Perturbation analysis of the M/M/1 queue in a Markovian environment via the matrix-geometric method

In this paper, the authors consider a family of queues in which customers according to nonhomogeneous Poisson processes with intensity , . They assume that is an irreducible finite-state Markov process. Based on matrix-geometric method, the authors use perturbation analysis to obtain the second order approximations for the expected queue length for two cases where is small and where is large. Using these approximations, they show that the expected waiting times are strictly decreasing in when is small. In the case where is large, the authors show that the expected waiting times are strictly decreasing in if the intensity process is dynamically reversible. These results partially answer a question posed by Rolski.