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On the transient delays of M/G/1 queues

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Chia-Li Wang
Chia-Li Wang*
Affiliation:
National Dong Hwa University, Taiwan
*
∗Postal address: Department of Applied Mathematics, National Dong Hwa University, Hualien, Taiwan, ROC. Email address: [email protected]
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Abstract

A recent study on the GI/G/1 queue derives the Maclaurin series for the moments of the waiting time and the delay to respect to some parameters. By the same approach, we obtain an identity on the moments of the transient delay of the M/G/1 queue. This identity allows us to understand the transient behavior of the process better. We apply the identity with other established results to study convergence rate and stochastic concavity of the transient delay process, and to derive bounds and approximations of the moments. Our approximation and bound both have simple closed forms and are asymptotically exact as either the traffic intensity goes to zero or the process approaches stationarity. Performance of the approximation of several M/G/1 queues is illustrated by numerical experiments. It is interesting to note that our results can also help to gain variance reduction in simulation.

Keywords

M/G/1 queuetransient delayapproximationsboundsconcavityladder epochssimulationvariance reduction

MSC classification

Primary: 60K25: Queueing theory
Secondary: 68U20: Simulation
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 882 - 893
Copyright
Copyright © Applied Probability Trust 1999 

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References

Asmussen, S. (1987). Applied Probability and Queues. Wiley, New York.Google Scholar
Asmussen, S., and Teugels, J. L. (1996). Convergence rates for M/G/1 queues and ruin problems with heavy tails. J. Appl. Prob. 33, 1183–1190.Google Scholar
Asmussen, S., and Wang, C-L. (1996). Variance reduction for simulating transient GI/G/1 behavior. Prob. Eng. Inf. Sci. 10, 197–205.CrossRefGoogle Scholar
Bertsimas, D. J., and Nakazato, D. (1992). Transient and busy period analysis of the GI/G/1 queue: the method of stages. Queueing Systems 10, 153–184.CrossRefGoogle Scholar
Gong, W-B., and Hu, J-Q. (1992). The Maclaurin series for the GIz/G/1 queue. J. Appl. Prob. 29, 176–184.CrossRefGoogle Scholar
Kelton, W. D., and Law, A. M. (1985). The transient behavior of the M/M/s queue, with applications for steady-state simulation. Operat. Res. 33, 378–395.Google Scholar
Murray, J. R., and Kelton, W. D. (1988). The transient behavior of the M/Ek /2 queue and steady-state simulation. Comput. Operat. Res. 15, 357–367.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1990). Convexity of a set of stochastically ordered random variables. Adv. Appl. Prob. 22, 160–177.Google Scholar
Wang, C-L. (1997). Light traffic approximations for regenerative queueing processes. Adv. Appl. Prob. 29, 1060–1080.Google Scholar
Wolff, R. W. (1989). Stochastic Modeling and the Theory of Queues. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar