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An Analytic Approach to a General Class of G/G/s Queueing Systems

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  • Dimitris Bertsimas

    (Massachusetts Institute of Technology, Cambridge, Massachusetts)

Abstract

We solve the queueing system C k /C m /s, where C k is the class of Coxian probability density functions (pdfs) of order k , which is a subset of the pdfs that have a rational Laplace transform. We formulate the model as a continuous-time, infinite-space Markov chain by generalizing the method of stages. By using a generating function technique, we solve an infinite system of partial difference equations and find closed-form expressions for the system-size, general-time, prearrival, post-departure probability distributions and the usual performance measures. In particular, we prove that the probability of n customers being in the system, when it is saturated is a linear combination of geometric terms. The closed-form expressions involve a solution of a system of nonlinear equations that involves only the Laplace transforms of the interarrival and service time distributions. We conjecture that this result holds for a more general model. Following these theoretical results we propose an exact algorithm for finding the system-size distribution and the system's performance measures. We examine special cases and apply this method for numerically solving the C 2 /C 2 /s and E k /C 2 /s queueing systems.

Suggested Citation

  • Dimitris Bertsimas, 1990. "An Analytic Approach to a General Class of G/G/s Queueing Systems," Operations Research, INFORMS, vol. 38(1), pages 139-155, February.
    • Handle: RePEc:inm:oropre:v:38:y:1990:i:1:p:139-155
    DOI: 10.1287/opre.38.1.139
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    Cited by:

    1. McGrory, C.A. & Pettitt, A.N. & Faddy, M.J., 2009. "A fully Bayesian approach to inference for Coxian phase-type distributions with covariate dependent mean," Computational Statistics & Data Analysis, Elsevier, vol. 53(12), pages 4311-4321, October.
    2. Drekic, Steve & Woolford, Douglas G., 2005. "A preemptive priority queue with balking," European Journal of Operational Research, Elsevier, vol. 164(2), pages 387-401, July.
    3. Winfried K. Grassmann, 2003. "The Use of Eigenvalues for Finding Equilibrium Probabilities of Certain Markovian Two-Dimensional Queueing Problems," INFORMS Journal on Computing, INFORMS, vol. 15(4), pages 412-421, November.
    4. Chaithanya Bandi & Dimitris Bertsimas & Nataly Youssef, 2015. "Robust Queueing Theory," Operations Research, INFORMS, vol. 63(3), pages 676-700, June.
    5. Adan, Ivo & de Kok, Ton & Resing, Jacques, 1999. "A multi-server queueing model with locking," European Journal of Operational Research, Elsevier, vol. 116(2), pages 249-258, July.

    More about this item

    Keywords

    queues: multichannel; Markovian queues;

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