Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-14T20:16:05.024Z Has data issue: false hasContentIssue false
  • English
  • Français

Computational analysis of single-server bulk-service queues, M/GY / 1

Published online by Cambridge University Press:  01 July 2016

G. Brière and
M. L. Chaudhry
G. Brière*
Affiliation:
Royal Military College of Canada
M. L. Chaudhry*
Affiliation:
Royal Military College of Canada
*
∗ Postal address: Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada K7K 5LO.
∗ Postal address: Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada K7K 5LO.
Get accessRights & Permissions [Opens in a new window]

Abstract

Algorithms are proposed for the numerical inversion of the analytical solutions obtained through classical transform methods. We compute steady-state probabilities and moments of the number of customers in the system (or in the queue) at three different epochs—postdeparture, random, and prearrival—for models of the type M/GY/1, where the capacity of the single server is a random variable. This implies first finding roots of the characteristic equation, which is detailed in an appendix for a general service time distribution. Numerical results, given a service time distribution, are illustrated through graphs and tables for cases covered in this study: deterministic, Erlang, hyperexponential, and uniform distributions. In all cases, the proposed method is computationally efficient and accurate, even for high values of the queueing parameters. The procedure is adaptable to other models in queueing theory (especially bulk queues), to problems in inventory control, transportation, flexible manufacturing process, etc. Exact results that can be obtained from the algorithms presented here will be found useful to test inequalities, bounds, or approximations.

Keywords

ROOTSGROUP-SERVICE QUEUERANDOM GROUP SIZESTEADY-STATE PROBABILITY DISTRIBUTION

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 1 , March 1989, pp. 207 - 225
Copyright
Copyright © Applied Probability Trust 1989 

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.)

Article purchase

Temporarily unavailable

References

Bagchi, T. P. and Templeton, J. G. C. (1972) Numerical Methods in Markov Chains and Bulk Queues. Lecture Notes in Economics and Mathematical Systems 72, Springer-Verlag, New York.CrossRefGoogle Scholar
Bailey, N. T. J. (1954) On queueing processes with bulk service. J. R. Statist. Soc. B 16, 80–87.Google Scholar
Chaudhry, M. L. and Templeton, J. G. C. (1983) A First Course in Bulk Queues. Wiley, New York.Google Scholar
Chaudhry, M. L., Madill, B. R. and Briere, G. (1987) Computational analysis of steady-state probabilities of M/Ga,b/1 and related nonbulk queues. Queueing Systems 2, 173–185.CrossRefGoogle Scholar
Conte, G. D. and Deboor, C. (1972) Elementary Numerical Analysis. McGraw-Hill, New York.Google Scholar
Downton, F. (1955) Waiting time in bulk service queues. J. R. Statist. Soc. B 17, 256–261.Google Scholar
Easton, G. D. (1981) Contributions to the Analysis of Some Bulk-Queueing Problems. M.Sc. Thesis, Royal Military College of Canada, Ontario.Google Scholar
Everett, J. L. (1953) State probabilities in congestion problems characterized by constant holding times. J. Operat. Res. Soc. Am. 1, 279–285.Google Scholar
Hillier, F. S. and Yu, O. S. (1981) Queueing Tables And Graphs. Elsevier North-Holland, New York.Google Scholar
Hirasawa, K. (1971) Numerical solutions of bulk queues via imbedded Markov chain. Elec. Eng. Jpn. 91, 127–136.Google Scholar
Jaiswal, N. K. (1964) A bulk-service queueing problem with variable capacity. J. R. Statist. Soc. B 26, 143–148.Google Scholar
Kotiah, T. C. T., Thompson, T. W. and Waugh, W. A. O’N. (1969) Use of Erlangian distribution for single-server queueing systems. J. Appl. Prob. 6, 584–593.CrossRefGoogle Scholar
Neuts, M. F. (1967) A general class of bulk queues with Poisson input. Ann. Math. Statist. 38, 759–770.CrossRefGoogle Scholar
Neuts, ?. F. (1979) Queues solvable without Rouché’s theorem. Operat. Res. 27, 767–781.CrossRefGoogle Scholar
Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. The Johns Hopkins University Press, Baltimore.Google Scholar
Pike, M. C. (1963) Some numerical results for the queueing system D/Ek/1. J. R. Statist. Soc. B 25, 477–488.Google Scholar
Powell, W. B. (1981) Stochastic Delays in Transportation Terminals: New Results in the Theory and Applications of Bulk Queues. Ph.D. Dissertation, MIT, Cambridge, MA.Google Scholar
Powell, W. B. (1985) Analysis of vehicle holding and cancellation strategies in bulk arrival, bulk service queues. Transportation Sci. 19, 352–377.CrossRefGoogle Scholar
Powell, W. B. (1986a) Iterative algorithms for bulk-service queues with Poisson and non-Poisson arrivals. Transportation Sci. 20, 65–79.CrossRefGoogle Scholar
Powell, W. B. (1986b) Approximate, closed form moment formulas for bulk arrival, bulk service queues. Transportation Sci. 20, 13–23.CrossRefGoogle Scholar
Seelen, L. P., Tijms, H. C. and Vanhorn, M. H. (1985) Tables for Multiserver Queues. North-Holland, Amsterdam.Google Scholar