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A relation between stationary queue and waiting time distributions

Published online by Cambridge University Press:  14 July 2016

Rasoul Haji  and
Gordon F. Newell
Rasoul Haji
Affiliation:
University of California, Berkeley
Gordon F. Newell
Affiliation:
University of California, Berkeley
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Abstract

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

Type
Short Communications
Information
Journal of Applied Probability , Volume 8 , Issue 3 , September 1971 , pp. 617 - 620
Copyright
Copyright © Applied Probability Trust 1971 

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References

[1] Little, J. D. C. (1961) A proof for the queueing formula: L= ?W, Operat. Res. 9, 383–387.Google Scholar
[2] Jewell, W. S. (1967) A simple proof of: L = ?W. Operat. Res. 15, 1109–1116.Google Scholar
[3] Beutler, F. J. and Leneman, O. A. Z. (1966) The theory of stationary point processes. Acta Math. 116, 159–197.Google Scholar
[4] Haji, R. (1970) The variance of the number of customers in a queue. Operations Research Center Report 70–18, University of California, Berkeley.Google Scholar