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Algorithms for the upper bound mean waiting time in the GI/GI/1 queue

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  • Yan Chen

    (Columbia University)

  • Ward Whitt

    (Columbia University)

Abstract

It has long been conjectured that the tight upper bound for the mean steady-state waiting time in the GI/GI/1 queue given the first two moments of the interarrival-time and service-time distributions is attained asymptotically by two-point distributions. The two-point distribution for the interarrival time has one mass point at 0, but the service-time distribution involves a limit; there is one mass point at a high value, but that upper mass point must increase to infinity while the probability on that point must decrease to 0 appropriately. In this paper, we develop effective numerical and simulation algorithms to compute the value of this conjectured tight bound. The algorithms are aided by reductions of the special queues with extremal interarrival-time and extremal service-time distributions to D/GI/1 and GI/D/1 models. Combining these reductions yields an overall representation in terms of a D/RS(D)/1 discrete-time model involving a geometric random sum of deterministic random variables (the RS(D)), where the two deterministic random variables in the model may have different values, so that the extremal steady-state waiting time need not have a lattice distribution. Efficient computational methods are developed. The computational results show that the conjectured tight upper bound offers a significant improvement over established bounds.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:queues:v:94:y:2020:i:3:d:10.1007_s11134-020-09649-9
    DOI: 10.1007/s11134-020-09649-9
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    References listed on IDEAS

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    1. Ward Whitt, 2005. "Engineering Solution of a Basic Call-Center Model," Management Science, INFORMS, vol. 51(2), pages 221-235, February.
    2. Teunis J. Ott, 1987. "Simple Inequalities for the D / G /1 Queue," Operations Research, INFORMS, vol. 35(4), pages 589-597, August.
    3. K. T. Marshall, 1968. "Some Inequalities in Queuing," Operations Research, INFORMS, vol. 16(3), pages 651-668, June.
    4. Ward Whitt, 1986. "Deciding Which Queue to Join: Some Counterexamples," Operations Research, INFORMS, vol. 34(1), pages 55-62, February.
    5. A. E. Eckberg, 1977. "Sharp Bounds on Laplace-Stieltjes Transforms, with Applications to Various Queueing Problems," Mathematics of Operations Research, INFORMS, vol. 2(2), pages 135-142, May.
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    Cited by:

    1. 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.
    2. Wouter van Eekelen & Dick den Hertog & Johan S.H. van Leeuwaarden, 2022. "MAD Dispersion Measure Makes Extremal Queue Analysis Simple," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1681-1692, May.
    3. Filip Lindskog & Mario V. Wuthrich, 2024. "Claims processing and costs under capacity constraints," Papers 2409.09091, arXiv.org.
    4. van Eekelen, Wouter, 2023. "Distributionally robust views on queues and related stochastic models," Other publications TiSEM 9b99fc05-9d68-48eb-ae8c-9, Tilburg University, School of Economics and Management.

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