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The MacLaurin series for the GI/G/1 queue

Part of: Special processes Approximations and expansions

Published online by Cambridge University Press:  14 July 2016

Wei-Bo Gong  and
Jian-Qiang Hu
Wei-Bo Gong*
Affiliation:
University of Massachusetts, Amherst
Jian-Qiang Hu*
Affiliation:
Boston University
*
∗ Postal address: Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003, USA.
∗∗ Postal address: Department of Manufacturing Engineering, Boston University, Boston, MA 02215, USA.
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Abstract

We derive the MacLaurin series for the moments of the system time and the delay with respect to the parameters in the service time or interarrival time distributions in the GI/G/1 queue. The coefficients in these series are expressed in terms of the derivatives of the interarrival time density function evaluated at zero and the moments of the service time distribution, which can be easily calculated through a simple recursive procedure. The light traffic derivatives can be obtained from these series. For the M/G/1 queue, we are able to recover the formulas for the moments of the system time and the delay, including the Pollaczek–Khinchin mean-value formula.

Keywords

SYSTEM TIMEDELAYPOLLACZEK–KHINCHIN FORMULA

MSC classification

Primary: 60K25: Queueing theory
Secondary: 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 1 , March 1992 , pp. 176 - 184
Copyright
Copyright © Applied Probability Trust 1992 

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References

[1]Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
[2]Beneš, V. (1965) Mathematical Theory of Connecting Networks and Telephone Traffic. Academic Press, New York.Google Scholar
[3]Blanc, J. P. C. (1990) A numerical approach to cyclic-server queueing models. QUESTA 6, 173–188.Google Scholar
[4]Hooghiemstra, G., Keane, M. and Van De Ree, S. (1988) Power series for stationary distributions of coupled processor models. SIAM J. Appl. Math. 48, 1159–1166.Google Scholar
[5]Kleinrock, L. (1975) Queueing Systems, Vol. 1. Wiley Interscience, New York.Google Scholar
[6]Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.Google Scholar
[7]Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. The Johns Hopkins University Press, Baltimore.Google Scholar
[8]Reiman, M. and Simon, B. (1989) Open queueing systems in light traffic. Math. Operat. Res. 14, 26–59.Google Scholar
[9]Reiman, M. and Weiss, A. (1989) Light traffic derivatives via likelihood ratios. IEEE Trans. Inf. Technol. 35, 648–654.Google Scholar
[10]Stoyan, D. (1983) Comparison Methods for Queueing and Other Stochastic Models. Wiley, New York.Google Scholar
[11]Takács, L. (1962) A single-server queue with Poisson input. Operat. Res. 10, 388–397.Google Scholar
[12]Fendick, W. and Whitt, W. (1989) Measurements and approximations to describe the offered traffic and predict the average workload in a single-server queue. Proc. IEEE 77, 171–194.Google Scholar