A functional central limit theorem for a Markov-modulated infinite-server queue

2013/09/16 by David F. Anderson, D. Anderson, J. Blom +11
Business, Management and Accounting · Mathematics · Physics and Astronomy · #60F05 #60F17 #60J60 #60K25 #60K37 #Advanced Queuing Theory Analysis #Complex Network Analysis Techniques #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60F05 #msc:60F17 #msc:60J60 #msc:60K25 #msc:60K37

paper · pdf · doi:10.48550/arxiv.1309.3962

4 figures

arxiv created 2013/09/16 · openalex publication_date 2013/09/16 · arxiv updated 2013/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

The production of molecules in a chemical reaction network is modelled as a Poisson process with a Markov-modulated arrival rate and an exponential decay rate. We analyze the distributional properties of M, the number of molecules, under specific time-scaling; the background process is sped up by N^α, the arrival rates are scaled by N, for N large. A functional central limit theorem is derived for M, which after centering and scaling, converges to an Ornstein-Uhlenbeck process. A dichotomy depending on α is observed. For α≤1 the parameters of the limiting process contain the deviation matrix associated with the background process.

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