Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T10:17:19.419Z Has data issue: false hasContentIssue false
  • English
  • Français

The heavy traffic approximation for single server queues in series

Published online by Cambridge University Press:  14 July 2016

J. Michael Harrison
Get access Rights & Permissions [Opens in a new window]

Abstract

A tandem queue with K single server stations and unlimited interstage storage is considered. Customers arrive at the first station in a renewal process, and the service times at the various stations are mutually independent i.i.d. sequences. The central result shows that the equilibrium waiting time vector is distributed approximately as a random vector Z under traffic conditions (meaning that the system traffic intensity is near its critical value). The weak limit Z is defined as a certain functional of multi-dimensional Brownian motion. Its distribution depends on the underlying interarrival and service time distributions only through their first two moments. The outstanding unsolved problem is to determine explicitly the distribution of Z for general values of the relevant parameters. A general computational approach is demonstrated and used to solve for one special case.

Keywords

TANDEM QUEUEHEAVY TRAFFIC APPROXIMATIONDIFFUSION APPROXIMATIONWEAK CONVERGENCE
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 3 , September 1973 , pp. 613 - 629
Copyright
Copyright © Applied Probability Trust 1973 

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] Billingsley, P. (1968) Convergence of Probability Measures. John Wiley and Sons, New York.Google Scholar
[2] Iglehart, D. L. and Kennedy, D. P. (1970) Weak convergence of the average of flag processes. J. Appl. Prob. 7, 747–753.Google Scholar
[3] Iglehart, D. L. and Whitt, W. (1971) The equivalence of functional central limit theorems for counting processes and associated partial sums. Ann. Math. Statist. 42, 1372–1378.Google Scholar
[4] Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic. I. Adv. Appl. Prob. 2, 150–177.Google Scholar
[5] Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic. II: sequences, networks, and batches. Adv. Appl. Prob. 2, 355–369.Google Scholar
[6] Kingman, J. F. C. (1961) The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 57, 902–904.Google Scholar
[7] Lindley, D. (1952) Theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.Google Scholar
[8] Parthasarathy, K. R. (1967) Probability Measures on Metric Spaces. Academic Press, New York.Google Scholar
[9] Prohorov, Y. V. (1956) Convergence of random processes and limit theorems in probability theory. Theor. Probability Appl. 1, 157–214.Google Scholar
[10] Saaty, T. L. (1961) Elements of Queueing Theory. McGraw-Hill, New York.Google Scholar
[11] Whitt, W. (1970) Weak convergence of probability measures on the function space D[0, 8). Technical Report, Department of Administrative Sciences, Yale University.Google Scholar
[12] Whitt, W. (1971) Obtaining convergence of limiting distributions from weak convergence of stochastic processes. Technical Report, Department of Administrative Sciences, Yale University.Google Scholar