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 $F_i (·)$ 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 $N^{1+\vareps}$ (for some $\vareps > 0$). 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.

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Queueing Systems
Evolutionary Intelligence

Blom, J., Kella, O., Mandjes, M., & Thorsdottir, H. (2014). Markov-modulated infinite-server queues with general service times
. Queueing Systems, 76(4), 403–424. doi:10.1007/s11134-013-9368-4