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.

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A. Dudin , K.E.E.S deTurck
Analytical & Stochastic Modelling Techniques & Applications
Evolutionary Intelligence

Blom, J., deTurck, K. E. E. S., & Mandjes, M. (2013). A Central Limit Theorem for Markov-Modulated Infinite-Server Queues . In A. Dudin & K. E. E. S. deTurck (Eds.), . doi:10.1007/978-3-642-39408-9_7