On approximating the conditional wait process of general multiserver queueing systems

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Abstract

In view of the extremely difficult task of analyzing G/G/K queueing systems, relatively few general results have been established. Most of the literature dealing with the G/G/K system has been concerned with the heavy traffic situation. Of special note are the bounds given by Kingman[7] on the wait process, the weak convergence theorems established by Iglehart and Whitt[4] and Loulou[8], and the many server approximations due to Newell[9].

In this paper a presentation will be made of new formulas constructed to approximate the mean and variance of the conditional wait process. Two key concepts play major roles behind the formulas to be presented. They are: (a) the use of a continuous (diffusion-type) process to approximate the steady-state probabilities of the number in the queue, and (b) the substitution of the original process of interdeparture times by a process of independent variables.

In order to assess the quality of the suggested approximation for the mean and variance of the conditional wait process, the approximation will be numerically compared with the results obtained from Kingman's[7] and Kendall's[5] exact formula for the special case of the G/M/K system, and will be tested against point estimate simulations.

References (11)

  • D.R. Cox et al.

    The Theory of Stochastic Processes

  • B. Halachmi et al.
  • B. Halachmi

    Diffusion approximations to the multiserver queueing systems

  • D. Iglehart et al.

    Multiple channel queues in heavy traffic—I & II

    Adv. appl. Probl.

    (1950–1977)
    D. Iglehart et al.

    Multiple channel queues in heavy traffic—I & II

    Adv. appl. Probl.

    (1955–1969)
  • D.G. Kendall

    Stochastic Processes Occuring in the Theory of Queues and Their Analysis by the Method of Imbedded Markov Chains

    Am. math. Stat.

    (1953)
There are more references available in the full text version of this article.

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Baruch Halachmi gained his B.A. in Statistics & Economics from The Hebrew University of Jerusalem, his M.A. in Operations Research & Statistics from Tel Aviv University, and Ph.D. in Computer Science from the University of Minnesota. Currently an Assistant Professor of Computer Science at the University of Kansas. Areas of interest include: Simulations and Modeling, Computer Performance Evaluation, Optimization and Queueing Theory.

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