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 \(\lambda _i\) 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.