Cell Loss Asymptotics in Buffers Fed by Heterogeneous Longtailed Sources

In this paper we consider a generalization of the so-called M=G=1 model where M types of long-tailed sessions enter a buffer . The instantaneous rates of the sessions are functions of the occupancy of an M=G=1 system with long-tailed G distributions. In particular we assume that a session of type i transmits ri cells per unit time and lasts for a random time with long tailed distribution given by P ( > x) ix (1+ i) where i > 0. We derive the mean cell loss asymptotics for large buffer size as well as the complementary distribution of the buffer occupancy exceeding a high level. When specialized to the homogeneous case we show that recent results on large buffer asymptotics, which have been shown under more restrictive assumptions, hold under more generally . In the heterogeneous case, we show that the asymptotics are not necessarily governed by the sources with the smallest i but also depend on the rates ri and it is the ratio of i to ri which is important. Finally it is a simple observation that light-tailed (exponential tails for example) sources have no influence on the asymptotics except that they contribute to reducing the capacity available to heavy tailed sources by their mean load.