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where f x is a function de ned for x P I; I . It is easy to verify that the above probability density functions are nonnegative and normalizable. The process Wn has signi cant impact on transient loss performance of the queueing system. To reach the steady state for realizing steady-state performance, the system needs time to forget its history characerized by the initial condition W1 w1 and the distributions of Wn for 1 < n < I. 13.3.2.2 Transient Loss Measures To compare the loss behavior of two-state Markov and LRD traf c in the transient state, we consider two transient loss measures de ned for on intervals. Recall that R n denotes R t for t P tn ; tn 1 . As we have pointed out already, for a two-state process, it is only necessary to consider on intervals, that is, tn ; tn 1 during which R n r1 . Our rst transient loss measure is the probability that loss of traf c occurs during the nth on interval due to buffer over ow, denoted by PflossjR n r1 g. The second transient loss measure is the expected traf c loss ratio of the nth interval, denoted by E ljR n r1 . Since W1 w1 is a constant, we can use PflossjR 1 r1 , W1 w1 g and E ljR 1 r1, W1 w1 directly as transient loss measures for n 1. Using Lemmas 13.3.1 and 13.3.3, we can easily prove the following theorem, which shows how to compute the loss measures. Theorem 13.3.11 8 n 1; > PfS > t w1 g; < B PflossjR n r1 g > PfS > t w gj w dw; n > 1; : n



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and 8 u0 u t w > > P S > 0 1 du; n 1; > < 0 u0 u E ljR n r1 > u t w > B u0 > : P S> 0 j w du dw; n > 1: u0 u n 0 0 COMMENT 13312 For n ! 2, the loss probability and the expected loss ratio depend on the distributions of both S and T, since jn w depends on the distributions of not only S but also T when n ! 2 13323 Investigating Transient Loss Behavior We can investigate the transient loss behavior of a general two-state uid-type traf c process, at least in principle, by computing the loss measures for any given n ! 1.

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By doing so, in fact, we have already broken, although in a ``brute-force'' way, the barrier of the Markov assumption in transient performance analysis of queueing systems Recall that the basis of transient performance analysis for a Markov model is the Chapman Kolmogorov equation governing the probabilistic evolution of the system state If the model is not Markovian, then the Chapman Kolmogorov equation does not necessarily hold In this case, some auxiliary Markov model may be constructed such that the original system can be analyzed approximately In contrast to this conventional approach, our method does not rely on the Markov assumption, so it can analyze non-Markov systems without approximation In addition, it can also analyze Markov systems without using the Chapman Kolmogorov equation However, if we compute the transient performance measures for each n ! 1 directly, then the computational complexity will gradually become prohibitive as n increases.

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But in the context of transient analysis, performance measures corresponding to a large n may be less interesting, since when n is large, the difference between transient and steady states may be less signi cant Consequently, for transient performance analysis, it may be suf cient to focus only on a relatively short time periods spanned by a number of on and off intervals By doing so, the computation involved will be feasible In fact, there is a simple way to explore the transient loss performance impact of long-range dependence in two-state uid traf c Recall that a two-state uid process can be viewed as an alternating renewal process By taking advantage of the renewal nature of two-state uids, we can simply focus only on a series of a small number of on and off intervals.

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