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Stability and heavy traffic results for the general bulk queue

Published online by Cambridge University Press:  01 July 2016

John Dagsvik
John Dagsvik*
Affiliation:
Central Bureau of Statistics, Oslo
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Abstract

In this paper we prove that the limiting distribution of the general bulk queue exists and is independent of the initial conditions if and only if the traffic intensity is less than one. We further generalize the following heavy traffic results of the GI/G/1 model to the general bulk queue model. When ρ > 1 or ρ = 1 the waiting time is distributed approximately as a Gaussian variable and the absolute value of a Gaussian variable, respectively. The exponential approximation is derived from the Wiener–Hopf matrix equations established in a previous paper while the unstable case ρ ≧ 1 is treated by means of functional central limit theorems for mixing processes.

Keywords

BULK QUEUESCHAIN-DEPENDENT VARIABLESMATRIX WIENER–HOPF EQUATIONSSEMI-MARKOV KERNELSHEAVY TRAFFIC APPROXIMATIONSWEAK CONVERGENCEFUNCTIONAL CENTRAL LIMIT THEOREMSMIXING PROCESSES
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 1 , March 1978 , pp. 213 - 231
Copyright
Copyright © Applied Probability Trust 1978 

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References

Arjas, E. (1972) On a fundamental identity in the theory of semi-Markov processes. Adv. Appl. Prob. 4, 271–284.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Dagsvik, J. (1975a) The general bulk queue as a matrix factorisation problem of the Wiener–Hopf type. Part I. Adv. Appl. Prob. 7, 636–646.CrossRefGoogle Scholar
Dagsvik, J. (1975b) The general bulk queue as a matrix factorisation problem of the Wiener-Hopf type. Part II. Adv. Appl. Prob. 7, 647–655.CrossRefGoogle Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and Its Applications. Wiley, New York.Google Scholar
Iglehart, D. L. and Whitt, W. (1970a) Multiple channel queues in heavy traffic, I. Adv. Appl. Prob. 2, 150–177.CrossRefGoogle Scholar
Iglehart, D. L. and Whitt, W. (1970b) Multiple channel queues in heavy traffic, II. Sequences, networks and batches. Adv. Appl. Prob. 2, 355–369.CrossRefGoogle Scholar
Kingman, J. F. C. (1961) The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 57, 902–904.CrossRefGoogle Scholar
Kingman, J. F. C. (1962) On queues in heavy traffic. J. R. Statist. Soc. B 24, 383–392.Google Scholar
Ito, K. and McKean, H. P. (1965) Diffusion Processes and their Sample Paths. Springer Verlag, Berlin.Google Scholar
Keilson, J. (1962) The general bulk queue as a Hilbert problem. J. R. Statist. Soc. B 24, 344–359.Google Scholar
Keilson, J. and Wishart, D. M. G. (1964) A central limit theorem for processes defined on a finite Markov chain. Proc. Camb. Phil. Soc. 60, 547–567.CrossRefGoogle Scholar
Lambotte, J. P. and Teghem, J. L. (1969) Modèles d'attente M/G/1 et GI/M/1 à arrivées et services en groupes. Springer Verlag, Berlin.Google Scholar
Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–283.CrossRefGoogle Scholar
Loynes, R. M. (1962) The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.CrossRefGoogle Scholar
Miller, H. D. (1961) A convexity property in the theory of random variables defined on a finite Markov-chain. Ann. Math. Statist. 32, 1260–1270.CrossRefGoogle Scholar
Miller, R. G. (1959) A contribution to the theory of bulk queues. J. R. Statist. Soc. B 21, 320–337.Google Scholar
O'Brien, G. L. and Denzel, G. E. (1975) Limit theorems for extreme values of chain-dependent processes. Ann. Prob. 773–779.Google Scholar