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The total waiting time in a busy period of a stable single-server queue, II

Published online by Cambridge University Press:  14 July 2016

D. J. Daley  and
D. R. Jacobs Jr.
D. J. Daley
Affiliation:
The Johns Hopkins University, Baltimore, Maryland
D. R. Jacobs Jr.
Affiliation:
The Johns Hopkins University, Baltimore, Maryland
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Extract

This paper is a continuation of Daley (1969), referred to as (I), whose notation and numbering is continued here. We shall indicate various approaches to the study of the total waiting time in a busy period2 of a stable single-server queue with a Poisson arrival process at rate λ, and service times independently distributed with common distribution function (d.f.) B(·). Let X'i denote3 the total waiting time in a busy period which starts at an epoch when there are i (≧ 1) customers in the system (to be precise, the service of one customer is just starting and the remaining i − 1 customers are waiting for service). We shall find the first two moments of X'i, prove its asymptotic normality for i → ∞ when B(·) has finite second moment, and exhibit the Laplace-Stieltjes transform of X'i in M/M/1 as the ratio of two Bessel functions.

Type
Research Papers
Information
Journal of Applied Probability , Volume 6 , Issue 3 , December 1969 , pp. 565 - 572
Copyright
Copyright © Applied Probability Trust 

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References

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