A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue |
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Authors: | D Anderson J Blom M Mandjes H Thorsdottir K de Turck |
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Institution: | 1.Department of Mathematics,University of Wisconsin – Madison,Madison,USA;2.Korteweg-de Vries Institute (KdVI) for Mathematics,University of Amsterdam,Amsterdam,the Netherlands;3.CWI,Amsterdam,the Netherlands;4.TELIN,Ghent University,Gent,Belgium |
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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(?). |
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