Buffer occupancy asymptotics in rate proportional sharing networks with heterogeneous long-tailed inputs

In this paper, we consider a network of rate proportional processor sharing servers in which sessions with long-tailed duration arrive as Poisson processes. In particular, we assume that a session of type n transmits at a rate rn bits per unit time and lasts for a random time τn with a generalized Pareto distribution given by IP{τn > x} ∼ αnxn for large x, where αn, βn > 0. The weights are taken to be the rates of the flows. The network is assumed to be loopfree with respect to source-destination routes. We characterize the order O−asymptotics of the complementary buffer occupancy distribution at each node in terms of the input characteristics of the sessions. In particular, we show that the distributions obey a power law whose exponent can be calculated via solving a fixed point and deterministic knapsack problem. The paper concludes with some canonical examples.