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 $J(\cdot)$.

Ornstein-Uhlenbeck processes, Markov modulation, Central limit theorems, Martingale methods
Queueing theory (msc 60K25), Processes in random environments (msc 60K37), Functional limit theorems; invariance principles (msc 60F17)
Life Sciences (theme 5)
Methodology and Computing in Applied Probability
Coarse grained stochastic methods for biochemical reactions
Life Sciences and Health

Anderson, D.F, Blom, J.G, Mandjes, M.R.H, Thorsdottir, H, & deTurck, K.E.E.S. (2016). A functional central limit theorem for a Markov-modulated infinite-server queue. Methodology and Computing in Applied Probability, 18(1), 153–168. doi:10.1007/s11009-014-9405-8