A functional central limit theorem for a Markov-modulated infinite-server queue

Abstract

The production of molecules in a chemical reaction network is modelled as a Poisson process with a Markov-modulated arrival rate and an exponential decay rate. We analyze the distributional properties of $M$, the number of molecules, under specific time-scaling; the background process is sped up by $N^{\hat{I}}\pm$, the arrival rates are scaled by $N$, for $N$ large. A functional central limit theorem is derived for $M$, which after centering and scaling, converges to an Ornstein-Uhlenbeck process. A dichotomy depending on $\hat{I}\pm$ is observed. For $\hat{I}\pm \le 1$ the parameters of the limiting process contain the deviation matrix associated with the background process.

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