Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T16:53:14.109Z Has data issue: false hasContentIssue false
  • English
  • Français

Complements to heavy traffic limit theorems for the GI/G/1 queue

Published online by Cambridge University Press:  14 July 2016

Ward Whitt
Ward Whitt*
Affiliation:
Yale University
Get access Rights & Permissions [Opens in a new window]

Abstract

A bound on the rate of convergence and sufficient conditions for the convergence of moments are obtained for the sequence of waiting times in the GI/G/1 queue when the traffic intensity is at the critical value ρ = 1.

Keywords

CONGESTIONQUEUESGI/G/1 QUEUEWAITING TIMESUNSTABLE QUEUESHEAVY TRAFFICLIMIT THEOREMSRATE OF CONVERGENCECONVERGENCE OF MOMENTS
Type
Short Communications
Information
Journal of Applied Probability , Volume 9 , Issue 1 , March 1972 , pp. 185 - 191
Copyright
Copyright © Applied Probability Trust 1972 

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. John Wiley and Sons, New York.Google Scholar
[2] Borovkov, A. (1964) Some limit theorems in the theory of mass service, I. Theor. Probability Appl. 9, 550–565.Google Scholar
[3] Borovkov, A. (1965) Some limit theorems in the theory of mass service, II. Theor. Probability Appl. 10, 375–400.Google Scholar
[4] Doob, J. (1953) Stochastic Processes. John Wiley and Sons, New York.Google Scholar
[5] Erdös, P. and Kac, M. (1946) On certain limit theorems in the theory of probability. Bull. Amer. Math. Soc. 52, 292–302.CrossRefGoogle Scholar
[6] Feller, W. (1966) An Introduction to Probability Theory and Its Applications. Vol. 2. John Wiley and Sons, New York.Google Scholar
[7] Heyde, C. C. (1969) On extended rate of convergence results for the invariance principle. Ann. Math. Statist. 40, 2178–2179.Google Scholar
[8] Iglehart, D. (1969) Multiple channel queues in heavy traffic, IV: law of the iterated logarithm. Technical Report No. 8, Department of Operations Research, Stanford University.Google Scholar
[9] Iglehart, D. and Whitt, W. (1970a) Multiple channel queues in heavy traffic, I. Adv. Appl. Prob. 2, 150–177.Google Scholar
[10] 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
[10a] Kennedy, D. P. (1972) Rates of convergence for queues in heavy traffic. I. Adv. Appl. Prob. (To appear).CrossRefGoogle Scholar
[11] Kingman, J. F. C. (1961) The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 57, 902–904.Google Scholar
[12] Kingman, J. F. C. (1965) The heavy traffic approximation in the theory of queues. Smith, W. and Wilkinson, W. (Eds.), Proceedings of the Symposium on Congestion Theory. The University of North Carolina Press, Chapel Hill, 137–159.Google Scholar
[13] Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.Google Scholar
[14] Pollaczek, F. (1952) Fonctions caractéristiques de certaines répartitions définies au moyen de la notion d'ordre. Applications à la théorie des attentes. C. R. Acad. Sci. 234, 2334–2336. (In French).Google Scholar
[15] Prohorov, Yu. (1956) Convergence of random processes and limit theorems in probability theory. Theor. Probability Appl. 1, 157–214.CrossRefGoogle Scholar
[16] Prohorov, Yu. (1963) Transient phenomena in processes of mass service. Litovsk. Mat. Sb. 3, 199–205. (In Russian).Google Scholar
[17] Rosenkrantz, W. (1968) On rates of convergence for the invariance principle. Trans. Amer. Math. Soc. 129, 542–552.Google Scholar
[18] Skorohod, A. (1965) Studies in the Theory of Random Processes. Addison-Wesley, Reading, Massachusetts.Google Scholar
[19] Whitt, W. (1969) Weak Convergence Theorems for Queues in Heavy Traffic. , Department of Operations Research, Cornell University. (Also, Technical Report No. 2, Department of Operations Research, Stanford University, 1968.) Google Scholar
[20] Whitt, W. (1970) Multiple channel queues in heavy traffic, III: random server selection. Adv. Appl. Prob. 2, 370–375.CrossRefGoogle Scholar
[21] Whitt, W. (1971a) Weak convergence theorems for priority queues: preemptive-resume discipline. J. Appl. Prob. 8, 74–94.Google Scholar
[22] Whitt, W. (1971b). Heavy traffic approximations for stable queues. Submitted to Adv. Appl. Prob. Google Scholar