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Customer average and time average queue lengths and waiting times

Published online by Cambridge University Press:  14 July 2016

Kneale T. Marshall and
Ronald W. Wolff
Kneale T. Marshall
Affiliation:
Naval Postgraduate School, Monterey, California
Ronald W. Wolff
Affiliation:
University of California, Berkeley
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Abstract

Bounds are obtained for the difference between the expected number in the queue found by an arrival and the time average expected number in the queue for the stationary GI/G/m queue. The lower bound is completely general but the upper bound requires that the class of inter-arrival distributions be restricted. When the upper bound applies, these quantities differ by at most one customer. Analogous results are obtained for the difference between the arrival average and time average number in the system for the GI/G/1 queue. An upper bound is also determined for the kth factorial moment of the number found in the queue by an arrival in terms of the kth. moment about the origin of the wait in the queue. Inequalities on the mean virtual wait are found in terms of the mean actual wait which show that under the same restrictions, these two measures of congestion differ by no more than half the mean inter-arrival time for the GI/G/1 queue.

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 3 , September 1971, pp. 535 - 542
Copyright
Copyright © Applied Probability Trust 1971 

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