Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T08:32:36.586Z Has data issue: false hasContentIssue false
  • English
  • Français

The stable M/G/1 queue in heavy traffic and its covariance function

Published online by Cambridge University Press:  01 July 2016

Teunis J. Ott
Teunis J. Ott*
Affiliation:
Case Western Reserve University
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X(t) be the virtual waiting-time process of a stable M/G/1 queue. Let R(t) be the covariance function of the stationary process X(t), B(t) the busy-period distribution of X(t); and let E(t) = P{X(t) = 0|X(0) = 0}.

For X(t) some heavy-traffic results are given, among which are limiting expressions for R(t) and its derivatives and for B(t) and E(t).

These results are used to find the covariance function of stationary Brownian motion on [0, ∞).

Keywords

QUEUEING THEORYM/G/1 QUEUEHEAVY TRAFFIC APPROXIMATIONSVIRTUAL WAITING-TIME PROCESSESCOVARIANCE FUNCTIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 1 , March 1977 , pp. 169 - 186
Copyright
Copyright © Applied Probability Trust 1977 

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] Abramowitz, M. and Stegum, I. A. Handbook of Mathematical Functions. U. S. Government Printing Office, Washington, D.C. Google Scholar
[2] Borovkov, A. A. (1967) On limit laws for service processes in multi-channel systems. Siberia J. Math. 8, 746–763.Google Scholar
[3] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1954) Tables of Integral Transforms, I. McGraw-Hill, New York.Google Scholar
[4] Feller, W. (1966) An Introduction to Probability Theory and its Applications, II. Wiley, New York.Google Scholar
[5] Iglehart, D. L. (1973) Weak convergence in queueing theory. Adv. Appl. Prob. 5, 570–594.Google Scholar
[6] Keilson, J. and Kooharian, A. (1960) On time-dependent queueing processes. Ann. Math. Statist. 31, 104–112.Google Scholar
[7] Kennedy, D. P. (1972) Rates of convergence for queues in heavy traffic, I. Adv. Appl. Prob. 4, 357–381.Google Scholar
[8] Loève, M. (1955) Probability Theory. Van Nostrand.Google Scholar
[9] Ott, T. J. (1977) The covariance function of the virtual waiting-time process in an M/G/1 queue. Adv. Appl. Prob. 9, 158–168.Google Scholar
[10] Ott, T. J. (1975) The stable M/G/1 queue in heavy traffic and its covariance function, with applications to stationary Brownian motion on [0, ∞]. Technical Memorandum No. 387, Department of Operations Research, Case Western Reserve University.Google Scholar
[11] Parthasarathy, K. R. (1967) Probability Measures on Metric Spaces. Academic Press, New York.Google Scholar
[12] Prohorov, Yu. V. (1963) Transient phenomena in processes of mass service. Litovsk Mat. Sb. 3, 199–205.Google Scholar