 Author
 Year
 2016
 Title
 A Functional Central Limit Theorem for a MarkovModulated InfiniteServer Queue
 Journal
 Methodology and Computing in Applied Probability
 Volume  Issue number
 18  1
 Pages (fromto)
 153168
 Document type
 Article
 Faculty
 Faculty of Science (FNWI)
 Institute
 Kortewegde Vries Institute for Mathematics (KdVI)
 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 timescaling. 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 (FCLT) for the number of molecules, in that, after centering and scaling, it converges to an OrnsteinUhlenbeck 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 FCLT is the usual N√,
whereas (ii) for α≤1 the background process is relatively slow, and the scaling in the FCLT is N 1−α/2. In the latter regime, the parameters of the limiting OrnsteinUhlenbeck process contain the deviation matrix associated with the background process J(⋅).
 URL
 go to publisher's site
 Language
 English
 Permalink
 http://hdl.handle.net/11245/1.536501
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