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Approximation of the queue-length distribution of an M/GI/s queue by the basic equations

Published online by Cambridge University Press:  14 July 2016

Masakiyo Miyazawa
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
*
∗Postal address: Department of Information Sciences, Faculty of Science and Technology, Science University of Tokyo, Noda City, Chiba 278, Japan.
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Abstract

We give a unified way of obtaining approximation formulas for the steady-state distribution of the queue length in the M/GI/s queue. The approximations of Hokstad (1978) and Case A of Tijms et al. (1981) are derived again. The main interest of this paper is in considering the theoretical meaning of the assumptions given in the literature. Having done this, we derive new approximation formulas. Our discussion is based on one version of the steady-state equations, called the basic equations in this paper. The basic equations are derived for M/GI/s/k with finite and infinite k. Similar approximations are possible for M/GI/s/k (k < +∞).

Keywords

APPROXIMATION FORMULAQUEUE LENGTHSTEADY-STATE DISTRIBUTIONPOISSON ARRIVAL
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 2 , June 1986 , pp. 443 - 458
Copyright
Copyright © Applied Probability Trust 1986 

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References

Boxma, O. J., Cohen, J. W. and Huffels, N. (1980) Approximations of the mean waiting time in an M/G/s queueing system. Operat. Res. 27, 1115–1127.Google Scholar
Burman, D. Y. and Smith, D. R. (1983) A light-traffic theorem for multi-server queues. Math. Operat. Res. 8, 15–25.Google Scholar
Groenevelt, H., Van Hoorn, M. H. and Tijms, H. C. (1984) Tables for M/G/c queueing systems with phase-type service. European. J. Operat. Res. 16, 257–269.Google Scholar
Haji, R. and Newell, G. F. (1971) A relation between stationary queue and waiting time distributions. J. Appl. Prob. 8, 617–620.Google Scholar
Hillier, F. S. and Yu, O. S. (1981) Queueing Tables and Graphs. North-Holland, New York.Google Scholar
Hokstad, P. (1978) Approximations for the M/G/m queue. Operat. Res. 26, 510–523.Google Scholar
Kimura, T. (1983) Diffusion approximation for an M/G/m queue. Operat. Res. 31, 304–321.Google Scholar
Köllerström, J. (1974) Heavy traffic theory for queues with several servers. I. J. Appl. Prob. 11, 544–552.Google Scholar
Miyazawa, M. (1979) A formal approach to queueing processes in the steady state and their applications. J. Appl. Prob. 16, 332–346.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
Nozaki, S. A. and Ross, S. M. (1978) Approximations in finite-capacity multi-server queues with Poisson arrivals. J. Appl. Prob. 15, 826–834.Google Scholar
Stoyan, D. (1976) Approximations for M/G/s queues. Math. Operationsforsch. Statist. Ser. Optimization 7, 587–594.CrossRefGoogle Scholar
Takahashi, Y. (1977) An approximation formula for the mean waiting time of an M/G/c queue. J. Operat. Res. Soc. Japan 20, 150–163.Google Scholar
Takahashi, Y. (1981) Asymptotic exponentiality of the tail of the waiting time distribution in a PH/PH/c queue. Adv. Appl. Prob. 13, 619–630.Google Scholar
Tijms, H. C., Van Hoorn, W. H. and Federgruen, A. (1981) Approximations for the steady-state probabilities in the M/G/c queue. Adv. Appl. Prob. 13, 186–206.Google Scholar
Van Hoorn, M. H. (1983) Algorithms and Approximations for Queueing Systems. , Vrije Universiteit te Amsterdam.Google Scholar