Decomposition and Aggregation in Queueing Networks

Abstract

This chapter considers the decomposition and aggregation of multiclass queueing networks with state-dependent routing. Combining state-dependent generalisations of quasi-reversibility and biased local balance, sufficient conditions are obtained under which the stationary distribution of the network is of product-form. This product-form factorises into one part that describes the nodes of the network in isolation, and one part that describes the routing and the global network state. It is shown that a decomposition holds for general nodes if the input-output behaviour of these nodes is suitably compensated by the state-dependent routing function. When only a subset of the nodes is of interest, it is shown that the other nodes may be aggregated into nodes that only capture their global behaviour. The results both unify and extend existing classes of product-form networks, as is illustrated by several cases and an example of an assembly network. IntroductionIn the analysis of queueing networks, two at first sight different techniques have been used to derive product form results: quasi-reversibility and local balance. Quasi-reversibility is a property of the nodes of the network, roughly stating that they should preserve input and output flows when they are considered in isolation and fed by a Poisson process. If such nodes are coupled into a network by Markov

Norton’s theorem and insensitivity

“…These examples illustrate the relation between Norton's theorem and insensitivity for open networks comprised of quasireversible stations. Norton's theorem may be extended to open and closed networks with more general detailed and global states [4]. Insensitivity for symmetric queues may be extended to insensitivity of the sojourn time distribution at a quasireversible sub-network, except for its mean, see [6] for an overview of such results.…”

A - and -invariant characterization of product form and decomposition in stochastic Petri nets

“…[30][31][32][33]). Recently, in a setting of general stochastic processes, these results have been unified and extended in [34,35].…”