Markov-modulated infinite-server queues with general service times

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 $$\lambda _i$$λi when an external Markov process (“background process”) is in state $$i$$i, (ii) service times are drawn from a distribution with distribution function $$F_i(\cdot )$$Fi(·) when the state of the background process (as seen at arrival) is $$i$$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 $$t\ge 0$$t≥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 $$N$$N, and the transition times by a factor $$N^{1+\varepsilon }$$N1+ε (for some $$\varepsilon >0$$ε>0). Under this scaling it turns out that the number of customers at time $$t\ge 0$$t≥0 obeys a central limit theorem; the convergence of the finite-dimensional distributions is proven.