## An M/M/1 Queue in a Semi-Markovian Environment

Authors
Philippe Nain <nain@sophia.inria.fr>
INRIA, Sophia Antipolis, France

Rudesindo Nunez-Queija <sindo@cwi.nl>
CWI, Amsterdam, The Netherlands

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 $\lambda_i$ and customers are served according to an exponential distribution with rate $\mu_i$. 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.

[Last updated Fri Mar 23 2001]

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