Skip to main content
SpringerLink
Log in

Asymptotic Behavior of Extreme Values of Queue Length in M / M / m Systems

  • Published:
Cybernetics and Systems Analysis Aims and scope

Abstract

The paper investigates the asymptotic behavior of almost surely maximum length in queueing systems. For a system M / M / m, 1≤ m< ∞, a statement of the type of the law of iterated logarithm is established. We also consider the case m = ∞ for which the asymptotic behavior is much different.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. V. Gnedenko and I. N. Kovalenko, Introduction to Queuing Theory [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  2. S. Karlin, A First Course in Stochastic Processes [Russian translation], Mir, Moscow (1971).

    Google Scholar 

  3. V. V. Anisimov, V. K. Zakusilo, and V. S. Donchenko, Elements of the Queuing Theory and Asymptotic Analysis of Systems [in Russian], Vysshaya Shkola, Kyiv (1987).

  4. J. W. Cohen, “Extreme values distribution for the M / G /1 and G / I / M / 1 queueing systems,” Ann. Inst. H. Poincare., Sect. B, Vol. 4, 83–98 (1968).

    MathSciNet  MATH  Google Scholar 

  5. C. W. Anderson, “Extreme value theory for a class of discrete distribution with application to some stochastic processes,” J. Appl. Prob., Vol. 7, 99–113 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  6. D. L. Iglehart, “Extreme values in the GI / G /1 queue,” Ann. Math. Statist., Vol. 43, 627–635 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  7. R. F. Serfozo, “Extreme values of birth and death processes and queues,” Stochastic Processes and Their Applications, Vol. 27, 291–306 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Asmussen, “Extreme values theory for queues via cycle maxima,” Extremes, Vol. 1, 137–168 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  9. O. K. Zakusylo and I. K. Matsak, “On extreme values of some regenerative processes,” Teoriya Jmovirn. ta Mat. Statystyka, No. 97, 58–71 (2017).

  10. I. K. Matsak, “Asymptotic behavior of extreme values of random variables. Discrete case,” Ukr. Math. J., Vol. 68, No. 7, 1077–1090 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Galambos, The Asymptotic Theory of Extreme Order Statistics, Wiley (1978).

  12. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, Wiley (1968).

Download references

Author information

Authors and Affiliations

  1. Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    B. V. Dovhai & I. K. Matsak

Authors
  1. B. V. Dovhai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. K. Matsak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. V. Dovhai.

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 2, March–April, 2019, pp. 171–179.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dovhai, B.V., Matsak, I.K. Asymptotic Behavior of Extreme Values of Queue Length in M / M / m Systems. Cybern Syst Anal 55, 321–328 (2019). https://doi.org/10.1007/s10559-019-00137-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10559-019-00137-4

Keywords

Search

Navigation