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On steady-state queue size distributions of the discrete-time GI/G/1 queue

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Tao Yang  and
M. L. Chaudhry
Tao Yang*
Affiliation:
Technical University of Nova Scotia
M. L. Chaudhry*
Affiliation:
Royal Military College of Canada
*
∗ Postal address: Department of Industrial Engineering, Technical University of Nova Scotia, Halifax, Nova Scotia B3J 2X4, Canada.
∗∗ Postal address: Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario K7K 5L0, Canada.
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Abstract

In this paper, we present results for the steady-state system length distributions of the discrete-time GI/G/1 queue. We examine the system at customer arrival epochs (customer departure epochs) and use the residual service time (residual interarrival time) as the supplementary variable. The embedded Markov chain is of GI/M/1 type if the embedding points are arrival epochs and is of M/G/1 type if the embedding points are departure epochs. Using the matrix analytic method, we identify the necessary and sufficient condition for both Markov chains to be positive recurrent. For the GI/M/1 type chain, we derive a matrix-geometric solution for its steady-state distribution and for the M/G/1 type chain, we develop a simple linear transformation that relates it to the GI/M/1 type chain and leads to a simple analytic solution for its steady-state distribution. We also show that the steady-state system length distribution at an arbitrary point in time can be obtained by a simple linear transformation of the matrix-geometric solution for the GI/M/1 type chain. A number of applications of the model to communication systems and numerical examples are also discussed.

Keywords

GI/G/1 DISCRETE-TIME QUEUESMATRIX ANALYTIC TECHNIQUE

MSC classification

Primary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 4 , December 1996 , pp. 1177 - 1200
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Forum, Atm (1992) ATM User-Network Interface Specification. Version 3.0.Google Scholar
[2] Ackroyd, M. H. (1980) Computing the waiting time distribution for the GI/G/1 queue by signal processing methods. IEEE Trans. Commun. 6, 52–58.Google Scholar
[3] Bruneel, H. and Kim, B. G. (1993) Discrete-Time Models for Communication Systems Including ATM. Kluwer, Boston.Google Scholar
[4] Chaudhry, M. L. (1993) Alternative numerical solutions of stationary queueing-time distributions in discrete-time queues: GI/G/1. J. Operat. Res. Soc. 44, 1035–1051.Google Scholar
[5] Chaudhry, M. L. and Templeton, J. G. C. (1983) A First Course in Bulk Queues. Wiley, New York.Google Scholar
[6] Fryer, M. J. and Winsten, C. B. (1986) An algorithm to compute the equilibrium distribution of a one dimensional bounded random walk. Operat. Res. 34, 449–454.CrossRefGoogle Scholar
[7] Gail, H. R. and Hantler, S. L. (1996) Solutions of the basic matrix equation for M/G/1 and G/M/1 type Markov chains. Stoch. Models. CrossRefGoogle Scholar
[8] Gail, H. R., Hantler, S. L., Konheim, A. G. and Taylor, B. A. (1996) An analysis of a class of telecommunications models. Perf. Eval. (to appear).Google Scholar
[9] Grassmann, W. K. and Jain, J. L. (1989) Numerical solutions of the waiting time distribution and idle time distribution of the arithmetic GI/G/1 queue. Operat. Res. 37, 141–150.CrossRefGoogle Scholar
[10] Gravey, A., Louvion, J.-R. and Boyer, P. (1990) On the Geo/D/1 and Geo/D/1/n queues. Perf. Eval. 11, 117–125.Google Scholar
[11] Gross, D. and Harris, C. M. (1974) Fundamentals of Queueing Theory. Wiley, New York.Google Scholar
[12] Hayes, J. F. (1984) Modeling and Analysis of Computer Communications Networks. Plenum, New York.CrossRefGoogle Scholar
[13] Hunter, J. J. (1983) Mathematical Techniques of Applied Probability. Vol. II: Discrete Time Models: Techniques and Applications. Academic Press, New York.Google Scholar
[14] Knopp, K. (1956) Infinite Sequences and Series. Dover, New York.Google Scholar
[15] Konheim, A. G. (1975) An elementary solution of the queueing system G/G/1. SIAM J. Comput. 4, 540–545.Google Scholar
[16] Leveque, W. J. (1956) Topics in Number Theory. Addison-Wesley, Reading, MA.Google Scholar
[17] Neuts, M. F. (1973) The single server queue in discrete-time-numerical analysis I. Naval Res. Lo gist. Quart. 20, 297–304.CrossRefGoogle Scholar
[18] Neuts, M. F. (1978) Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector. Adv. Appl. Prob. 10, 185–212.Google Scholar
[19] Neuts, M. F. (1980) The probabilistic significance of the rate matrix in matrix-geometric invariant vectors. J. Appl. Prob. 17, 291–296.CrossRefGoogle Scholar
[20] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. The Johns Hopkins University Press, Baltimore.Google Scholar
[21] Neuts, M. F. (1989) Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker, New York.Google Scholar
[22] Neuts, M. F. and Klimko, E. M. (1973) The single server queue in discrete-time-numerical analysis III. Naval Res. Logist. Quart. 20, 557–567.CrossRefGoogle Scholar
[23] Ponstein, J. (1974) Theory and solution of a discrete queueing problem. Statist. Neerland. 20, 139–152.Google Scholar
[24] Roberts, J. W. (1992) Performance Evaluation and Design of Multiservice Networks. ECSC-EEC-EAEC, Brussels.Google Scholar
[25] Sakasegawa, H., Miyazawa, M. and Yamazaki, G. (1993) Evaluating the overflow probability using the infinite queue. Management Sci. 39, 1238–1245.Google Scholar
[26] Tsuchiya, T. and Takahashi, Y. (1993) On discrete-time single-server queues with Markov modulated batch Bernoulli input and finite capacity. J. Operat. Res. Soc. Japan 36, 29–45.Google Scholar
[27] Van Ommeren, J. C. W. (1991) The discrete-time single-server queue. Queueing Systems: Theory Appl. 8, 279–294.CrossRefGoogle Scholar
[28] Yang, T. (1993) A note on GI/M/1 type Markov chains. Working Paper 93-08. Technical University of Nova Scotia, Canada.Google Scholar