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The matrix M/M/∞ system: retrial models and Markov Modulated sources

Part of: Special processes Computer system organization

Published online by Cambridge University Press:  01 July 2016

J. Keilson  and
L. D. Servi
J. Keilson*
Affiliation:
University of Rochester
L. D. Servi*
Affiliation:
GTE Laboratories
*
∗Postal address: University of Rochester, Dept of Statistics, Rochester, NY 14627, USA.
∗∗Postal address: GTE Laboratories Incorporated, 40 Sylvan Road, Waltham, MA, USA.
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Abstract

The matrix-geometric work of Neuts could be viewed as a matrix variant of M/M/1. A 2 × 2 matrix counterpart of Neuts for M/M/∞ is introduced, the stability conditions are identified, and the ergodic solution is solved analytically in terms of the ten parameters that define it. For several cases of interest, system properties can be found from simple analytical expressions or after easy numerical evaluation of Kummer functions. When the matrix of service rates is singular, a qualitatively different solution is derived. Applications to telecommunications include some retrial models and an M/M/∞ queue with Markov-modulated input.

Keywords

RETRIAL QUEUEING MODELSM/M/∞MATRIX METHODS

MSC classification

Primary: 68M20: Performance evaluation; queueing; scheduling
Secondary: 60K25: Queueing theory
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 2 , June 1993 , pp. 453 - 471
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

This work was conducted while JK was a senior staff scientist at GTE Laboratories.

References

Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC.Google Scholar
Aleksandrov, A. M. (1974) A queueing system with repeated orders. Engineering Cybernet. Rev. 12(3), 1–4.Google Scholar
Cohen, J. W. (1957) On the fundamental problem of telephone traffic theory and the influence of repeated calls. Philips Telecomm. Rev. 18, 49–100.Google Scholar
Falin, G. I. (1975) Multi-phase servicing in a single-channel system for automation of experiments with repeated calls. Problems of Automation of Scientific Investigations in Radio Engineering and Electronics. USSR Academy of Science, Moscow.Google Scholar
Falin, G. I. (1980) An M/M/1 queue with repeated calls in the presence of persistence function. All-Union Institute for Sci. and Technical Information, Moscow, 1606–1980.Google Scholar
Falin, G. I. (1990) A survey on retrial queues. QUESTA 7, 127–168.Google Scholar
Hanschhke, T. (1987) Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts. J. Appl. Prob. 24, 484–494.CrossRefGoogle Scholar
Jonin, G. L. and Sedol, J. J. (1970a) Investigation of telephone systems with repeated calls. Latvian Math. Yearbook 7, 71–83.Google Scholar
Jonin, G. L. and Sedol, J. J. (1970b) Telephone systems with repeated calls. Proc. 6th Internat. Telecommunication Congress , pp. 435/1–435/5.Google Scholar
Keilson, J., Cozzolino, J. and Young, H. (1968) A service system with unfilled requests repeated. Operat. Res. 16, 1126–1137.Google Scholar
Kosten, L. (1947). On the influence of repeated calls in the theory of probabilities of blocking. De Ingenieur 59, (in Dutch).Google Scholar
Le Gall, F. (1976) Trafics généraux de télécommunications sans attente. Commutation et Electronique 55, 5–24.Google Scholar
Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach . The John Hopkins University Press, Baltimore, MD.Google Scholar
O'Cinneide, C. A. and Purdue, P. (1986) The M/M/8 queue in a random environment. J. Appl. Prob. 23, 175–184.Google Scholar
Slater, L. J. (1960) Confluent Hypergeometric Functions. Cambridge University Press.Google Scholar
Tricomi, F. G. (1954) Funzioni Ipergeometriche Confluenti. Monografie Matematiche, Rome, I.Google Scholar
Wolff, R. W. (1989) Stochastic Modeling and the Theory of Queues. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Yang, T. and Templeton, J. C. G. (1987) A survey on retrial queues. QUESTA 2, 201–233.Google Scholar