Functional continuity and large deviations for the behavior of single-class queueing networks

Abstract

We consider a single-class queueing network in which the functional network primitives describe the cumulative exogenous arrivals, service times and routing decisions of the queues. The behavior of the network consisting of the cumulative total arrival, cumulative idle time, and queue length developments for each node is specified by conditions which relate the network primitives to the network behavior. For a broad class of network primitives, including discrete customer and fluid models, a network behavior exists, but need not be unique. Nevertheless, the mapping from network primitives to the set of associated network behavior is upper semicontinuous at network primitives with continuous routing. As an application we consider a sequence of random network primitives satisfying a sample path large deviation principle. We take advantage of the partial functional set-valued upper semicontinuity in order to derive a large deviation principle for the sequence of associated random queue length processes and to identify the rate function. This extends the results of Puhalskii (Markov Process. Relat. Fields 13(1), 99---136, 2007) about large deviations for the tail probabilities of generalized Jackson networks. Since the analysis is carried out on the doubly-infinite time axis ?, we can directly treat stationary situations.