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The Relation between Customer and Time Averages in Queues

Author

Listed:
  • Daniel P. Heyman

    (Bell Laboratories, Holmdel, New Jersey)

  • Shaler Stidham

    (North Carolina State University, Raleigh, North Carolina)

Abstract

Brumelle has generalized the queueing formula L = λ W to H = λ G , where λ is the arrival rate and H and G are respectively time and customer averages of some queue statistics which have a certain relationship to each other but are otherwise arbitrary. Stidham has developed a simple proof of L = λ W for each sample path, in which the only requirement is that λ and W be finite. In this note it is shown that Stidham's proof applies directly to the more general case of H = λ G , provided λ and G are finite and a simple technical assumption is satisfied. The result is used to obtain time average probabilities in the queue GI / M / c / K . Finally, a counterexample is given to demonstrate that the technical assumption is not superfluous, even in the special case where H and G can be interpreted, respectively, as the time average number of units in the system and the average time spent by a unit in the system, as is the case with both L = λ W and the application to the queue GI / M / c / K .

Suggested Citation

  • Daniel P. Heyman & Shaler Stidham, 1980. "The Relation between Customer and Time Averages in Queues," Operations Research, INFORMS, vol. 28(4), pages 983-994, August.
    • Handle: RePEc:inm:oropre:v:28:y:1980:i:4:p:983-994
    DOI: 10.1287/opre.28.4.983
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    Cited by:

    1. John D. C. Little, 2011. "OR FORUM---Little's Law as Viewed on Its 50th Anniversary," Operations Research, INFORMS, vol. 59(3), pages 536-549, June.
    2. Muhammad El-Taha, 2017. "A general workload conservation law with applications to queueing systems," Queueing Systems: Theory and Applications, Springer, vol. 85(3), pages 361-381, April.
    3. Ken’ichi Kawanishi & Tetsuya Takine, 2016. "MAP/M/c and M/PH/c queues with constant impatience times," Queueing Systems: Theory and Applications, Springer, vol. 82(3), pages 381-420, April.
    4. Vinck, Bart & Bruneel, Herwig, 2006. "System delay versus system content for discrete-time queueing systems subject to server interruptions," European Journal of Operational Research, Elsevier, vol. 175(1), pages 362-375, November.

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