Markov-modulated infinite-server queues with general service times

Abstract

This paper analyzes several aspects of the Markov-modulated infinite-server queue. In the system considered (i) particles arrive according to a Poisson process with rate $\lambda _i$ when an external Markov process (“background process”) is in state $i$, (ii) service times are drawn from a distribution with distribution function $F_i(\cdot )$ when the state of the background process (as seen at arrival) is $i$, (iii) there are infinitely many servers. We start by setting up explicit formulas for the mean and variance of the number of particles in the system at time $t\ge 0$, given the system started empty. The special case of exponential service times is studied in detail, resulting in a recursive scheme to compute the moments of the number of particles at an exponentially distributed time, as well as their steady-state counterparts. Then we consider an asymptotic regime in which the arrival rates are sped up by a factor $N$, and the transition times by a factor $N^{1+\varepsilon }$ (for some $\varepsilon >0$). Under this scaling it turns out that the number of customers at time $t\ge 0$ obeys a central limit theorem; the convergence of the finite-dimensional distributions is proven.