Central Limit Theorems for Markov-modulated infinite-server queues

This paper 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 an independently evolving Markovian background process. Scaling the arrival rates $\lambda_i$ by a factor $N$ and the rates $q_{ij}$ of the background process by a factor $N^\alpha$, with $\alpha \in \mathbb R^+$, we establish a central limit theorem as $N$ tends to $\infty$. We find different scaling regimes, which depend on the specific value of $\alpha$. Remarkably, for $\alpha<1$, we find a central limit theorem in which the centered process has to be normalized by $N^{{1-}\alpha/2}$ rather than $\sqrt{N}$; in the expression for the variance deviation matrices appear.