The distribution of network states during residence times in product form queueing networks

Abstract

Product form queueing networks are so named because their equilibrium state probabilities have a simple product form. This simple form has lead to computationally efficient techniques for obtaining solutions of these networks. Recent work on the probability of network states at those instants when a customer completes service at a service center (departure instants) or arrives at a service center (arrival instants) has revealed a similar product form. The arrival instant result provides a simple, intuitive explanation of the Mean Value Analysis solution technique for product form networks with First-Come-First-Served service centers. In this paper we derive the distribution of network states seen by a particular customer while resident at a particular service center. This distribution too has a relatively simple product form. We use this information to explain in an intuitive way the MVA solution technique for a more general class of network, those containing load independent Processor Sharing and last-Come-First-Served-Preemptive-Resume service centers. It is hoped that, just as the intuitive explanation of the response time formula for FCFS centers has led to approximation techniques for non-separable FCFS centers, this new information may provide approximation techniques for non-separable centers with other scheduling disciplines such as preemptive and non-preemptive priority.