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 λi when an external Markov process ("background process") is in state i , (ii) service times are drawn from a distribution with distribution function Fi(⋅) 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≥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 N1+ε (for some ε>0 ). Under this scaling it turns out that the number of customers at time t≥0 obeys a central limit theorem; the convergence of the finite-dimensional distributions is proven.