Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-07-01T00:59:52.749Z Has data issue: false hasContentIssue false
  • English
  • Français

On some limit theorems for the GI/G/1 queue

Published online by Cambridge University Press:  14 July 2016

John A. Hooke
John A. Hooke*
Affiliation:
Cornell University
Get access Rights & Permissions [Opens in a new window]

Extract

For a GI/G/1 queue a limit theorem is obtained for the total input of work during (0, t]. This result is then used to obtain similar theorems for the waiting and idle times and for busy periods initiated by large service loads. Some of the results contained in the paper have recently been proved by other authors in more general settings. The intent of this work is to show how they may be obtained in lesser generality using simpler techniques.

Type
Research Papers
Information
Journal of Applied Probability , Volume 7 , Issue 3 , December 1970 , pp. 634 - 640
Copyright
Copyright © Applied Probability Trust 1970 

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

Beneš, V. E. (1963) General Stochastic Processes in the Theory of Queues. Addison-Wesley, Reading.Google Scholar
Borovkov, A. A. (1965) Some limit theorems in the theory of mass service, II. Theor. Probability Appl. 10, 375–400.Google Scholar
Brody, S. M. (1963) On a limiting theorem of the theory of queues. Ukrain. Mat. Z. 15, 76–79.Google Scholar
Doob, J. L. (1948) Renewal theory from the point of view of the theory of probability. Trans. Amer. Math. Soc. 63, 422–438.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York.Google Scholar
Iglehart, D. and Kennedy, D. (1970) Weak convergence of the average of flag processes. J. Appl. Prob. 7, 747–753.Google Scholar
Iglehart, D. and Whitt, W. (1970a) Multiple channel queues in heavy traffic. I. Adv. Appl. Prob. 2, 150–177.Google Scholar
Iglehart, D. and Whitt, W. (1970b). Multiple channel queues in heavy traffic, II: sequences, networks and batches. Adv. Appl. Prob. 2, 355–369.Google Scholar
Kingman, J. F. C. (1962) Some inequalities for the queue GI/G/1. Biometrika 49, 315.Google Scholar
Prabhu, N. U. (1965a) Stochastic Processes. MacMillan, New York.Google Scholar
Prabhu, N. U. (1965b) Queues and Inventories. John Wiley, New York.Google Scholar
Prabhu, N. U. (1968) Some new results in storage theory. J. Appl. Prob. 5, 452–460.Google Scholar
Prabhu, N. U. (1969) Limit theorems for the single server queue with traffic intensity one. Tech. Report 71, Dept. of Operations Research, Cornell University.Google Scholar
Reich, E. (1958) On the integro-differential equation of Takács. Ann. Math. Statist. 29, 563–570.Google Scholar
Takács, L. (1967) Combinational Methods in the Theory of Stochastic Processes. John Wiley and Sons, New York.Google Scholar
Whitt, W. (1969) Weak convergence theorems for queues in heavy traffic. Submitted to Adv. Appl. Prob. Google Scholar
Whitt, W. (1970) Multiple channel queues in heavy traffic, III: random server selection. Adv. Appl. Prob. 2, 370–375.CrossRefGoogle Scholar
Whitt, W. (1971) Weak convergence theorems for priority queues: pre-emptive-resume discipline. To appear in J. Appl. Prob. CrossRefGoogle Scholar