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Characterizing the idle-period distribution of GI/G/1 queues

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Jingwen Li  and
Jihong Ou
Jingwen Li
Affiliation:
National University of Singapore
Jihong Ou*
Affiliation:
National University of Singapore
*
∗ Postal address: Department of Decision Sciences, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511.
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Abstract

A variety of performance measures of a GI/G/1 queue are explicitly related to the idle-period distribution of the queue, suggesting that the system analysis can be accomplished by the analysis of the idle period. However, the ‘stand-alone' relationship for the idle-period distribution of the GI/G/1 queue (i.e. the counterpart of Lindley's equation) has not been found in the literature. In this paper we develop a non-linear integral equation for the idle period distribution of the GI/G/1 queue. We also show that this non-linear system defines a unique solution. This development makes possible the analysis of the GI/G/1 queue in a different perspective.

Keywords

PREEMPTIVE-RESUME LIFONON-LINEAR INTEGRAL EQUATIONRENEWAL PROCESS

MSC classification

Primary: 60K25: Queueing theory

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 247 - 255
Copyright
Copyright © Applied Probability Trust 1995 

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References

Cooper, R. B. and Niu, S.-C. (1986) Benes' formula for M/G/1-FIFO explained by preemptiveresume LIFO. J. Appl. Prob. 23, 550–554.Google Scholar
Fakinos, D. (1981) The G/G/1 queueing system with a particular queue discipline. J. R. Statist. Soc. B43, 190–196.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory And Its Applications. Volume 2, 2nd edn. Wiley, New York.Google Scholar
Grassmann, W. K. and Jain, J. L. (1989) Numerical solution of the waiting time distribution and idle time distribution of the arithmetic GI/G/1 queue. Operat. Res. 37, 141–150.Google Scholar
Gross, D. and Harris, C. M. (1985) Fundamentals of Queueing Theory, 2nd edn. Wiley, New York.Google Scholar
Heyman, D. P. and Sobel, M. J. (1982) Stochastic Models in Operations Research, Volume 2. McGraw-Hill, New York.Google Scholar
Lindley, D. V. (1952) On the theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.Google Scholar
Marshall, K. T. (1968) Some inequalities in queueing. Operat. Res. 16, 651–665.Google Scholar
Niu, S.-C. (1988) Representing workloads in GI/G/1 queues through the preemptive-resume LIFO queue discipline. Queueing Systems 3, 157–178.Google Scholar
Smith, W. L. (1953) On the distribution of queueing times. Proc. Camb. Phil. Soc. 49, 449–461.Google Scholar