Hostname: page-component-6898cfcb55-w5hgh Total loading time: 0 Render date: 2026-11-11T21:16:26.775Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get accessRights & Permissions [Opens in a new window]

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

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 1 , March 1978, pp. 213 - 231
Copyright
Copyright © Applied Probability Trust 1978 

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.)

Article purchase

Temporarily unavailable

References

Arjas, E. (1972) On a fundamental identity in the theory of semi-Markov processes. Adv. Appl. Prob. 4, 271–284.Google 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.Google 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.Google 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.Google 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.Google Scholar
Kingman, J. F. C. (1961) The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 57, 902–904.Google 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.Google 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.Google 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.Google 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.Google 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