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 λ i by a factor N and the rate q ij of the background process by a factor N α , with α ∈ ℝ + , we establish a central limit theorem as N tends to ∞. We find different scaling regimes, based on the value of α. Remarkably, for α < 1, we find a central limit theorem with a non-square-root scaling but rather with N α/2; in the expression for the variance deviation matrices appear.