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 F0(x) and F1(x). When in state i = 0, 1, customers are generated according to a Poisson process with intensity λi and customers are served according to an exponential distribution with rate μi. Using the theory of Riemann-Hilbert boundary value problems we compute the z-transform of the queue-length distribution when either F0(x) or F1(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.

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