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Overflow probabilities for Markov-modulated infinite-server queues: a large-deviations approach

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Abstract
In this paper we consider an infinite-server queue in a random environment. The distinguishing feature of the model is the presence of two irreducible Markov chains: one Markov chain modulates the arrival rates, while the other modulates the service times. We are interested in the probability that the number of jobs in the system becomes unusually large, i.e. we are interested in overflow. Because arrival rates and service times are stochastically varying over time, the number of jobs in the system has a Poisson distribution with random parameters rather than a 'classical' Poisson distribution. In this case we cannot use a CLT-type result to analyze the system. However, basic large-deviations techniques provide an alternative approach. Scaling the arrival rates linearly, we prove a large-deviations principle by conditioning on paths of the background processes that are very likely to lead to overflow. We show that, conditional on these paths, we are in a situation in which Cramér's Theorem may be applied. This gives us the rate function for the number of jobs in the system and we use it to describe overflow probabilities. A nice observation is that we do not need to know the transition probabilities of the Markov chains to say something about overflow probabilities.
Keywords
Markov modulation, overflow, infinite-server systems, queues, large deviations

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MLA
Jansen, Hermanus Marinus, Koen De Turck, Michel Mandjes, et al. “Overflow Probabilities for Markov-modulated Infinite-server Queues: a Large-deviations Approach.” Booklet of Abstracts of the First European Conference on Queueing Theory. Ed. Herwig Bruneel et al. 2014. 35–35. Print.
APA
Jansen, H. M., De Turck, K., Mandjes, M., & Wittevrongel, S. (2014). Overflow probabilities for Markov-modulated infinite-server queues: a large-deviations approach. In H. Bruneel, O. Boxma, J. Walraevens, & S. Wittevrongel (Eds.), Booklet of Abstracts of the First European Conference on Queueing Theory (pp. 35–35). Presented at the First European Conference on Queueing Theory (ECQT 2014).
Chicago author-date
Jansen, Hermanus Marinus, Koen De Turck, Michel Mandjes, and Sabine Wittevrongel. 2014. “Overflow Probabilities for Markov-modulated Infinite-server Queues: a Large-deviations Approach.” In Booklet of Abstracts of the First European Conference on Queueing Theory, ed. Herwig Bruneel, Onno Boxma, Joris Walraevens, and Sabine Wittevrongel, 35–35.
Chicago author-date (all authors)
Jansen, Hermanus Marinus, Koen De Turck, Michel Mandjes, and Sabine Wittevrongel. 2014. “Overflow Probabilities for Markov-modulated Infinite-server Queues: a Large-deviations Approach.” In Booklet of Abstracts of the First European Conference on Queueing Theory, ed. Herwig Bruneel, Onno Boxma, Joris Walraevens, and Sabine Wittevrongel, 35–35.
Vancouver
1.
Jansen HM, De Turck K, Mandjes M, Wittevrongel S. Overflow probabilities for Markov-modulated infinite-server queues: a large-deviations approach. In: Bruneel H, Boxma O, Walraevens J, Wittevrongel S, editors. Booklet of Abstracts of the First European Conference on Queueing Theory. 2014. p. 35–35.
IEEE
[1]
H. M. Jansen, K. De Turck, M. Mandjes, and S. Wittevrongel, “Overflow probabilities for Markov-modulated infinite-server queues: a large-deviations approach,” in Booklet of Abstracts of the First European Conference on Queueing Theory, Ghent, Belgium, 2014, pp. 35–35.
@inproceedings{6847836,
  abstract     = {In this paper we consider an infinite-server queue in a random environment. The distinguishing feature of the model is the presence of two irreducible Markov chains: one Markov chain modulates the arrival rates, while the other modulates the service times. We are interested in the probability that the number of jobs in the system becomes unusually large, i.e. we are interested in overflow.
Because arrival rates and service times are stochastically varying over time, the number of jobs in the system has a Poisson distribution with random parameters rather than a 'classical' Poisson distribution. In this case we cannot use a CLT-type result to analyze the system. However, basic large-deviations techniques provide an alternative approach. Scaling the arrival rates linearly, we prove a large-deviations principle by conditioning on paths of the background processes that are very likely to lead to overflow. We show that, conditional on these paths, we are in a situation in which Cramér's Theorem may be applied. This gives us the rate function for the number of jobs in the system and we use it to describe overflow probabilities. A nice observation is that we do not need to know the transition probabilities of the Markov chains to say something about overflow probabilities.},
  author       = {Jansen, Hermanus Marinus and De Turck, Koen and Mandjes, Michel and Wittevrongel, Sabine},
  booktitle    = {Booklet of Abstracts of the First European Conference on Queueing Theory},
  editor       = {Bruneel, Herwig and Boxma, Onno and Walraevens, Joris and Wittevrongel, Sabine},
  isbn         = {9789461972095},
  keywords     = {Markov modulation,overflow,infinite-server systems,queues,large deviations},
  language     = {eng},
  location     = {Ghent, Belgium},
  pages        = {35--35},
  title        = {Overflow probabilities for Markov-modulated infinite-server queues: a large-deviations approach},
  year         = {2014},
}