Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-26T05:58:21.568Z Has data issue: false hasContentIssue false
  • English
  • Français

Light traffic approximations in many-server queues

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

D. J. Daley  and
T. Rolski
D. J. Daley*
Affiliation:
Australian National University
T. Rolski*
Affiliation:
University of Wrocław
*
∗Postal address: Statistics Research Section, SMS, Australian National University, GPO Box 4, Canberra ACT 2601, Australia.
∗∗Postal address: Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2–4, 50–384 Wrocław, Poland.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper complements two previous studies (Daley and Rolski (1984), (1991)) by investigating limit properties of the waiting time in k-server queues with renewal arrival process under ‘light traffic' conditions. Formulae for the limits of the probability of waiting and the waiting time moments are derived for the two approaches of dilation and thinning of the arrival process. Asmussen's (1991) approach to light traffic limits applies to the cases considered, of which the Poisson arrival process (i.e. M/G/k) is a special case and for which formulae are given.

Keywords

GI/GI/k QUEUEM/G/k QUEUESTATIONARY POINT PROCESS WITH MULTIPLE POINTSMULTIVARIATE ABELIAN LEMMA

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G55: Point processes
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 1 , March 1992 , pp. 202 - 218
Copyright
Copyright © Applied Probability Trust 1992 

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.)

Footnotes

Research carried out in part while visiting the Mathematical Institute, University of Wrocław.

Research carried out in part while visiting Department of Mathematics and Statistics, Case Western Reserve University at Cleveland, Ohio.

References

Asmussen, S. (1991) Light traffic equivalence in single server queues. Ann. Appl. Prob. 1.CrossRefGoogle Scholar
Boxma, O. J., Cohen, J. W. and Huffels, N. (1979) Approximation of the mean waiting time in an M/G/s queueing system. Operat. Res. 27, 1115–1127.CrossRefGoogle Scholar
Burman, D. Y. and Smith, D. R. (1983) A light traffic theorem for multi-server queues. Math. Operat. Res. 8, 15–25.CrossRefGoogle Scholar
Daley, D. J. and Rolski, T. (1984) A light traffic approximation for a single-server queue. Math. Operat. Res. 9, 624–628.CrossRefGoogle Scholar
Daley, D. J. and Rolski, T. (1991) Light traffic approximations in queues. Math. Operat. Res. 16, 57–71.CrossRefGoogle Scholar
Daley, D. J. and Rolski, T. (1992) Light traffic approximations in general stationary single-server queues. Submitted.CrossRefGoogle Scholar
Kiefer, J. and Wolfowitz, J. (1955) On the theory of queues with many servers. Trans. Amer. Math. Soc. 78, 1–18.CrossRefGoogle Scholar
Kiefer, J. and Wolfowitz, J. (1956) On the characteristics of the general queueing process with application to random walks. Ann. Math. Statist. 27, 147–161.CrossRefGoogle Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Processes. Wiley, Chichester.Google Scholar
Whitt, W. (1989) An interpolation approximation for the mean workload in a GI/G/1 queue. Operat. Res. 37, 936–952.CrossRefGoogle Scholar