Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T18:09:29.321Z Has data issue: false hasContentIssue false
  • English
  • Français

The continuity of the single server queue

Published online by Cambridge University Press:  14 July 2016

Douglas P. Kennedy
Douglas P. Kennedy*
Affiliation:
University of Sheffield
Get access Rights & Permissions [Opens in a new window]

Abstract

In many applications of queueing theory assumptions of either Poisson arrivals or exponential service times are made. The implicit assumption is that if the actual arrival process approximates a Poisson process and the service times are close to exponential, then the quantities of interest in the real queueing system (viz. the virtual waiting time, queue length, idle times, etc.), will approximate those of the idealized model. The continuity of the single server queue acting as functionals of the arrival and service processes is established. The proof involves an application of the theory of weak convergence of probability measures on metric spaces.

Keywords

WEAK CONVERGENCEQUEUESFUNCTIONAL LIMIT THEOREMS
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 2 , June 1972 , pp. 370 - 381
Copyright
Copyright © Applied Probability Trust 1972 

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

[1] Benes, V. (1963) General Stochastic Processes in the Theory of Queues. Addison-Wesley, Reading, Massachusetts.Google Scholar
[2] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[3] Kennedy, D. (1970) Weak convergence for the superposition and thinning of point processes. Technical Report No. 11, Department of Operations Research, Stanford University.Google Scholar
[4] Khintchine, A. (1955) Mathematical Methods in the Theory of Queueing. Translation (1960) Griffin, London.Google Scholar
[5] Schassberger, R. (1970) On the waiting time in the queueing system GI/G/1. Ann. Math. Statist. 41, 182–187.CrossRefGoogle Scholar
[6] Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar
[7] Whitt, W. (1968) Weak Convergence Theorems for Queues in Heavy Traffic. , Cornell University. (Technical Report No. 2, Department of Operations Research, Stanford University.) Google Scholar
[8] Whitt, W. (1970) Weak convergence of probability measures on the function space D[0, ∞). Technical Report, Yale University.Google Scholar