), . . . ,(Λ k   i ,P k   i )}. Each trace τ k  represents a different characteristic scale of the self-similar network traffic at a different scale where Λ k   i =time stamp of leading of the packet and P k   i =the packet/burst size. This process applies a constructive or bottom-up approach (vs. a deconstructive or analytical approach, which is top-down.)  
         [0050]    The input stream of each packet (i) is analyzed as shown in FIG. 4 (algorithm τ A ).  
         [0051]    Burst inter-arrival time, λ 1   i =Λ 1   i −Λ 1   i−1 , as follows:  
         [0052]    λ 1   i =the inter-arrival time between packets P 1   i ,  
         [0053]    λ 2   i =the inter-arrival time between bursts of packets P 2   i ,  
         [0054]    λ 3   i =the inter-arrival time between clusters (a burst of bursts) P 3   i ,  
         [0055]    λ j   i =the inter-arrival time between bursts of clusters P j   i    
         [0056]    The trace τ 1  is generated at  402  as follows. An incoming packet (i) is read at  404  and i is set to a value of 1. The leading edge of the packet i is detected at  406 . If the leading edge is not found, a second attempt is made to detect it. If detected, the arrival time is stored at  408  as time stamp t=Λ i . The arrival of the trailing edge of the packet is detected at  410  and the formula P i =t−Λ i  representing the time interval between the detection of the leading edge and the trailing edge and the time stamp is calculated and recorded. If the trailing edge is not detected the first time, the process is repeated until detected. The packet size is then stored at  412 . This process is repeated for each packet until the end of the trace is reached at  414 . If the end is not reached, then the process is repeated for the next packet, i=i+1. The end of the trace is signaled at  418 .  
         [0057]    The four test cases are as follows:  
         [0058]    Case 1. λ i &lt;λ B   max  and λ i &lt;λ I   min : P i  is detected to belong to burst state P B .  
         [0059]    Case 2. λ i &gt;λ B   max  and λ i &gt;λ I   min : P i  is detected to belong to idle state P I .  
         [0060]    Case 3. λ i &gt;λ B   max  and λ i &lt;λ I   min : P i  is detected to be inside of the transition window [λ B   max , λ I   min ]. In this case, the next state transition selected is dependent on the user process requirements.  
         [0061]    Case 4. λ i &lt;λ B   max  and λ i &gt;λ I   min : This is not a valid combination because both can not occur.  
         [0062]    The following trace is recorded into an ordered set with sequential format. Assume the following:  
         τ 1 ={(Λ 1   1   ,P   1   1 ),(Λ 1   2   ,P   1   2 ), . . . ,(Λ 1   i   ,P   1   i )},  
         [0063]    Where,  
         [0064]    Λ 1   i =the time stamp of leading of the packet,  
         [0065]    P 1   i =the packet size in microseconds.  
         [0066]    Next, trace τ j  can be analyzed as shown in FIG. 5 (algorithm τ B ). The trace generation starts at  502  as follows. The previous algorithm τ j−l  is read at  504  and i is set to a value of 1. The leading edge of the packet j is determined at  506 . If the leading edge is not found, a second attempt is made to detect it. If detected, the arrival time is stored at 508 as time stamp t=Λ j . With the arrival time of the leading edge stored, the arrival of the trailing edge is detected at  510  and the formula P i =t−Λ j  representing the time interval between the detection of the leading edge and the trailing edge and the time stamp is recorded. The packet arrival is then tested λ j−1   max &lt;λ j   min &lt;λ j   i &lt;λ j   max &lt;λ j+1   min  at  512 . If the packet is determined to be the same pulse at  514 , the packet is again sent to  510  to detect the trailing edge. If it is not the same pulse, the value  Pj =t−Λ j  is stored at  516 . This procedure is repeated for each packet until the end of the trace is reached at 518. If this does not represent the end of the trace, the process is repeated for the next packet j=j+1 at  522 . The end of the trace is signaled at  518 .  
         [0067]    From the analysis, the following trace is derived:  
         τ 2   32  {(Λ 2   1   ,P   2   1 ),(Λ 2   2   ,P   2   2 ), . . . , (Λ 2   1   ,P   2   1 )},  
         [0068]    Each trace can thus be analyze to produce a higher level trace. In general, the following set of traces are derived:  
         τ 1 , τ 2 , τ 3 , . . . , τ k ,  
         [0069]    where trace τ k ={(Λ k   l ,P k   1 ), (Λ k   2 ,P k   2 ), . . . , (Λ k   i ,P k   i )}, represents a different characteristic scale of the self-similar network traffic at a different scale.  
         [0070]    Since the scale for each consecutive level is approximated by an exponential distribution, the following ordering is established:  
         λ j−1   max &lt;λ j   min &lt;λ j   i &lt;λ j   max &lt;λ j+1   min    
       EXAMPLE 1  
       [0071]    This relates to a 3 level, 6 state MMPP model to simulate heavy traffic that is bursty and self-similar as shown in FIG. 6. The packet bursts are shown by the vertical arrow clusters  602 , and the first time level is shown as  604 , the second time level as  606 , and the third time level as  608 .  
         [0072]    1. Use algorithm τ A  to analyze input stream and generate trace τ 1 ={(Λ 1   1 , P 1   1 ), (Λ 1   2 ,P 1   2 ), . . . ,(Λ 1   i ,P 1   i )}.  
         [0073]    2. Analyze parameters Λ 1   i  (the time stamp of leading of the packet), and P 1   i  (the packet size) from trace τ 1 . From this the inter-arrival time λ i  is computed. Thus, there are four possible cases:  
         [0074]    Case 1. λ i &lt;λ 1   B   max  and λ i &lt;λ 1   I   min : P i  is detected to belong to burst state P B .  
         [0075]    Case 2. λ i &gt;λ 1   B   max  and λ i &gt;λ 1   I   min : P i  is detected to belong to idle state P I .  
         [0076]    Case 3. λ i &gt;λ 1   B   max  and λ i &lt;λ 1   I   min : P i  is detected to be inside of the transition window [λ 1   B   max , λ 1   I   min ]. In this case, the next state transition selected is dependent on the user process requirements.  
         [0077]    Case 4. λ i &lt;λ 1   B   max  and λ i &gt;λ 1   I   1min : This is not a valid combination because both can not occur.  
         [0078]    These four cases are used as the test criteria in algorithm τ B  to generate Λ 2   i  (the time stamp of leading of the burst), and P 2   i  (the burst size). This analysis of trace τ 1  generates trace τ 2 ={(Λ 2   1 , P 2   1 ), (Λ 2   2 ,P 2   2 ), . . . , (Λ 2   i ,P 2   i )}.  
         [0079]    2. Analyze parameters Λ 2   i  (the time stamp of leading of the packet), and P 2   i  (the packet size) from trace τ 2 . From this the inter-arrival time λ i  is computed.  
         [0080]    Case 1. λ i &lt;λ 2   B   max  and λ i &lt;λ 2   I   min : P i  is detected to belong to burst state P B .  
         [0081]    Case 2. λ i &gt;λ 2   B   max  and λ i &gt;λ 2   I   min : P i  is detected to belong to idle state P I .  
         [0082]    Case 3. λ i &gt;λ 2   B   max  and λ i &lt;λ 2   I   min : P i  is detected to be inside of the transition window [λ B   max , λ I   min ]. In this case, the next state transition selected is dependent on the user process requirements.  
         [0083]    Case 4. λ i &lt;λ 2   B   max  and λ i &gt;λ 2   I   min : This is not a valid combination because both can not occur.  
         [0084]    These four cases are used as the test criteria in algorithm τ B  to generate Λ 3   i  (the time stamp of leading of the burst), and P 3   i  (the burst size). This analysis of trace τ 2  generates trace τ 3 ={(Λ 3   1 ,P 3   1 ),(Λ 3   2 ,P 3   2 ), . . . ,(Λ 3   i ,P 3   i )},  
         [0085]    3. Analyze parameters Λ 3   i  (the time stamp of leading of the packet), and P 3   i  (the packet size) from trace τ 3 . From this, inter-arrival time λ i  is computed. As explained before, four cases are possible:  
         [0086]    Case 1. λ i &lt;λ 3   B   max  and λ i &lt;λ 3   I   min : P i  is detected to belong to burst state P B .  
         [0087]    Case 2. λ i &gt;λ 3   B   max  and λ i &gt;λ 3   I   min : P i  is detected to belong to idle state P I .  
         [0088]    Case 3. λ i &gt;λ 3   B   max  and λ i &lt;λ 3   I   min , P i  is detected to be inside of the transition window [λ B   max , λ I   min ]. In this case, the next state transition selected is dependent on the user process requirements.  
         [0089]    Case 4. λ i &lt;λ 3   B   max  and λ i &gt;λ 3   I   min : This is not a valid combination because both can not occur.  
         [0090]    These four cases are used as the test criteria in algorithm τ B  to generate Λ 4   i  (the time stamp of leading of the burst), and P 4   i  (the burst size). This analysis of trace τ 3  generates trace τ 4 ={(Λ 4   I ,P 4   1 ), (Λ 4   2 ,P 4   2 ), . . . , (Λ 4   i ,P 4   i )},  
       EXAMPLE 2  
       [0091]    This relates to a 2-level, 4-state MMPP model to simulate heavy traffic that is bursty and self-similar, as shown in FIG. 7. The packet bursts are shown by the vertical arrow clusters  702 . The first level is shown as  704  and the second level as  706 .  
         [0092]    Use algorithm τ A  to analyze input stream and generate trace τ 1 ={(Λ 1   1 ,P 1   1 ),(Λ 1   2 ,P 1   2 ), . . . ,(Λ 1   i ,P 1   i )}.  
         [0093]    Analyze parameters Λ 1   i  (the time stamp of leading of the packet), and P 1   i  (the packet size) from trace τ 1 . From this the inter-arrival time is λ i  computed. As explained before, there are four cases are possible:  
         [0094]    Case 1. λ i &lt;λ 1   B   max  and λ i &lt;λ 1   I   min : P i  is detected to belong to burst state P B .  
         [0095]    Case 2. λ i &gt;λ 1   B   max  and λ i &gt;λ 1   I   min : P i  is detected to belong to idle state P I .  
         [0096]    Case 3, 4: These cases are the same as for Example 1.  
         [0097]    These four cases are used as the test criteria in algorithm s to generate Λ 2   i  (the time stamp of leading of the burst), and P 2   i  (the burst size). This analysis of trace τ 1  generates trace τ 2 ={(Λ 2   1 ,P 2   1 ), (Λ 2   2 ,P 2   2 ), . . . ,(Λ 2   i ,P 2   i )}.  
         [0098]    Analyze parameters Λ 2   i  (the time stamp of leading of the packet), and P 2   i  (the packet size) from trace τ 2 . From this the inter-arrival time is λ i  computed. As explained before, four cases are possible:  
         [0099]    Case 1. λ i &lt;λ 2   B   max  and λ i &lt;λ 2   I   min : P i  is detected to belong to burst state P B .  
         [0100]    Case 2. λ i &gt;λ 2   B   max  and λ i &gt;λ 2   I   min : P i  detected to belong to idle state P I .  
         [0101]    Case 3, 4: These cases are the same as for Example 1.  
         [0102]    These four cases are used as the test criteria in algorithm τ B  to generate Λ 3   i  (the time stamp of leading of the burst), and P 3   i  (the burst size). This analysis of trace τ 2  generates trace τ 3 ={(Λ 3   1 ,P 3   1 ),(Λ 3   2 ,P 3   2 ), . . . ,(Λ 3   i ,P 3   i )}.  
       EXAMPLE 3  
       [0103]    This is directed to a 1-level, 2-state MMPP model to simulate heavy traffic that is bursty and self-similar, and is shown in FIG. 8. The packet bursts are shown by the vertical arrow clusters  802 , and the only time level is shown as  804 .  
         [0104]    First, use algorithm τ A  to analyze input stream and generate trace τ 1 ={(Λ 1   1 ,P 1   1 ), (Λ 1   2 ,P 1   2 ), . . . ,(Λ 1   i ,P 1   i )}.  
         [0105]    Then, analyze parameters Λ 1   i  (the time stamp of leading of the packet), and P 1   i  (the packet size) from trace τ 1 . From this the inter-arrival time is λ i  computed. As explained before, there are four cases are possible:  
         [0106]    Case 1. λ i &lt;λ 1   B   max  and λ i &lt;λ 1   I   min : P i  is detected to belong to burst state P B .  
         [0107]    Case 2. λ i &gt;λ 1   B   max  and λ i &gt;λ 1   I   min : P i  is detected to belong to idle state P I .  
         [0108]    Case 3, 4: These cases are the same as for Example 1.  
         [0109]    These four cases are used as the test criteria in algorithm τ B  to generate Λ 2   i  (the time stamp of leading of the burst), and P 2   i  (the burst size). This analysis of trace τ 1  generates trace τ 2 ={(Λ 2   1 ,P 2   1 ),(Λ 2   2 ,P 2   2 ), . . . ,(Λ 2   i ,P 2   i )}.  
       EXAMPLE 4  
       [0110]    [0110]FIG. 9 shows a 1-level, 1-state MMPP model to simulate light traffic that is not bursty, with the packet arrivals shown as  902 , and the time line as  904 . It involves the following steps:  
         [0111]    1. Use algorithm τ A  to analyze input stream and generate trace τ 1 ={(Λ 1   1 ,P 1   1 ),(Λ 1   2 ,P 1   2 ), . . . ,(Λ 1   i ,P 1   i )} 
         [0112]    2. Analyze parameters Λ 1   i  (the time stamp of leading of the packet), and P 1   i  (the packet size) from trace τ 1 .  
         [0113]    3. From this the inter-arrival time is λ i  computed. As explained before, there are four possible cases:  
         [0114]    Case 1. λ i &lt;λ 1   B   max  and λ i &lt;λ 1   I   min : There are no packets detected that belong to burst state P B .  
         [0115]    Case 2. λ i &gt;λ 1   B   max  and λ i &gt;λ 1   I   min  and: All packet inter-arrivals are detected as belonging to idle state P I .  
         [0116]    Case 3, 4: These cases are the same as for Example 1.  
         [0117]    These four cases are used as the test criteria in algorithm τ B  to generate Λ 2   1  (the time stamp of leading of the burst), and P 2   i  (the burst size). Since there are no bursts in this traffic, trace τ 2  does not exist.  
       EXAMPLE 5  
       [0118]    A K-level, N-state MMPP model to simulate variable traffic is shown in FIG. 10. The packet bursts are shown by the vertical arrows  160 . Traffic pattern  1  ( 130 ) is not bursty and shows a low density of arrows  170 . Traffic pattern  2  ( 140 ) shows 2 states of bursty and idle traffic represented by arrows  170 ; and traffic pattern n ( 150 ) represents bursts of bursts shown by multiple clusters of arrows  180 .  
         [0119]    The conditions of the traffic are such that a high variability can be expected from one time period to the other (minute to minute, hour to hour, day to day, week to week, etc.). It is important for the model used by the methodology to be able to capture this variability in a dynamic way, real-time. The model of the invention assumes bursty self-similar traffic, but it adapts to changes in the traffic from very light traffic (pattern  1 ) when traffic is not bursty, to more heavy bursty traffic (pattern  2 ), on to very heavy peaks (pattern  3 ) where the traffic acquires self-similar characteristics. The model assumes the structure of a 2-state multilevel model, or a simpler 2-state single-level model or a single-state model, depending on the traffic. Between the two extremes, there is a continuum of conditions that are represented by the model. The model restructures itself adaptively to the changing conditions of the network traffic.  
         [0120]    [0120]FIG. 11 shows a computer-readable medium in the form of a floppy disc 160 for containing the software implementation of the program to carry out the various steps of modeling the network traffic according to the present invention. Other machine readable storage mediums are fixed hard drives, optical discs, magnetic tapes, semiconductor memories such as read-only memories (ROMs), programmable (PROMs), etc. The article containing this computer readable code is utilized by executing the code directly from the storage device, or by copying the code from one storage device to another storage device, or by transmitting the code on a network for remote execution.  
         [0121]    While the invention has been described in combination with specific embodiments thereof, there are many alternatives, modifications, and variations that are likewise deemed to be within the scope thereof. Accordingly, the invention is intended to embrace all such alternatives, modifications and variations as fall within the spirit and scope of the appended claims.