Cell loss asymptotics in buffers fed by heterogeneous long-tailed sources

In this paper we consider a generalization of the so-called M/G/∞ 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/∞ system with long-tailed G distributions. In particular we assume that a session of type i transmits r_i cells per unit time and lasts for a random time r with long-tailed distribution given by P(τ>x)~α_ix(-(1+β_i)) where β_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 more generally. In the heterogeneous case, we show that the asymptotics are not necessarily governed by the sources with the smallest β_i but also depend on the rates r_i and it is the ratio of β_i to r_i 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