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The unobserved waiting customer approximation

Author

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  • Kevin Granville

    (University of Waterloo
    University of Western Ontario)

  • Steve Drekic

    (University of Waterloo)

Abstract

We introduce a new approximation procedure to improve the accuracy of matrix analytic methods when using truncated queueing models to analyse infinite buffer systems. This is accomplished through emulating the presence of unobserved waiting customers beyond the finite buffer that are able to immediately enter the system following an observed customer’s departure. We show that this procedure results in exact steady-state probabilities at queue lengths below the buffer for truncated versions of the classic M/M/1, $$M/M/1+M$$ M / M / 1 + M , $$M/M/\infty $$ M / M / ∞ , and M/PH/1 queues. We also present two variants of the basic procedure for use within a $$M/PH/1+M$$ M / P H / 1 + M queue and a N-queue polling system with exhaustive service, phase-type service and switch-in times, and exponential impatience timers. The accuracy of these two variants in the context of the polling model are compared through several numerical examples.

Suggested Citation

  • Kevin Granville & Steve Drekic, 2021. "The unobserved waiting customer approximation," Queueing Systems: Theory and Applications, Springer, vol. 99(3), pages 345-396, December.
    • Handle: RePEc:spr:queues:v:99:y:2021:i:3:d:10.1007_s11134-021-09706-x
    DOI: 10.1007/s11134-021-09706-x
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    References listed on IDEAS

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    1. Chakravarthy, Srinivas R., 2012. "Maintenance of a deteriorating single server system with Markovian arrivals and random shocks," European Journal of Operational Research, Elsevier, vol. 222(3), pages 508-522.
    2. Gertsbakh, Ilya, 1984. "The shorter queue problem: A numerical study using the matrix-geometric solution," European Journal of Operational Research, Elsevier, vol. 15(3), pages 374-381, March.
    3. Kostia Avrachenkov & Efrat Perel & Uri Yechiali, 2016. "Finite-buffer polling systems with threshold-based switching policy," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(3), pages 541-571, October.
    4. Kevin Granville & Steve Drekic, 2020. "A 2-class maintenance model with dynamic server behavior," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(1), pages 34-96, April.
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