A unified view of product-form approximation techniques for general closed queueing networks

Abstract

Two major approximate techniques have been proposed for the analysis of general closed queueing networks, namely the aggregation method and Marie's method. The idea of the aggregation technique is to replace a subsystem (a subnetwork) by a flow equivalent single-server with load-dependent service rates. The parameters of the equivalent server are obtained by analyzing the subsystem in isolation as a closed system with different populations. The idea of Marie's method is also to replace a subsystem by an equivalent exponential service station with load-dependent service rates. However, in this case, the parameters of the equivalent server are obtained by analyzing the subsystem in isolation under a load-dependent Poisson arrival process. Moreover, in Marie's case, the procedure is iterative. In this paper we provide a general and unified view of these two methods. The contributions of this paper are the following. We first show that their common principle is to partition the network into a set of subsystems and then to define an equivalent product-form network. To each subsystem is associated a load-dependent exponential station in the equivalent network. We define a set of rules in order to partition any general closed network with various features such as general service time distributions, pupulation constraints, finite buffers, state-dependent routing. We then show that the aggregation method and Marie's method are two ways of obtaining the parameters of the equivalent network associated with a given partition. Finally, we provide a discussion pertaining to the comparison of the two methods with respect to their accuracy and computational complexity.