A functional central limit theorem for a Markov-modulated infinite-server queue
- Author
- David Anderson, Joke Blom, Michel Mandjes, Halldora Thorsdottir and Koen De Turck (UGent)
- Organization
- 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 lambda (i) when an external Markov process J(a <...) is in state i. It is assumed that molecules decay after an exponential time with mean mu (-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 (alpha) , for some alpha > 0, whereas the arrival rates become N lambda (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 alpha > 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 , whereas (ii) for alpha a parts per thousand currency sign1 the background process is relatively slow, and the scaling in the F-CLT is N (1-alpha/2). In the latter regime, the parameters of the limiting Ornstein-Uhlenbeck process contain the deviation matrix associated with the background process J(.).
- Keywords
- REACTION NETWORKS, RANDOM ENVIRONMENT, MATRIX, MODELS, Ornstein-Uhlenbeck processes, Markov modulation, Central limit theorems, Martingale methods
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU-5651799
- MLA
- Anderson, David, et al. “A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue.” METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY, vol. 18, no. 1, 2016, pp. 153–68, doi:10.1007/s11009-014-9405-8.
- APA
- Anderson, D., Blom, J., Mandjes, M., Thorsdottir, H., & De Turck, K. (2016). A functional central limit theorem for a Markov-modulated infinite-server queue. METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY, 18(1), 153–168. https://doi.org/10.1007/s11009-014-9405-8
- Chicago author-date
- Anderson, David, Joke Blom, Michel Mandjes, Halldora Thorsdottir, and Koen De Turck. 2016. “A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue.” METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY 18 (1): 153–68. https://doi.org/10.1007/s11009-014-9405-8.
- Chicago author-date (all authors)
- Anderson, David, Joke Blom, Michel Mandjes, Halldora Thorsdottir, and Koen De Turck. 2016. “A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue.” METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY 18 (1): 153–168. doi:10.1007/s11009-014-9405-8.
- Vancouver
- 1.Anderson D, Blom J, Mandjes M, Thorsdottir H, De Turck K. A functional central limit theorem for a Markov-modulated infinite-server queue. METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY. 2016;18(1):153–68.
- IEEE
- [1]D. Anderson, J. Blom, M. Mandjes, H. Thorsdottir, and K. De Turck, “A functional central limit theorem for a Markov-modulated infinite-server queue,” METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY, vol. 18, no. 1, pp. 153–168, 2016.
@article{5651799,
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 lambda (i) when an external Markov process J(a <...) is in state i. It is assumed that molecules decay after an exponential time with mean mu (-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 (alpha) , for some alpha > 0, whereas the arrival rates become N lambda (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 alpha > 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 , whereas (ii) for alpha a parts per thousand currency sign1 the background process is relatively slow, and the scaling in the F-CLT is N (1-alpha/2). In the latter regime, the parameters of the limiting Ornstein-Uhlenbeck process contain the deviation matrix associated with the background process J(.).}},
author = {{Anderson, David and Blom, Joke and Mandjes, Michel and Thorsdottir, Halldora and De Turck, Koen}},
issn = {{1387-5841}},
journal = {{METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY}},
keywords = {{REACTION NETWORKS,RANDOM ENVIRONMENT,MATRIX,MODELS,Ornstein-Uhlenbeck processes,Markov modulation,Central limit theorems,Martingale methods}},
language = {{eng}},
number = {{1}},
pages = {{153--168}},
title = {{A functional central limit theorem for a Markov-modulated infinite-server queue}},
url = {{http://doi.org/10.1007/s11009-014-9405-8}},
volume = {{18}},
year = {{2016}},
}
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