13.2 TRAFFIC MODELS AND TRANSIENT LOSS MEASURES

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system. Finally, for a multiple-state arrival process, we may further limit the analysis to an arbitrary single state of the arrival process and compare transient performance measures computed under different modeling assumptions. With this approach, we have gained some insights into QoS impact of long-range dependence in network traf c. The rest of the chapter is organized as follows. In Section 13.2, we rst introduce a framework for traf c modeling that captures the essential property of long-range dependence. Within this framework, traf c is modeled by multistate, uid-type stochastic processes. When such a process is in a given state, the underlying traf c source generates traf c at a constant rate. The time spent by the process in a state is a random variable. For the purpose of this chapter, we let the distribution of the random variable be arbitrary. As a result, we can construct Markov and LRD traf c models as we wish. Then we de ne loss performance measures in the transient state. In Section 13.3, we compare transient loss performance between the traditional Markov models and the LRD models. To keep the comparison reasonable, for the Markov and LRD models, except for the distributions of the times spent by the traf c processes in their respective states, we let all other traf c parameters be the same. By doing so, the difference in loss behavior between Markov and LRD traf c is only due to the modeling assumption on the underlying traf c process. We then compare transient loss of Markov and LRD traf c for two cases. In the rst case, we assume that both traf c processes are in the same state with the same initial condition characterized by the amount of traf c left in the system when the processes enter the state. In the second case, we consider two-state Markov and LRD uids. To examine whether it is appropriate to predict loss performance computed according to Markov models in steady state for LRD traf c, in Section 13.4, we show how to compute steady-state limits of transient loss measures for general two-state uids, and compare transient loss against loss in steady state. In Section 13.5, we discuss the impact of long-range dependence in network traf c, based on the analytical and numerical results obtained. We conclude this chapter in Section 13.6, with a summary of the ndings of our study, and a brief discussion on the challenge posed by transient performance guarantee in the presence of longrange dependence and some extension of this work. Section 13.7 contains two appendixes. 13.2 TRAFFIC MODELS AND TRANSIENT LOSS MEASURES

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We adopt general uid-type stochastic processes with multiple states as a framework for traf c modeling. The state of such a process is associated with the bit rate of the underlying traf c source. When the process is in a given state, the source generates traf c at a constant bit rate. The bit rates are different for different states. Such uidtype traf c models have been used in many previous studies for traf c engineering. A well-known example is the Markov-modulated uid model [5]. However, the traf c model in our study is essentially different from traditional uid traf c models: our traf c model is not necessarily Markovian, which allows us to capture the

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TRANSIENT LOSS PERFORMANCE IMPACT OF LRD IN NETWORK TRAFFIC

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property of long-range dependence in traf c. A special case of our traf c model is the general two-state uid, which can capture the most important traf c properties such as long-range dependence and burstiness. An important part of this work is based on the two-state uid model. Various on=off uid models are special cases of the general two-state uid and have widely been used for traf c modeling. For example, on=off sources with heavy-tailed on=off periods are proposed to explain long-range dependence or self-similarity in traf c [18]. For an on=off uid, no traf c is generated in the off state. In this book, on=off traf c models are also considered in s 5, 7, 11, and 17. Let us denote a uid-type traf c process by R t . The physical meaning of R t is the time-dependent bit rate of the underlying traf c source. Denote R t by R n for t P tn ; tn 1 , where tn is the instant at which the nth transition of the state of R t occurs. Accordingly, tn ; tn 1 is an interval during which R t remains unchanged. Suppose that the bit rate of the traf c source is r during the interval, that is, R n r. Denote the length of the interval tn ; tn 1 by Dtn. Clearly Dtn is a random variable, representing the time spent by R t in the state in which the bit rate of the traf c source is r. Suppose that Dtn obeys a distribution FDtn s PfDtn sg. We assume that the distributions of Dtn are the same when the traf c process is in the same state but may differ for different states. For a two-state process, we use on and off to refer to the states. When the state is on, the bit rate is denoted by r1, and the bit rate corresponding to the off state is r0 , where r1 > r0 ! 0. Denote the lengths of the nth on and off intervals, respectively, by Sn and Tn . We assume that for n ! 1, Sn are independent and identically distributed (i.i.d.) random variables as are Tn . Since both Sn and Tn are i.i.d., we can drop the subscript n in Sn and Tn . The general two-state uid model is appealing from an analysis point of view, since it can capture the essential property of long-range dependence in network traf c while still permitting an exact analysis without approximation. For a Markov uid, Dtn is of course exponentially distributed. To capture the property of long-range dependence in traf c, we can assume that Dtn obeys some heavy-tailed distribution. Readers can nd a simple formal proof in Grossglauser and Bolot [9] for a special case of the general uid traf c model, which shows that if for all n ! 1, Dtn are i.i.d. with respect to both n and R n , and are drawn from a common heavy-tailed distribution, then the corresponding traf c process R t is an LRD process or, more exactly, an asymptotically second-order self-similar process, with autocorrelation function C t $ t a 1 as t 3 I, where the symbol $ represents an asymptotic relation. To de ne transient loss measures, we assume that traf c loss is caused only by buffer over ow. For a multistate uid, the loss measures are the expected traf c loss ratio and the probability that loss of traf c occurs in interval tn ; tn 1 , conditioned on w, the amount of traf c left in the system at tn , where n ! 1. The quantity w is a random variable. In general, for a multistate uid, it is dif cult to obtain the distribution of w. Therefore, we have to treat w as a given condition. However, in the special case of a two-state uid, we only need to treat w1 as a given condition, where w1 is the initial amount of traf c in the system when n 1. For any n > 1, we can compute the distribution of w by recurrence. So for the special case of a two-state

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