Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-06T22:17:50.385Z Has data issue: false hasContentIssue false
  • English
  • Français

Convex ordering of the attained waiting times in single-server queues and related problems

Published online by Cambridge University Press:  14 July 2016

Masakiyo Miyazawa  and
Genji Yamazaki
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
Genji Yamazaki*
Affiliation:
Tokyo Metropolitan Institute of Technology
*
∗Postal address: Department of Information Sciences, Science University of Tokyo, Noda-city, Chiba 278, Japan.
∗∗Postal address: Department of Engineering Management, Tokyo Metropolitan Institute of Technology, 6–6 Asahigaoka, Hino-city, Tokyo 191, Japan.
Get access Rights & Permissions [Opens in a new window]

Abstract

The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

Keywords

WORK-CONSERVING DISCIPLINEFCFSLCFSFINITENESS OF MOMENTS
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 2 , June 1991 , pp. 433 - 445
Copyright
Copyright © Applied Probability Trust 1991 

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

Franken, P., König, D., Arndt, U. and Schmidt, V, (1982) Queues and Point Processes. Wiley, Chichester.Google Scholar
Ghahramani, S. and Wolff, R. W. (1989) Finite moment conditions for GI/G/1 busy periods. Queueing Systems 4, 171–178.Google Scholar
Köllerström, J. (1976) Stochastic bounds for the single-server queue. Math. Proc. Camb. Phil. Soc. 80, 521–525.10.1017/S0305004100053135Google Scholar
Loynes, R. M. (1962) The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
Miyazawa, M. (1983) The derivation of invariance relations in complex queueing systems with stationary inputs. Adv. Appl. Prob. 15, 874–885.10.2307/1427329Google Scholar
Miyazawa, M. (1985) The intensity conservation law for queues with randomly changed service rate. J. Appl. Prob. 22, 408–418.Google Scholar
Prabhu, N. U. (1965) Queues and Inventories: A Study of their Basic Stochastic Processes. Wiley, New York.Google Scholar
Righter, R., Shanthikumar, J. G. and Yamazaki, G. (1990) On extremal service discipline in single server queueing systems. J. Appl. Prob. 27, 409–416.Google Scholar
Sakasegawa, H. and Wolff, R. W. (1990) The equality of the virtual delay and attained waiting time distributions. Adv. Appl. Prob. 22, 257–259.10.2307/1427611Google Scholar
Sengupta, B. (1989) An invariance relationship for the G/G/1 queue. Adv. Appl. Prob. 21, 956–957.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1990) Convexity of a set of stochastically ordered random variables. Adv. Appl. Prob. 22, 160–177.10.2307/1427603Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and other Stochastic Models. Edited with revision by Daley, D. J., Wiley, New York.Google Scholar
Wolff, R. W. (1970) Work conserving priorities. J. Appl. Prob. 7, 327–337.Google Scholar
Wolff, R. W. (1989) Stochastic Modeling and the Theory of Queues. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
Yamazaki, G. and Miyazawa, M. (1991) The equality of the workload and total attained waiting time on average. J. Appl. Prob. 28, 238–244.Google Scholar