A Central Limit Theorem 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 a Markovian background process. Scaling the arrival rates $\lambda_i$ by a factor $N$ and the rate $q_{ij}$ 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, based on the value of $\alpha$. Remarkably, for $\alpha < 1$, we find a central limit theorem with a non-square-root scaling but rather with $N^{\alpha/2}$; in the expression for the variance deviation matrices appear