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Simple Inequalities for the D / G /1 Queue

Author

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  • Teunis J. Ott

    (Bell Communications Research, Morristown, New Jersey)

Abstract

During the last quarter century a large number of papers, starting in 1962 with J. F. C. Kingman, have been written about inequalities for queues. Their objective has been to eliminate the need for detailed distributional analysis by providing upper and lower bounds for such entities as the expected waiting time and the probability of no waiting. Notable results include Kingman's upper bound for the expected waiting time in a GI / G /1 queue and K. T. Marshall's lower bound for the expected waiting time in a D / G /1 queue. In this paper we present a new method for obtaining bounds for the stationary D / G /1 queue. The basic idea is to translate crude bounds on the probability that the system is empty at time t into sharp bounds for the expected waiting time. The crudest possible bounds on the probability that the system is empty at time t reproduce the Kingman upper bound and the Marshall lower bound, and show that the Marshall lower bound is tight for the D / G /1 queue and is attained if and only if the service times are integer (random) multiples of the interarrival time.

Suggested Citation

  • Teunis J. Ott, 1987. "Simple Inequalities for the D / G /1 Queue," Operations Research, INFORMS, vol. 35(4), pages 589-597, August.
    • Handle: RePEc:inm:oropre:v:35:y:1987:i:4:p:589-597
    DOI: 10.1287/opre.35.4.589
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    Cited by:

    1. Yan Chen & Ward Whitt, 2021. "Extremal GI/GI/1 queues given two moments: exploiting Tchebycheff systems," Queueing Systems: Theory and Applications, Springer, vol. 97(1), pages 101-124, February.
    2. 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.
    3. Hai Wang & Amedeo Odoni, 2016. "Approximating the Performance of a “Last Mile” Transportation System," Transportation Science, INFORMS, vol. 50(2), pages 659-675, May.

    More about this item

    Keywords

    684 inequalities for the D/G/1 queue;

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