Operational analysis of stochastic closed queueing networks

Abstract

The purpose of this article is to provide alternative derivations and explanations of some classical results on stochastic closed queueing networks with a single class of customers, using operational analysis. The results we deal with are mainly the product-form solution, the Sevcik-Mitrani theorem, the aggregation property, and Marie's approximate method. We first extend operational analysis in order to analyze the asymptotic behaviour of stochastic closed queueing networks. We then show that any realization of a closed single-class BCMP network satisfies all operational assumptions with probability one. For queueing networks with exponential service times, we provide a simple proof of the Sevcik-Mitrani theorem. As opposed to the classical derivation of the aggregation property which is based on some algebraic manipulations, we provide a direct proof of this property by showing that the aggregate network satisfies the operational assumptions yielding the product-form solution. Finally, we provide an alternative explanation of Marie's approximate method for general closed queueing networks.