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A conservation law for single-server queues and its applications

Published online by Cambridge University Press:  14 July 2016

Genji Yamazaki ,
Hirotaka Sakasegawa  and
J. George Shanthikumar
Genji Yamazaki*
Affiliation:
Tokyo Metropolitan Institute of Technology
Hirotaka Sakasegawa*
Affiliation:
University of Tsukuba
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
*
∗Postal address: Department of Engineering Management, Tokyo Metropolitan Institute of Technology, Hino-city, Tokyo 191, Japan.
∗∗Postal address: Institute of Socio-Economic Planning, University of Tsukuba, Tsukuba City, Ibaraki 305, Japan.
∗∗∗Postal address: School of Business Administration, University of California, Berkeley, CA 94720, USA.
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Abstract

We establish a conservation law for G/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in a G/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in a GI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

Keywords

FCFSLIPS DISCIPLINESNBUENWUEAPPROXIMATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 1 , March 1991 , pp. 198 - 209
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

Supported in part by a grant from the Tokyo Government.

Supported in part by a grant from the International Information Foundation of Japan.

References

Avi-Itzhak, B. and Halfin, S. (1987) Server sharing with a limited number of service positions and symmetric queues. J. Appl. Prob. 24, 990–1000.Google Scholar
Cox, D. R. (1955) A use of complex probabilities in the theory of stochastic processes. Proc. Camb. Phil. Soc. 51, 313–319.Google Scholar
Franken, P., König, D., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes, Academie-Verlag Berlin; Wiley, New York.Google Scholar
Halfin, S. and Whitt, W. (1990) An extremal property of the FIFO discipline via an ordinal version of L = ?W. Stochastic Models. To appear.CrossRefGoogle Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, Chichester.Google Scholar
Miyazawa, M. (1983) The derivation of invariance relations in complex queueing systems with stationary inputs. Adv. Appl. Prob. 15, 874–885.Google Scholar
Miyazawa, M. (1986) Approximation of the queue-length distribution of an M/GI/s queue by the basic equations. J. Appl. Prob. 23, 443–458.CrossRefGoogle Scholar
Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore.Google Scholar
Rege, K. M. and Sengupta, B. (1985) A priority-based admission scheme for a multiclass queueing system. AT&T Tech. J. 64, 1731–1753.Google Scholar
Righter, R., Shanthikumar, J. G. and Yamazaki, G. (1990) On extremal service disciplines in single stage queueing systems. J. Appl. Prob. 27, 409–416.Google Scholar
Shanthikumar, J. G. and Sumita, U. (1986) On G/G/1 queue with LIFO-P service discipline. J. Operat. Res. Soc. Japan 29, 220–231.Google Scholar
Yamazaki, G. and Sakasegawa, H. (1986) The effect of multiplicity in limited processor-sharing systems (in Japanese). Trans. Inf. Process. Soc. Japan 27, 1–9.Google Scholar
Yamazaki, G, and Sakasegawa, H. (1987a) Limited processor-sharing discipline in GI/GI/1 models, Discussion Paper 321, Institute of Socio-Economic Planning, University of Tsukuba.Google Scholar
Yamazaki, G, and Sakasegawa, H. (1987b) An optimal design problem for limited processor-sharing systems. Management Sci. 33, 1010–1019.Google Scholar