Heavy traffic approximations of large deviations of feedforward queueing networks

Abstract

We consider a multi-class feedforward queueing network with first come first serve queueing discipline and deterministic services and routing. The large deviation asymptotics of tail probabilities of the distribution of the workload vector can be characterized by convex path space minimization problems with non-linear constraints. So far there exists no numerical algorithm which could solve such minimization problems in a general way. When the queueing network is heavily loaded it can be approximated by a reflected Brownian motion. The large deviation asymptotics of tail probabilities of the distribution of this heavy traffic limit can again be characterized by convex path space minimization problems with non-linear constraints. However, due to their less complicated structure there exist algorithms to solve such minimization problems. In this paper we show that, as the network tends to a heavy traffic limit, the solution of the large deviation minimization problems of the network approaches the solution of the corresponding minimization problems of a reflected Brownian motion. Stated otherwise, we show that large deviation and heavy traffic asymptotics can be interchanged. We prove this result in the case when the network is initially empty. Without proof we state the corresponding result in the stationary case. We present an illuminating example with two queues.