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On the relationship between the distribution of maximal queue length in the M/G/1 queue and the mean busy period in the M/G/1/n queue

Published online by Cambridge University Press:  14 July 2016

Robert B. Cooper  and
Borge Tilt
Robert B. Cooper
Affiliation:
Georgia Institute of Technology
Borge Tilt
Affiliation:
Georgia Institute of Technology
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Abstract

Takács has shown that, in the M/G/1 queue, the probability P(k | i) that the maximum number of customers present simultaneously during a busy period that begins with i customers present is P(k | i) = Qk–i/Qk, where the Q's are easily calculated by recurrence in terms of an arbitrary Q0 ≠ 0. We augment Takács's theorem by showing that P(k | i) = bk–i/bk, where bn is the mean busy period in the M/G/1 queue with finite waiting room of size n; that is, if we take Q0 equal to the mean service time, then Qn =bn.

Keywords

MAXIMAL QUEUE LENGTH DISTRIBUTIONQUEUES WITH FINITE WAITING ROOM
Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 1 , March 1976 , pp. 195 - 199
Copyright
Copyright © Applied Probability Trust 1976 

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References

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