A M/M/ queue in a semi-Markovian environment

Abstract

We consider an M/M/1 queue in a semi-Markovian environment. The environment is modeled by a two-state semi-Markov process with arbitrary sojourn time distributions F 0 ( x ) and F 1 ( x ). When in state i = 0, 1, customers are generated according to a Poisson process with intensity λ i and customers are served according to an exponential distribution with rate μ i . Using the theory of Riemann-Hilbert boundary value problems we compute the z -transform of the queue-length distribution when either F 0 ( x ) or F 1 ( x ) has a rational Laplace-Stieltjes transform and the other may be a general --- possibly heavy-tailed --- distribution. The arrival process can be used to model bursty traffic and/or traffic exhibiting long-range dependence, a situation which is commonly encountered in networking. The closed-form results lend themselves for numerical evaluation of performance measures, in particular the mean queue-length.