Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris

Little's law in distributional form

  • L. D. Servi
Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_549

Since Little's Law first appeared in 1961, its simplicity and importance have established it as a basic tool of queueing theory. Little's Law relates the average number of customers in a system, N, with the average time in the system, T, under very broad conditions. For example, Keilson and Servi (1988) have demonstrated that for many systems, the relationship between the queue length and the time in the system can be characterized beyond just their average value.

This is possible, however, if a class of customers arrives according to a Poisson process, is served first-in, first-out (FIFO) within the class, and is processed as either
  1. (i)

    an ordinary single-server queue,

     
  2. (ii)

    a single-server queue with one or more classes of priority which processes each class according to a preemptive-resume, preemptive-repeat, or nonpreemptive discipline,

     
  3. (iii)

    a vacation model system, where the server takes one or more “vacations” when the queue is depleted,

     
  4. (iv)

    a polling system, where a...

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References

  1. [1]
    Abramowitz, M. and Stegun, I.A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, U.S. Government Printing Office. 824–825. Google Scholar
  2. [2]
    Bertsimas, D. and Mourtzinou, G. (1997). “Transient laws of non-stationary queueing systems and their applications.” Queueing Systems, 25, 115–155.Google Scholar
  3. [3]
    Keilson, J. and Servi, L.D. (1988). “A distributional form of Little's Law,” Operation Research Letters, 7, 223–227.Google Scholar
  4. [4]
    Keilson, J. and Servi, L.D. (1990). “The distributional form of Little's Law and the Fuhrmann-Cooper decomposition.” Operations Research Letters, 9, 239–247.Google Scholar
  5. [5]
    Little, J. (1961). “A proof of the theorem L = λ W.” Operations Research, 8, 383–387.Google Scholar
  6. [6]
    Szczotka, W. (1992). “A distributional form of Little's law in heavy traffic.” Annals Probability, 20, 790–800.Google Scholar
  7. [7]
    Takahashi, Y. and Miyazawa, M. (1994). “Relationship between queue-length and waiting time distributions in a priority queue with batch arrivals.” Jl. Operations Research Society Japan, 37, 48–63.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • L. D. Servi
    • 1
  1. 1.GTE LaboratoriesWalthamUSA