Moments and tails in monotone-separable stochastic networks

Abstract

A network belongs to the monotone separable class if its state variables are homogeneous and monotone functions of the epochs of the arrival process. This framework, which was first introduced to derive the stability region for stochastic networks with stationary and ergodic driving sequences, is revisited. It contains several classical queueing network models, including generalized Jackson networks, max-plus networks, polling systems, multiserver queues, and various classes of stochastic Petri nets. Our purpose is the analysis of the tails of the stationary state variables in the particular case of i.i.d. driving sequences. For this, we establish general comparison relationships between networks of this class and the GI/GI/1/\infty queue. We first use this to show that two classical results of the asymptotic theory for GI/GI/1/\infty queues can be directly extended to this framework. The first one concerns the existence of moments for the stationary state variables. We establish that for all α\geq 1, the (α+1)-moment condition for service times is necessary and sufficient for the existence of the α-moment for the stationary maximal dater (typically the time to empty the network when stopping further arrivals) in any network of this class. The second one is a direct extension of Veraverbeke's tail asymptotic for the stationary waiting times in the GI/GI/1/\infty queue.