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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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