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The virtual waiting time of the GI/G/1 queue in heavy traffic

Published online by Cambridge University Press:  01 July 2016

E. Kyprianou
E. Kyprianou*
Affiliation:
University of Manchester
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Extract

Investigations in the theory of heavy traffic were initiated by Kingman ([5], [6] and [7]) in an effort to obtain approximations for stable queues. He considered the Markov chains {W n i } of a sequence {Q i } of stable GI/G/1 queues, where W n i is the waiting time of the nth customer in the ith queueing system, and by making use of Spitzer's identity obtained limit theorems as first n → ∞ and then ρi ↑ 1 as i → ∞. Here &rHi is the traffic intensity of the ith queueing system. After Kingman the theory of heavy traffic was developed by a number of Russians mainly. Prohorov [10] considered the double sequence of waiting times {W n i } and obtained limit theorems in the three cases when n 1/2(ρi -1) approaches (i) - ∞, (ii) -δ and (iii) 0 as n → ∞ and i → ∞ simultaneously. The case (i) includes the result of Kingman. Viskov [12] also studied the double sequence {W n i } and obtained limits in the two cases when n 1/2(ρi − 1) approaches + δ and + ∞ as n → ∞ and i → ∞ simultaneously.

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Type
Research Article
Information
Advances in Applied Probability , Volume 3 , Issue 2 , Autumn 1971 , pp. 249 - 268
Copyright
Copyright © Applied Probability Trust 1971 

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References

[1] Billingsley, P. (1968) Convergence of Probability Measures. John Wiley, New York.Google Scholar
[2] Cox, D. R. and Miller, H. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
[3] Hooke, J. A. (1970) On some limit theorems for the GI/G/1 queue. J. Appl. Prob. 7, 634–640.CrossRefGoogle Scholar
[4] Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic I. Adv. Appl. Prob. 2, 150–177.CrossRefGoogle Scholar
[5] Kingman, J. F. C. (1961) The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 57, 902–904.CrossRefGoogle Scholar
[6] Kingman, J. F. C. (1962) On queues in heavy traffic. J. R. Statist. Soc. B 24, 383–392.Google Scholar
[7] Kingman, J. F. C. (1965) The heavy traffic approximation in the theory of queues. Proceedings of the Symposium on Congestion Theory. Smith, W. and Wilkinson, W. (Eds). The University of N. California Press, Chapel Hill, 137–159.Google Scholar
[8] Prabhu, N. U. (1968) Some new results in storage theory. J. Appl. Prob. 5, 452–460.CrossRefGoogle Scholar
[9] Prohorov, Yu. (1956) Convergence of random processes and limit theorems in probability. Theor. Probability Appl. 1, 157–214.CrossRefGoogle Scholar
[10] Prohorov, Yu. (1963) Transient phenomena in processes of mass service. Litovsk. Mat. Sb. 3, 199–205. (In Russian.) Google Scholar
[11] Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. John Wiley, New York.Google Scholar
[12] Viskov, O. (1964) Two asymptotic formulae in the theory of mass service. Theor. Probability Appl. 9, 177–178.CrossRefGoogle Scholar
[13] Whitt, W. (1968) Weak Convergence Theorems for Queues in Heavy Traffic. , Stanford University.Google Scholar