Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-24T02:58:14.364Z Has data issue: false hasContentIssue false
  • English
  • Français

Controlled queues in heavy traffic

Published online by Cambridge University Press:  01 July 2016

John H. Rath
John H. Rath*
Affiliation:
Bell Telephone Laboratories, Holmdel, New Jersey
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies a controlled queueing system in which the decisionmaker may change servers according to rules which depend only on the queue length. It is proved that for a given control policy a properly normalised sequence of these controlled queue length processes converges weakly to a controlled diffusion process as the queueing systems approach a state of heavy traffic.

Keywords

CONTROLLED QUEUESDIFFUSION APPROXIMATIONSWEAK CONVERGENCEHEAVY TRAFFIC
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 3 , September 1975 , pp. 656 - 671
Copyright
Copyright © Applied Probability Trust 1975 

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] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[2] Breiman, L. (1968) Probability. Addison-Wesley, New York.Google Scholar
[3] Crabill, T. (1969) Optimal Control of a Queue with Variable Service Rates. Technical Report No. 75, Department of Operations Research, Cornell University.Google Scholar
[4] Crabill, T. (1972) Optimal control of a service facility with variable exponential service time and constant arrival rate. Management Science, 18, 560–566.CrossRefGoogle Scholar
[5] Harrison, J. M. (1970) Queueing Models for Assembly-Like Systems. Technical Report No. 7, Department of Operations Research, Stanford University.Google Scholar
[6] Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic, I. Adv. Appl. Prob. 2, 150–177.CrossRefGoogle Scholar
[7] Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic, II. Adv. Appl. Prob. 2, 355–369.CrossRefGoogle Scholar
[8] Lindvall, T. (1973) Weak convergence of probability measures and random functions in the function space D[0, ∞). J. Appl. Prob. 10, 109–121.CrossRefGoogle Scholar
[9] Prabhu, N. U. and Stidham, S. (1973) Optimal Control of Queueing Systems. Technical Report No. 186, Department of Operations Research, Cornell University.Google Scholar
[10] Rath, J. (1973) Limit Theorems for Controlled Queues. Technical Report No. 21, Department of Operations Research, Stanford University.Google Scholar
[11] Whitt, W. (1968) Weak Convergence Theorems for Queues in Heavy Traffic. , Cornell University (Technical Report No. 2, Department of Operations Research, Stanford University).Google Scholar
[12] Whitt, W. (1970) Weak convergence of probability measures on the function space C[0, ∞). Ann. Math. Statist. 41, 939–944.CrossRefGoogle Scholar
[13] Whitt, W. (1971) Weak convergence of first passage time processes. J. Appl. Prob. 8, 417–422.CrossRefGoogle Scholar
[14] Whitt, W. (1971) Weak Convergence Involving a Random Time Change. Technical Report, Department of Administrative Sciences, Yale University.Google Scholar
[15] Whitt, W. (1974) ‘Heavy Traffic Limit Theorems for Queues: A Survey’, in Mathematical Methods in Queueing Theory. Springer-Verlag, New York.Google Scholar