Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T04:20:52.190Z Has data issue: false hasContentIssue false
  • English
  • Français

Idle period approximations and bounds for the GI/G/1 queue

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Ronald W. Wolff  and
Chia-Li Wang
Ronald W. Wolff*
Affiliation:
University of California, Berkeley
Chia-Li Wang*
Affiliation:
National Dong Hwa University
*
∗ Postal address: Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720, USA.
∗∗ Postal address: Department of Applied Mathematics, National Dong Hwa University, Hualien, Taiwan, ROC. Email address: cwang@mail.ndhu.edu.tw
Get access Rights & Permissions [Opens in a new window]

Abstract

The average delay for the GI/G/1 queue is often approximated as a function of the first two moments of interarrival and service times. For highly irregular arrivals, however, it varies widely among queues with the same first two moments, even in moderately heavy traffic. Empirically, it decreases as the interarrival time third moment increases. For GI/M/1 queues, a heavy-traffic expression for the average delay with this property has been previously obtained. The method, however, sheds little light on why the third moment arises. We analyze the equilibrium idle-period distribution in heavy traffic using real-variable methods. For GI/M/1 queues, we derive the above heavy-traffic result and also obtain conditions under which it is either an upper or lower bound. Our approach provides an intuitive explanation for the result and also strongly suggests that similar results should hold for general service. This is supported by empirical evidence. For any given service distribution, it has been conjectured that the expected delay under pure-batch arrivals, where interarrival times are scaled Bernoulli random variables, is an upper bound on the average delay over all interarrival distributions with the same first two moments. We investigate this conjecture and show, among other things, that pure-batch arrivals have the smallest third moment. We obtain conditions under which this conjecture is true and present a counterexample where it fails. Arrivals that arise as overflows from other queues can be highly irregular. We show that interoverflow distributions in a certain class have decreasing failure rate.

Keywords

GI/G/1 queueidle periodapproximationboundIMRLDMRLIFRDFRequilibrium distributionbatch arrivaloverflow process

MSC classification

Primary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 3 , September 2003 , pp. 773 - 792
Copyright
Copyright © Applied Probability Trust 2003 

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] Daley, D. J. (1977). Inequalities for moments of tails of random variables, with queueing applications. Z. Wahrscheinlichkeitsth. 41, 139–143.Google Scholar
[2] Daley, D. J. and Rolski, T. (1984). A light traffic approximation for a single-server queue. Math. Operat. Res. 9, 624–628.Google Scholar
[3] Daley, D. J. and Trengove, C. D. (1977). Bounds for Mean Waiting Times in Single-Server Queues: A Survey. Statistics Department, Australian National University, Canberra.Google Scholar
[4] Daley, D. J., Kreinin, A. Ya. and Trengove, C. D. (1992). Inequalities concerning the waiting-time in single-server queues: a survey. In Queuing and Related Models (Oxford Statist. Sci. Ser. 9), eds Bhat, U. N. and Basawa, I. V., Oxford University Press, pp. 177–233.Google Scholar
[5] Guerin, R. and Lien, L. Y.-C. (1990). Overflow analysis for finite waiting room systems. IEEE Trans. Commun. 38, 1569–1577.Google Scholar
[6] Halfin, S. (1985). Delays in queues, properties and approximations. In Teletraffic Issues in an Advanced Information Society, ed. Akiyama, M., North-Holland, Amsterdam, pp. 47–52.Google Scholar
[7] Kingman, J. F. C. (1962). Some inequalities for the GI/G/1 queue. Biometrika 49, 315–324.CrossRefGoogle Scholar
[8] Kingman, J. F. C. (1970). Inequalities in the theory of queues. J. R. Statist. Soc. B 32, 102–110.Google Scholar
[9] Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
[10] Marshall, K. T. (1968). Some inequalities in queuing. Operat. Res. 16, 651–665.CrossRefGoogle Scholar
[11] Wang, C.-L. and Wolff, R. W. (2003). Efficient simulation of queues in heavy traffic. ACM Trans. Modeling Comput. Simulation 13, 62–81.Google Scholar
[12] Whitt, W. (1989). An interpolation approximation for the mean workload in a GI/G/1 queue. Operat. Res. 6, 936–952.CrossRefGoogle Scholar
[13] Wolff, R. W. (1989). Stochastic Modeling and the Theory of Queues. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar