Brownian Models of Feedforward Queueing Networks: Quasireversibility and Product Form Solutions

Abstract

We consider a very general type of $d$-station open queueing network, with multiple customer classes and a more or less arbitrary service discipline at each station, but restricted by the requirement that customers always flow from lower numbered stations to higher numbered ones. To approximate teh behavior of such a queueing network under heavy traffic conditions, a corresponding \emph{Brownian network model} is proposed and it is shown that the approximating Brownian model reduced to a $d$-dimensional reflected Brownian motion $W$ whose state space is the nonnegative orthant. A necessary and sufficient condition for $W$ to have a product form stationary distribution (that is, a stationary distribuition with independent components) and a probabilistic interpretation for that condition are given. Our interpretation involves a notion of \emph{quasireversibility} analogous to that introduced by Kelly and elaborated by Walrand in their brilliant analysis of product form solutions for conventional queueing network models. Three illustrative queueing network models are discussed in detail and the analysis of these examples show how a Brownian network approximation may have a product form stationary distribution even when the original or exact model is intractable. Particularly intriguing in that regard are two examples involving non-Poisson inputs, deterministic routing, deterministic service times and processor-sharing service disciplines.