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

We consider a model in which the production of new molecules in a chemical reaction network occurs in a seemingly stochastic fashion, and can be modeled as a Poisson process with a varying arrival rate: the rate is $\lambda_i$ when an external Markov process $J(\cdot)$ is in state $i$. It is assumed that molecules decay after an exponential time with mean $\mu^{-1}$. The goal of this work is to analyze the distributional properties of the number of molecules in the system, under a specific time-scaling. In this scaling, the background process is sped up by a factor $N^{\alpha}$, for some $\alpha>0$, whereas the arrival rates become $N\lambda_i$, for $N$ large. The main result of this paper is a functional central limit theorem ({\sc f-clt}) for the number of molecules, in that the number of molecules, after centering and scaling, converges to an Ornstein-Uhlenbeck process. An interesting dichotomy is observed: (i)~if $\alpha>1$ the background process jumps faster than the arrival process, and consequently the arrival process behaves essentially as a (homogeneous) Poisson process, so that the scaling in the {\sc f-clt} is the usual $\sqrt{N}$, whereas (ii)~for $\alpha\leq1$ the background process is relatively slow, and the scaling in the {\sc f-clt} is $N^{1-\alpha/2}.$ In the latter regime, the parameters of the limiting Ornstein-Uhlenbeck process contain the deviation matrix associated with the background process $J(\cdot)$.