

An M/M/1 Queue in a SemiMarkovian Environment

Authors

Philippe Nain <nain@sophia.inria.fr>
INRIA, Sophia Antipolis, France
Rudesindo NunezQueija <sindo@cwi.nl>
CWI, Amsterdam, The Netherlands

Abstract

We consider an M/M/1 queue in a semiMarkovian environment. The
environment is modeled by a twostate semiMarkov 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 RiemannHilbert boundary value problems we compute
the $z$transform of the queuelength distribution when either
$F_0(x)$ or $F_1(x)$ has a rational LaplaceStieltjes transform
and the other may be a general  possibly heavytailed 
distribution. The arrival process can be used to model bursty
traffic and/or traffic exhibiting longrange dependence, a
situation which is commonly encountered in networking. The
closedform results lend themselves for numerical evaluation of
performance measures, in particular the mean queuelength.

