Perturbation analysis of the m/m/1 queue in a markovian environment via the matrix-geometric method

Abstract

In this paper, we consider a family of M(t)/M/1 queues in which customers arrive according to nonhomogenous Poisson processes with intensity λt(ε) = λet, 0 < ε < ∞. We assume that λt(ε) is an irreducible finite-state Markov process. Based on the matrix-geometric method, we use perturbation analysis to obtain the second order approximations for the expected queue length for two cases where ε is small and where ε is large. Using these approximations, we show that the expected waiting times are strictly decreasing in ε when ε is small. In the case where ε is large, we show that the expected waiting times are strictly decreasing in ε if the intensity process is dynamically reversible. These results partially answer a question posed by Rolski. © 1993, Taylor & Francis Group, LLC. All rights reserved.