Rare event analysis of Markov-modulated infinite-server queues: A Poisson limit

This article studies an infinite-server queue in a Markov environment, that is, an infinite-server queue with arrival rates and service times depending on the state of a Markovian background process. Scaling the arrival rates $\lambda_i$ by a factor $N$ and the rates $\nu_{ij}$ of the background process by $N^{1+\vareps}$ (for some $\vareps > 0$), the focus is on the tail probabilities of the number of customers in the system, in the asymptotic regime that $N$ tends to $\infty$. In particular, it is shown that the logarithmic asymptotics correspond to those of a Poisson distribution with an appropriate mean