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Rates of convergence for queues in heavy traffic. I

Published online by Cambridge University Press:  01 July 2016

Douglas P. Kennedy
Douglas P. Kennedy*
Affiliation:
University of Sheffield
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Abstract

Estimates are given for the rates of convergence in functional central limit theorems for quantities of interest in the GI/G/1 queue and a general multiple channel system. The traffic intensity is fixed ≧ 1. The method employed involves expressing the underlying stochastic processes in terms of Brownian motion using the Skorokhod representation theorem.

Keywords

RATES OF CONVERGENCEERROR TERMSWEAK CONVERGENCEFUNCTIONAL LIMIT THEOREMSHEAVY TRAFFICMULTIPLE CHANNEL QUEUESQUEUES
Type
Research Article
Information
Advances in Applied Probability , Volume 4 , Issue 2 , August 1972 , pp. 357 - 381
Copyright
Copyright © Applied Probability Trust 1972 

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References

[1] Billingsley, P. (1968) Convergence of Probability Measures. John Wiley, New York.Google Scholar
[2] Borovkov, A. (1965) Some limit theorems in the theory of mass service, II. Theor. Probability Appl. 10, 375–400.Google Scholar
[3] Breiman, L. (1968) Probability. Addison-Wesley, Reading, Massachusetts.Google Scholar
[4] Dharmadhikari, S. W., Fabian, V. and Jogdeo, K. (1968) Bounds on the moments of martingales. Ann. Math. Statist. 39, 1719–1723.Google Scholar
[5] Feller, W. (1957) An Introduction to Probability Theory and its Applications. Vol. I (Second edition). John Wiley, New York.Google Scholar
[6] Heyde, C. C. (1969) On extended rate of convergence results for the invariance principle. Ann. Math. Statist. 40, 2178–2179.Google Scholar
[7] Iglehart, D. L. and Kennedy, D. P. (1970) Weak convergence for the average of flag processes. J. Appl. Prob. 7, 747–753.Google Scholar
[8] Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic. I. Adv. Appl. Prob. 2, 150–177.Google Scholar
[9] Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic. II: sequences, networks and batches. Adv. Appl. Prob. 2, 355–369.Google Scholar
[10] Loève, M. (1963) Probability Theory. Van Nostrand, Princeton, New Jersey.Google Scholar
[11] Rosenkrantz, W. A. (1967) On rates of convergence for the invariance principle. Trans. Amer. Math. Soc. 129, 542–552.Google Scholar
[12] Skorokhod, A. V. (1965) Studies in the Theory of Random Processes. Addison-Wesley, Reading, Massachusetts.Google Scholar
[13] Tomko, J. (1972) The rate of convergence in central limit theorems for service systems with finite queue capacity. J. Appl. Prob. 9, 87–102.Google Scholar
[14] von Bahr, B. and Esseen, C. G. (1965) Inequalities for the rth absolute moment of a sum of random variables, 1 ≦ r ≦ 2. Ann. Math. Statist. 36, 299–303.Google Scholar
[15] Whitt, W. (1968) Weak convergence theorems for queues in heavy traffic. Technical Report No. 2, Department of Operations Research, Stanford University.Google Scholar
[16] Whitt, W. (1970) Multiple channel queues in heavy traffic. III: random server selection. Adv. Appl. Prob. 2, 370–375.Google Scholar
[17] Whitt, W. (1971) Weak convergence theorems for priority queues: preemptive-resume discipline. J. Appl. Prob. 8, 74–94.Google Scholar
[18] Whitt, W. (1972) Complements to heavy traffic limit theorems for the GI/G/1 queue. J. Appl. Prob. 9, 185–191.Google Scholar