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

D. Anderson (), J. Blom (), M. Mandjes (), H. Thorsdottir () and K. Turck ()
Additional contact information
D. Anderson: University of Wisconsin – Madison
J. Blom: CWI
M. Mandjes: University of Amsterdam
H. Thorsdottir: University of Amsterdam
K. Turck: Ghent University

Methodology and Computing in Applied Probability, 2016, vol. 18, issue 1, 153-168

Abstract: 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 N $\sqrt {N}$ , 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(⋅).

Keywords: Ornstein-Uhlenbeck processes; Markov modulation; Central limit theorems; Martingale methods; Primary 60K25; 60K37; 60F17; Secondary 60G15 60G99 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (13)

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DOI: 10.1007/s11009-014-9405-8

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