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

Author

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  • 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
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    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.

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