IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v28y1980i4p983-994.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.28.4.983
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.28.4.983?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. 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.
    3. 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.
    4. 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.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:oropre:v:28:y:1980:i:4:p:983-994. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.