Skip to main content
SpringerLink
Log in

Some light-traffic and heavy-traffic results for the GI/G/n/0 queue using the GM Heuristic

  • Published:
Cybernetics and Systems Analysis Aims and scope

Abstract

In this paper we carry out both light-traffic and heavy-traffic analyses for the calculation of steady-state loss probabilities in the general multi-server queuing loss system, the GI/G/n/0 queue. The analysis makes use of a heuristic approach called the GM Heuristic, for which a detailed analysis in normal traffic has previously been published. Sufficient conditions are given for the GM Heuristic to be asymptotically exact in light traffic. The heuristic is also shown to be asymptotically exact in heavy-traffic when the number of servers n tends to infinity. These results are illustrated numerically using two-phase Coxian distributions for both the inter-arrival time and service time.

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. J. B. Atkinson, “Two new heuristics for the GI/G/n/0 queueing loss system with examples based on the two-phase Coxian distribution,” J. Oper. Res. Soc., 60, 818–830 (2009).

    Article  MATH  Google Scholar 

  2. D. R. Cox, “A use of complex probabilities in the theory of stochastic processes,” Proc. Cambridge Philosophical Society, 51, 313–319 (1955).

    Article  MATH  Google Scholar 

  3. T. Kimura, “Approximations for multi-server queues: System interpolations,” Queueing Syst., 17, 347–382 (1994).

    Article  MATH  Google Scholar 

  4. I. N. Kovalenko and J. B. Atkinson, “Conditions for the light-traffic insensitivity to loss probability in a GI/G/m/0 queuing system,” Cybern. Syst. Analysis, 38, No. 6, 846–854 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  5. I. N. Kovalenko, J. B. Atkinson, and K. V. Mykhalevych, “Three cases of light-traffic insensitivity of the loss probability in a GI/G/m/0 loss system to the shape of the service time distribution,” Queueing Syst., 45, 245–271 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  6. L. Takács, Introduction to the Theory of Queues, Chapter 4, Oxford University Press, New York (1962).

    MATH  Google Scholar 

  7. H. C. Tijms, A first Course in Stochastic Modelling, John Wiley & Sons, Chichester (2003).

    Book  Google Scholar 

  8. W. Whitt, “An interpolation approximation for the mean workload in a GI/G/1 queue,” Oper. Research, 37, 936–952 (1989).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. STORM Research Centre, London Metropolitan University, 166-200 Holloway Road, London, N 78 DB, UK

    J. B. Atkinson

  2. V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    I. N. Kovalenko

Authors
  1. J. B. Atkinson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. N. Kovalenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. B. Atkinson.

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 92–100, May–June 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Atkinson, J.B., Kovalenko, I.N. Some light-traffic and heavy-traffic results for the GI/G/n/0 queue using the GM Heuristic . Cybern Syst Anal 46, 426–435 (2010). https://doi.org/10.1007/s10559-010-9217-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10559-010-9217-y

Keywords

Search

Navigation