Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-25T12:33:03.427Z Has data issue: false hasContentIssue false
  • English
  • Français

Heavy traffic theory for queues with several servers. II

Published online by Cambridge University Press:  14 July 2016

Julian Köllerström
Julian Köllerström*
Affiliation:
University of Kent at Canterbury
*
∗Postal address: Mathematical Institute, Cornwallis Building, The University, Canterbury, Kent, CT2 7NF, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

The queues being studied here are of the type GI/G/k in statistical equilibrium (with traffic intensity less than one). The exponential limiting formula for the waiting time distribution function in heavy traffic, conjectured by Kingman (1965) and established by Köllerström (1974), is extended here. The asymptotic properties of the moments are investigated as well as further approximations for the characteristic function and error bounds for the limiting foemulae.

Keywords

QUEUE GI/G/kWAITING TIMEEQUILIBRIUM DISTRIBUTIONEXPONENTIAL LIMIT FORMULAMOMENTSERROR BOUNDSCHARACTERISTIC FUNCTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 2 , June 1979 , pp. 393 - 401
Copyright
Copyright © Applied Probability Trust 1979 

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

Blomqvist, N. (1974) A simple derivation of the GI/G/1 waiting time distribution in heavy traffic. Scand. J. Statist. 1, 39–40.Google Scholar
Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
Daley, D. J. (1978) Bounds on mean waiting times in GI/G/1 queues. Z. Wahrscheinlichkeitsth. Google Scholar
Kiefer, J. and Wolfowitz, J. (1955) On the theory of queues with many servers. Trans. Amer. Math. Soc. 78, 1–18.Google Scholar
Kiefer, J. and Wolfowitz, J. (1956) On the characteristics of the general queueing process, with applications to random walk. Ann. Math. Statist. 27, 147–161.Google 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. (1962a) On queues in heavy traffic. J. R. Statist. Soc. B, 24, 383–392.Google Scholar
Kingman, J. F. C. (1962b) Some inequalities for the queue GI/G/1. Biometrika 49, 315–324.Google Scholar
Kingman, J. F. C. (1965) The heavy traffic approximation in the theory of queues. In Proc. Symp. Congestion Theory, ed. Smith, W. L. and Wilkinson, W. E. University of North Carolina Press, Chapel Hill, N. C. Google Scholar
Kingman, J. F. C. (1966) On the algebra of queues. J. Appl. Prob. 3, 285–326.CrossRefGoogle Scholar
Köllerström, J. (1974) Heavy traffic theory for queues with several servers. I. J. Appl. Prob. 11, 544–552.Google Scholar
Köllerström, J. (1976a) Some Analysis in the Theory of Queues with Several Servers. D. Phil. Thesis, University of Oxford.Google Scholar
Köllerström, J. (1976b) Stochastic bounds for the single-server queue. Math. Proc. Camb. Phil. Soc. 80, 521–525.Google Scholar
Lemoine, A. J. (1976) On random walks and stable GI/G/1 queues. Maths. Opns. Res. 1, 159–164.Google Scholar
Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.Google Scholar
Loeve, M. (1963) Probability Theory. Van Nostrand, Princeton, N.J.Google Scholar
Loulou, R. (1971) Weak Convergence for Multichannel Queues in Heavy Traffic. Ph.D. Thesis, University of California, Berkeley.Google Scholar
Loulou, R. (1973) Multiple-channel queues in heavy traffic. J. Appl. Prob. 10, 769–777.Google Scholar
Pollaczek, F. (1961) Théorie analytique des problèmes stochastiques relatifs à un groupe de lignes téléphoniques avec dispositif d'attente. Mémor. Sci. Math. 150.Google Scholar
Whitt, W. (1973a) Exponential heavy traffic approximations for multi-server queues. Private communication.Google Scholar
Whitt, W. (1973b) Heavy traffic limit theorems for queues: a survey. Prepared for the Conference on Mathematical Methods in Queueing Theory at Western Michigan University, Kalamazoo, Michigan, 1973; Administrative Sciences, Yale.Google Scholar
Yu, O. S. (1972) On the steady-state solution of a GI/Ek/r queue with heterogeneous servers. T.R. No. 38, Dept. of Operations Research, Stanford University.Google Scholar
Yu, O. S. (1973) Stochastic bounding relations for a heterogeneous-server queue with Erlang service times. Stanford Research Institute, Menlo Park, California.Google Scholar
Zygmund, A. (1947) A remark on characteristic functions. Ann. Math. Statist. 18, 272–276.Google Scholar