Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-16T10:24:25.363Z Has data issue: false hasContentIssue false
  • English
  • Français

On the virtual and actual waiting time distributions of a GI/G/1 queue

Published online by Cambridge University Press:  14 July 2016

J. Michael Harrison  and
Austin J. Lemoine
J. Michael Harrison
Affiliation:
Stanford University
Austin J. Lemoine
Affiliation:
Systems Control, Inc.
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a stable GI/G/1 queue with non-lattice interarrival time distribution. Let G and H be the limiting actual and virtual waiting time distributions respectively. Two separate statements of the relationship between G and H are found in a classical theorem of Takàcs and a more recent (and previously unpublished) theorem of Hooke. A simplified proof of Takàcs's theorem, based on a sample path relationship between the virtual and actual waiting time processes, has recently been advanced. This paper gives a similar proof of Hooke's theorem, based on the same sample path relationship, and demonstrates the utility of the result in analyzing the special case of Poisson input. In particular, by combining the Takàcs and Hooke results one can obtain the Pollaczek–Khintchine formula without any reference to the imbedded Markov chain.

Keywords

GI/G/1 QUEUEWAITING TIMEVIRTUAL WAITING TIMERENEWAL THEORYBUSY CYCLE
Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 4 , December 1976 , pp. 833 - 836
Copyright
Copyright © Applied Probability Trust 1976 

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

[1] Cohen, J. W. (1969) The Single Server Queue. North Holland, Amsterdam.Google Scholar
[2] Hooke, J. A. (1969) Some limit theorems for priority queues. Technical Report No. 91, Department of Operations Research, Cornell University.Google Scholar
[3] Lemoine, A. J. (1974) Limit theorems for generalized single server queues. Adv. Appl. Prob. 6, 159–174.CrossRefGoogle Scholar
[4] Lemoine, A. J. (1974) On two stationary distributions for the stable GI/G/1 queue. J. Appl. Prob. 11, 849–852.CrossRefGoogle Scholar
[5] Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.CrossRefGoogle Scholar
[6] Stidham, S. (1972) Regenerative processes in the theory of queues, with applications to the alternating-priority queue. Adv. Appl. Prob. 4, 542–577.CrossRefGoogle Scholar
[7] Takàcs, L. (1963) The limiting distribution of the virtual waiting time and the queue size for a single-server queue with recurrent input and general service times. Sankhya A 25, 91–100.Google Scholar
[8] Takàcs, L. (1973) On a method of Pollaczek. Stoch. Proc. Appl. 1, 1–9.CrossRefGoogle Scholar