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 $r_n$ bits per unit time and lasts for a random time $\tau_n$ with a generalized Pareto distribution given by $P \{\tau_n > x\} \sim \alpha_n x^{-(1+\beta_n)}$ for large $x$, where $\alpha_n, \beta_n > 0$. The weights are taken to be the rates of the flows. The network is assumed to be loop-free 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.