Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T05:31:28.815Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Daley, D. J. (1969) The total waiting time in a busy period of a stable single-server queue, I. J. Appl. Prob. 6, 550–564.Google Scholar
Gaver, D. P. (1969) Highway delays resulting from flow-stopping incidents. J. Appl. Prob. 6, 137–153.Google Scholar
Gleser, L. J. (1965) On the asymptotic theory of fixed-size sequential confidence bounds for linear regression parameters. Ann. Math. Statist. 36, 463–467.Google Scholar
LoèVe, M. (1963) Probability Theory. Van Nostrand, Princeton.Google Scholar
Mcneil, D. R. (1968) A solution to the fixed-cycle traffic light problem for compound Poisson arrivals. J. Appl. Prob. 5, 624–635.Google Scholar
Prabhu, N. U. (1965) Queues and Inventories. Wiley, New York.Google Scholar
Wall, H. S. (1948) Analytic Theory of Continued Fractions. Van Nostrand, New York.Google Scholar
Whittaker, E. T. and Watson, G. N. (1927) A Course in Modern Analysis. Cambridge University Press, Cambridge.Google Scholar