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


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.


Queues Infinite-server systems Markov modulation  Large deviations 

Mathematics Subject Classification




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.


  1. 1.
    Blom, J., Mandjes, M.: A large-deviations analysis of Markov-modulated inifinite-server queues. Oper. Res. Lett. 41, 220–225 (2013)CrossRefGoogle Scholar
  2. 2.
    Blom, J., de Turck, K., Mandjes, M.: Rare-event analysis of Markov-modulated infinite-server queues: a Poisson limit. Stoch. Models 29, 463–474 (2013)CrossRefGoogle Scholar
  3. 3.
    Blom, J., de Turck, K., Mandjes, M.: A central limit theorem for Markov-modulated infinite-server queues. In: Proceedings ASMTA 2013, Ghent, Belgium. Lecture Notes in Computer Science (LNCS) Series, vol. 7984, pp. 81–95 (2013)Google Scholar
  4. 4.
    Blom, J., Kella, O., Mandjes, M., Thorsdottir, H.: Markov-modulated infinite server queues with general service times. Queueing Syst. 76, 403–424 (2014)CrossRefGoogle Scholar
  5. 5.
    Blom, J., Mandjes, M., Thorsdottir, H.: Time-scaling limits for Markov-modulated infinite-server queues. Stoch. Models 29, 112–127 (2013)CrossRefGoogle Scholar
  6. 6.
    D’Auria, B.: M/M/\(\infty \) queues in semi-Markovian random environment. Queueing Syst. 58, 221–237 (2008)CrossRefGoogle Scholar
  7. 7.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1998)CrossRefGoogle Scholar
  8. 8.
    de Turck, K., Mandjes, M.: Large deviations of an infinite-server system with a linearly scaled background process. Perform. Eval. 75–76, 36–49 (2014)CrossRefGoogle Scholar
  9. 9.
    Fralix, B., Adan, I.: An infinite-server queue influenced by a semi-Markovian environment. Queueing Syst. 61, 65–84 (2009)CrossRefGoogle Scholar
  10. 10.
    Keilson, J., Servi, L.: The matrix M/M/\(\infty \) system: retrial models and Markov modulated sources. Adv. Appl. Probab. 25, 453–471 (1993)CrossRefGoogle Scholar
  11. 11.
    Norris, J.: Markov Chains. Cambridge University Press, Cambridge (1998)Google Scholar
  12. 12.
    O’Cinneide, C., Purdue, P.: The M/M/\(\infty \) queue in a random environment. J. Appl. Probab. 23, 175–184 (1986)CrossRefGoogle Scholar

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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