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 $J(\cdot)$.

Additional Metadata
Keywords Ornstein-Uhlenbeck processes, Markov modulation, Central limit theorems, Martingale methods
MSC Queueing theory (msc 60K25), Processes in random environments (msc 60K37), Functional limit theorems; invariance principles (msc 60F17)
THEME Life Sciences (theme 5)
Persistent URL dx.doi.org/10.1007/s11009-014-9405-8
Journal Methodology and Computing in Applied Probability
Project Coarse grained stochastic methods for biochemical reactions
Citation
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