Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-30T21:05:15.886Z Has data issue: false hasContentIssue false
  • English
  • Français

The mean waiting time of a GI/G/1 queue in light traffic via random thinning

Part of: Operations research and management science Approximations and expansions Special processes

Published online by Cambridge University Press:  14 July 2016

Soracha Nananukul  and
Wei-Bo Gong
Soracha Nananukul*
Affiliation:
University of Massachusetts at Amherst
Wei-Bo Gong*
Affiliation:
University of Massachusetts at Amherst
*
∗Postal address: Department of Electrical and Computer Engineering, University of Massachusetts at Amherst, Amherst, MA 01003, USA.
∗Postal address: Department of Electrical and Computer Engineering, University of Massachusetts at Amherst, Amherst, MA 01003, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we derive the MacLaurin series of the mean waiting time in light traffic for a GI/G/1 queue. The light traffic is defined by random thinning of the arrival process. The MacLaurin series is derived with respect to the admission probability, and we prove that it has a positive radius of convergence. In the numerical examples, we use the MacLaurin series to approximate the mean waiting time beyond light traffic by means of Padé approximation.

Keywords

MACLAURIN SERIESWAITING TIMEPADÉ APPROXIMATION

MSC classification

Primary: 60K25: Queueing theory
Secondary: 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series) 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 256 - 266
Copyright
Copyright © Applied Probability Trust 1995 

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] Baker, G. A. and Graves-Morris, P. (1981) Padé Approximants, Part I: Basic Theory. Addison-Wesley, Reading, MA.Google Scholar
[2] Chen, C. T. (1975) Analysis and Synthesis of Linear Control Systems. Holt, Rinehart and Winston, New York.Google Scholar
[3] Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[4] Daley, D. J. and Rolski, T. (1984) A light traffic approximation for a single-server queue. Math. Operat. Res. 9, 624–628.Google Scholar
[5] Daley, D. J. and Rolski, T. (1991) Light traffic approximations in queues. Math. Operat. Res. 16, 57–71.Google Scholar
[6] Daley, D. J. and Rolski, T. (1992) Light traffic approximations in many-server queues. Adv. Appl. Prob. 24, 202–218.Google Scholar
[7] Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Springer-Verlag, Berlin.Google Scholar
[8] Gong, W. B. and Hu, J. Q. (1992) The MacLaurin series for the GI/G/1 queues. J. Appl. Prob. 29, 176–184.Google Scholar
[9] Gong, W. B., Nananukul, S. and Yan, A. (1992) Padé approximation for stochastic discrete event systems. Proc. 31st CDC, Tucson, Arizona, 3209–3214.Google Scholar
[10] Hu, J. Q. (1994) Analyticity of single-server queues in light traffic. Unpublished.Google Scholar
[11] Reiman, M. I. and Simon, B. (1989) Open queueing systems in light traffic. Math. Operat. Res. 14, 26–59.CrossRefGoogle Scholar
[12] Reiman, M. L, Simon, B. and Willie, J. S. (1992) Simterpolation: A simulation based interpolation approximation for queueing systems. Operat. Res. 40, 706–723.Google Scholar
[13] Whitt, W. (1989) An interpolation approximation for the mean workload in a GI/G/1 queue. Operat. Res. 37, 936–952.CrossRefGoogle Scholar
[14] Wolff, R. W. (1988) Stochastic Modeling and the Theory of Queues. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[15] Wolfram, S. (1991) Mathematica: A System for Doing Mathematics by Computer, 2nd edn. Addison-Wesley, Reading, MA.Google Scholar