IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v8y1960i6p755-772.html
   My bibliography  Save this article

Time-Dependent Solution of the Many-Server Poisson Queue

Author

Listed:
  • Thomas L. Saaty

    (Office of Naval Research, Washington, D. C.)

Abstract

In this paper we obtain the Laplace transform of the transient probabilities of the ordered queuing problem, with Poisson inputs, multiple channels, and exponential service times. Explicit expressions are derived for the two-channel case and known equilibrium conditions are shown to hold. The proof proceeds in two stages. The first obtains the Laplace transform of the generating function of the system and the second solves a first order linear partial differential equation in a restricted generating function introduced to determine the Laplace transform of the probability functions appearing in the first generating function. In the two-channel case the solution is decomposed into two components, one of which is immediately related to the well-known solution of the single-channel queue. We also study the problem with different service distributions for the two-channel case and compute the distribution of a busy period for that case.

Suggested Citation

  • Thomas L. Saaty, 1960. "Time-Dependent Solution of the Many-Server Poisson Queue," Operations Research, INFORMS, vol. 8(6), pages 755-772, December.
    • Handle: RePEc:inm:oropre:v:8:y:1960:i:6:p:755-772
    DOI: 10.1287/opre.8.6.755
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.8.6.755
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.8.6.755?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. Li, Na & Stanford, David A., 2016. "Multi-server accumulating priority queues with heterogeneous servers," European Journal of Operational Research, Elsevier, vol. 252(3), pages 866-878.
    2. Charles Knessl & Johan Leeuwaarden, 2015. "Transient analysis of the Erlang A model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(2), pages 143-173, October.
    3. Kahraman, Aykut & Gosavi, Abhijit, 2011. "On the distribution of the number stranded in bulk-arrival, bulk-service queues of the M/G/1 form," European Journal of Operational Research, Elsevier, vol. 212(2), pages 352-360, July.
    4. Jain, Apurva, 2007. "Value of capacity pooling in supply chains with heterogeneous customers," European Journal of Operational Research, Elsevier, vol. 177(1), pages 239-260, February.

    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:8:y:1960:i:6:p:755-772. 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.