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/∞ queue. We first use this to show that two classical results of the asymptotic theory for GI/GI/1/∞ 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≥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/∞ queue. We show that under subexponential assumptions for service times, the stationary maximal dater in any such network has tail asymptotics which can be bounded from below and from above by a multiple of the integrated tails of service times. In general, the upper and the lower bounds do not coincide. Nevertheless, exact asymptotics can be obtained along the same lines for various special cases of networks, providing direct extensions of Veraverbeke”s tail asymptotic for the stationary waiting times in the GI/GI/1/∞ queue. We exemplify this on tandem queues (maximal daters and delays in stations) as well as on multiserver queues.