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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Work partially done while D. Anderson was visiting CWI and KdVI, and K. de Turck was visiting KdVI; financial support from NWO cluster STAR (Anderson) and Fonds Wetenschappelijk Onderzoek / Research Foundation – Flanders (De Turck) is greatly appreciated. Anderson was also supported under NSF grant DMS-100975 and DMS-1318832.
M. Mandjes is also with EURANDOM, Eindhoven University of Technology, Eindhoven, the Netherlands, and IBIS, Faculty of Economics and Business, University of Amsterdam, Amsterdam, the Netherlands.
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Anderson, D., Blom, J., Mandjes, M. et al. A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue. Methodol Comput Appl Probab 18, 153–168 (2016). https://doi.org/10.1007/s11009-014-9405-8
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DOI: https://doi.org/10.1007/s11009-014-9405-8