×

Computing stationary queueing-time distributions of \(GI/D/1\) and \(GI/D/c\) queues. (English) Zbl 0757.60093

Summary: This article gives closed-form analytic expressions as well as the exact computational analysis of stationary queueing-time distribution for the \(GI/D/1\) queue. By exploiting the relationship between the distributions of queueing times of \(GI/D/1\) and \(GI/D/c\) queues, the computational analysis of the queueing-time distribution of \(GI/D/c\) queue is also done. Numerical results are presented for (i) the first two moments of queueing time and (ii) the probability that queueing time is zero. Also, comments are made regarding the graphs of the distribution functions for particular cases of \(E_ m/D/1\) and \(HE_ 2/D/1\). Some further properties such as computing the pre- and postdeparture probabilities for \(GI/D/1\) are also discussed. The results discussed here should prove to be useful to practitioners and queueing theorists dealing with inequalities, bounds, et cetera.

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bertsimas, Queueing Systems 3 pp 305– (1988)
[2] Botta, Communication in Statistics–Stochastic Models 3 pp 115– (1987)
[3] Brière, Advances in Applied Probability 21 pp 207– (1987)
[4] ”Stationary Queueing-Time Distribution in GI/R/1 Systems: Explicit Closed-Form Analysis and Exact Computational Results”, presented at the ORSA/TIMS Conference, Philadelphia, October 1990.
[5] QPACK Software Package, A & A Publications, 395 Carrie Cres., Kingston, Ontario, K7M 5{\(\times\)}7, Canada, 1991.
[6] Chaudhry, Queueing Systems 10 pp 351– (1992)
[7] Chaudhry, ORSA Journal on Computing 2 pp 273– (1990) · Zbl 0760.60081
[8] and , A First Course in Bulk Queues, Wiley, New York, 1983. · Zbl 0559.60073
[9] The Single Server Queue, North Holland, Amsterdam, 1982. · Zbl 0481.60003
[10] and , ”Queueing Systems Having Phase-Dependent Arrival and Service Rates”, in (Ed.), Numerical Solutions of Markov Chains, Marcel Dekker, New York, 1991, pp. 161–202. · Zbl 0735.60097
[11] De Smit, Operations Research Letters 2 pp 217– (1983)
[12] De Smit, Journal of Applied Probability 22 pp 214– (1985)
[13] An Introduction to Probability Theory and Its Applications (3rd ed.), Wiley, New York 1968, Vol. 1.
[14] and , Fundamentals of Queueing Theory (2nd ed.), Wiley, New York, 1985.
[15] and , Queueing Tables and Graphs, North-Holland, New York, 1981.
[16] Ishikawa, Journal of the Operations Research Society of Japan 27 pp 130– (1984)
[17] Iversen, Operations Research Letters 2 pp 20– (1983)
[18] Kimura, Management Science 32 pp 751– (1986)
[19] Queueing Systems, Vol. I: Theory, Wiley, New York, 1975.
[20] Lindley, Proceedings of the Cambridge Philosophical Society 48 pp 277– (1952)
[21] Matrix-Geometric Solutions in Stochastic Models–An Algorithmic Approach, The Johns Hopkins University Press, Baltimore, 1981. · Zbl 0469.60002
[22] Queues and Inventories–A Study of Their Basic Stochastic Processes, Wiley, New York, 1965.
[23] Rosberg, Journal of Applied Probability 24 pp 749– (1987)
[24] Elements of Queueing Theory with Applications, McGraw-Hill, New York, 1961.
[25] , and , Tables for Multi-Server Queues, North-Holland, New York, 1985.
[26] Stochastic Modeling and Analysis: A Computational Approach, Wiley, New York, 1986.
[27] Van Hoorn, Zeitschrift für Operations Research 30A pp 15– (1986)
[28] Van Hoorn, Journal of Applied Probability 23 pp 484– (1986) · Zbl 0601.90051
[29] The Fourier Transform of Probability Distributions, Baltimore, 1947.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.