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For an asymptotically heavy-tailed distribution, it is only necessary for us to consider the case that the value of S is large enough to be in the heavy tail, since only the heavy-tailed effect appears essentially different from Markov traf c modeling and hence is of great interest for the purpose of this study To be speci c, we consider the following heavy-tailed distribution: PfS sg 1 gs 1 a ; 0 s < I; g > 0; 1 < a < 2; 13:6.

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which is a variant of the conventional Pareto distribution. The reason for us to consider this variant is that the range of the random variable of interest in our study is 0; I while for the conventional Pareto distribution, the range of the random variable is o; I , where o > 0. As we can see, the tail of the distribution becomes heavier and heavier as a decreases toward 1. In fact, a smaller a corresponds to a stronger LRD effect [18]. We are concerned with transient loss performance of the underlying traf c process R t predicted by M t . In other words, we want to know the impact of long-range dependence on the transient loss performance predicted by the Markov model. For convenience of exposition, when necessary, M and L, representing respectively the Markov and LRD traf c models, will substitute for R in the notation

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R n for distinction of the use of the notation For example, L n represents the bit rate of L t in the nth interval between transitions of the state of L t and PflossjL n r, w 0 wg is the conditional loss probability of L t Our approach is to compare PflossjM n r, w 0 wg and E ljM n r, w 0 w with PflossjL n r, w 0 wg and E ljL n r, w 0 w , respectively, for given B < I, C < I, r > C, and w < B, under the assumption that E S is the same for Markov model M t and LRD model L t That is, the comparison is made such that M t and L t are in the same state with the same initial amount of traf c left in the buffer.

With such a comparison, we believe that the difference in transient loss performance predicted by the Markov model and the LRD model is only due to the different modeling assumptions on the underlying traf c process R t According to Lemmas 1331 and 1333, the comparison is straightforward To exclude the trivial cases, we consider only r > C and w < B Theorem 1336 Supose that S obeys the Pareto distribution (136) for L t , and E S is the same for L t and M t .

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For given B < I, C < I, r > C, and w < B, we have PflossjM n r, w 0 wg ! PflossjL n r, w 0 wg if t w g 1 z and PflossjM n r, w 0 wg < PflossjL n r, w 0 wg otherwise, where t w is given by Eq (132) and z > 0 is the solution of ebx x 1 for x P 0; I and b 1 a 1 Proof Under the assumptions that S obeys the Pareto distribution (136) for L t , and E S is the same for M t and L t , we have E S a 1 1 g 1 for both L t and M t , and according to Eq (13.

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