A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue

Abstract

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 λ _i when an external Markov process J(⋅) is in state i. It is assumed that molecules decay after an exponential time with mean μ ⁻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 ^α, for some α>0, whereas the arrival rates become N λ _i , for N large. The main result of this paper is a functional central limit theorem (F-CLT) for the number of molecules, in that, after centering and scaling, it converges to an Ornstein-Uhlenbeck process. An interesting dichotomy is observed: (i) if α > 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 F-CLT is the usual $\sqrt {N}$, whereas (ii) for α≤1 the background process is relatively slow, and the scaling in the F-CLT is N ^1−α/2. In the latter regime, the parameters of the limiting Ornstein-Uhlenbeck process contain the deviation matrix associated with the background process J(⋅).