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Queueing Systems

, Volume 78, Issue 4, pp 337–357 | Cite as

Tail asymptotics of a Markov-modulated infinite-server queue

  • J. BlomEmail author
  • K. De Turck
  • O. Kella
  • M. Mandjes
Article

Abstract

This paper analyzes large deviation probabilities related to the number of customers in a Markov-modulated infinite-server queue, with state-dependent arrival and service rates. Two specific scalings are studied: in the first, just the arrival rates are linearly scaled by \(N\) (for large \(N\)), whereas in the second in addition the Markovian background process is sped up by a factor \(N^{1+\varepsilon }\), for some \(\varepsilon >0\). In both regimes (transient and stationary) tail probabilities decay essentially exponentially, where the associated decay rate corresponds to that of the probability that the sample mean of i.i.d. Poisson random variables attains an atypical value.

Keywords

Queues Infinite-server systems Markov modulation  Large deviations 

Mathematics Subject Classification

60K25 

Notes

Acknowledgments

We thank Rami Atar (Technion) for useful discussions related to the proof of Theorem 1. The work of O. Kella is partially supported by Israel Science Foundation Grant No. 1462/13 and The Vigevani Chair in Statistics.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • J. Blom
    • 1
    Email author
  • K. De Turck
    • 2
  • O. Kella
    • 3
  • M. Mandjes
    • 1
    • 4
    • 5
    • 6
  1. 1.CWIAmsterdamThe Netherlands
  2. 2.Telin, Ghent UniversityGentBelgium
  3. 3.Department of StatisticsThe Hebrew University of JerusalemJerusalemIsrael
  4. 4.Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  5. 5.Eurandom, Eindhoven University of TechnologyEindhovenThe Netherlands
  6. 6.Ibis, Faculty of Economics and BusinessUniversity of AmsterdamAmsterdamThe Netherlands

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