Decomposition and Customer Streams of Feedback Networks of Queues in Equilibrium

Abstract

Burke and Reich independently showed that the output of an M/M/1 queue in equilibrium is a Poisson process. Consequently, analysis of series (tandem) exponential servers with a Poisson input stream can be reduced to consideration of a series of M/M/1 queues. This work generalizes the above results to so-called Jackson networks, consisting of exponential servers with mutually independent Poisson exogenous inputs and random customer routings permitting customer feedback. We prove that traffic on all exit arcs of a network in equilibrium is Poisson; moreover, the customer streams leaving any exit set are mutually independent. Here an exit arc is a path from server node i such that a customer moving along the arc cannot return to i; an exit set V is a set of server nodes such that customers departing V can never re-enter V. As a special case, the traffic streams leaving a Jackson network in equilibrium are mutually independent Poisson processes. In contrast, traffic on non-exit arcs is non-Poisson, and indeed non-renewal. A canonical decomposition of the server nodes is defined as a partition of the server node set into components C*q, each consisting of communicating servers. We show that the set of all exit arcs coincides with the arcs emanating from the C*q. Finally, for a Jackson network in equilibrium, each C*q is itself a Jackson network in equilibrium.