IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v9y1961i3p383-387.html
   My bibliography  Save this article

A Proof for the Queuing Formula: L = (lambda) W

Author

Listed:
  • John D. C. Little

    (Case Institute of Technology, Cleveland, Ohio)

Abstract

In a queuing process, let 1/(lambda) be the mean time between the arrivals of two consecutive units, L be the mean number of units in the system, and W be the mean time spent by a unit in the system. It is shown that, if the three means are finite and the corresponding stochastic processes strictly stationary, and, if the arrival process is metrically transitive with nonzero mean, then L = (lambda) W .

Suggested Citation

  • John D. C. Little, 1961. "A Proof for the Queuing Formula: L = (lambda) W," Operations Research, INFORMS, vol. 9(3), pages 383-387, June.
  • Handle: RePEc:inm:oropre:v:9:y:1961:i:3:p:383-387
    DOI: 10.1287/opre.9.3.383
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.9.3.383
    Download Restriction: no

    File URL: https://libkey.io/10.1287/opre.9.3.383?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. Andreas Klein, 2011. "Die Entwicklung eines agentenbasierten Basismodells zur Bestimmung der deckungsbeitragsmaximierenden Anzahl von Außendienstmitarbeitern," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 21(2), pages 189-210, January.
    2. Lonnie Turpin & Barron Brown, 2021. "On Reworks in a Serial Process with Flexible Windows of Time," SN Operations Research Forum, Springer, vol. 2(2), pages 1-13, June.
    3. 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.
    4. Jelmer P. Gaast & Ivo J. B. F. Adan & René B. M. Koster, 2017. "The analysis of batch sojourn-times in polling systems," Queueing Systems: Theory and Applications, Springer, vol. 85(3), pages 313-335, April.
    5. Moshe Haviv & Binyamin Oz, 2021. "A busy period approach to some queueing games," Queueing Systems: Theory and Applications, Springer, vol. 97(3), pages 261-277, April.
    6. W. Rogiest & K. Laevens & J. Walraevens & H. Bruneel, 2015. "Random-order-of-service for heterogeneous customers: waiting time analysis," Annals of Operations Research, Springer, vol. 226(1), pages 527-550, March.
    7. A. Azadeh & M. S. Naghavi lhoseiny & V. Salehi, 2018. "Optimum alternatives of tandem G/G/K queues with disaster customers and retrial phenomenon: interactive voice response systems," Telecommunication Systems: Modelling, Analysis, Design and Management, Springer, vol. 68(3), pages 535-562, July.
    8. Dieter Armbruster & Daniel E. Marthaler & Christian Ringhofer & Karl Kempf & Tae-Chang Jo, 2006. "A Continuum Model for a Re-entrant Factory," Operations Research, INFORMS, vol. 54(5), pages 933-950, October.
    9. Shaler Stidham, 2002. "Analysis, Design, and Control of Queueing Systems," Operations Research, INFORMS, vol. 50(1), pages 197-216, February.
    10. Marco Cello & Giorgio Gnecco & Mario Marchese & Marcello Sanguineti, 2015. "Narrowing the Search for Optimal Call-Admission Policies Via a Nonlinear Stochastic Knapsack Model," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 819-841, March.
    11. Simon Lidberg & Tehseen Aslam & Leif Pehrsson & Amos H. C. Ng, 2020. "Optimizing real-world factory flows using aggregated discrete event simulation modelling," Flexible Services and Manufacturing Journal, Springer, vol. 32(4), pages 888-912, December.
    12. Kevin Granville & Steve Drekic, 2020. "A 2-class maintenance model with dynamic server behavior," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(1), pages 34-96, April.
    13. M. A. C. Almeida & F. R. B. Cruz & F. L. P. Oliveira & G. Souza, 2020. "Bias correction for estimation of performance measures of a Markovian queue," Operational Research, Springer, vol. 20(2), pages 943-958, June.
    14. Juan S. Borrero & Colin Gillen & Oleg A. Prokopyev, 2017. "Fractional 0–1 programming: applications and algorithms," Journal of Global Optimization, Springer, vol. 69(1), pages 255-282, September.
    15. Yifan Liu & Lawrence M. Wein, 2008. "A Queueing Analysis to Determine How Many Additional Beds Are Needed for the Detention and Removal of Illegal Aliens," Management Science, INFORMS, vol. 54(1), pages 1-15, January.
    16. Mehrdad Moshtagh & Jafar Fathali & James MacGregor Smith & Nezam Mahdavi-Amiri, 2019. "Finding an optimal core on a tree network with M/G/c/c state-dependent queues," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(1), pages 115-142, 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:9:y:1961:i:3:p:383-387. 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.