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Applying optimization theory to study extremal GI/GI/1 transient mean waiting times

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  • Yan Chen

    (Columbia University)

  • Ward Whitt

    (Columbia University)

Abstract

We study the tight upper bound of the transient mean waiting time in the classical GI/GI/1 queue over candidate interarrival-time distributions with finite support, given the first two moments of the interarrival time and the full service-time distribution. We formulate the problem as a non-convex nonlinear program. We derive the gradient of the transient mean waiting time and then show that a stationary point of the optimization can be characterized by a linear program. We develop and apply a stochastic variant of the Frank and Wolfe (Naval Res Logist Q 3:95–110, 1956) algorithm to find a stationary point for any given service-time distribution. We also establish necessary conditions and sufficient conditions for stationary points to be three-point distributions or special two-point distributions. We illustrate by applying simulation together with optimization to analyze several examples.

Suggested Citation

  • Yan Chen & Ward Whitt, 2022. "Applying optimization theory to study extremal GI/GI/1 transient mean waiting times," Queueing Systems: Theory and Applications, Springer, vol. 101(3), pages 197-220, August.
    • Handle: RePEc:spr:queues:v:101:y:2022:i:3:d:10.1007_s11134-021-09725-8
    DOI: 10.1007/s11134-021-09725-8
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    References listed on IDEAS

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    1. James E. Smith, 1995. "Generalized Chebychev Inequalities: Theory and Applications in Decision Analysis," Operations Research, INFORMS, vol. 43(5), pages 807-825, October.
    2. Henry Lam & Clementine Mottet, 2017. "Tail Analysis Without Parametric Models: A Worst-Case Perspective," Operations Research, INFORMS, vol. 65(6), pages 1696-1711, December.
    3. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
    4. Yan Chen & Ward Whitt, 2020. "Algorithms for the upper bound mean waiting time in the GI/GI/1 queue," Queueing Systems: Theory and Applications, Springer, vol. 94(3), pages 327-356, April.
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