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Extended continued fractions, recurrence relations and two-dimensional Markov processes

Published online by Cambridge University Press:  01 July 2016

C. E. M. Pearce
C. E. M. Pearce*
Affiliation:
University of Adelaide
*
∗Postal address: Applied Mathematics Department, The University of Adelaide, GPO Box 498, SA 5001, Australia.
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Abstract

Connections between Markov processes and continued fractions have long been known (see, for example, Good [8]). However the usefulness of extended continued fractions in such a context appears not to have been explored. In this paper a convergence theorem is established for a class of extended continued fractions and used to provide well-behaved solutions for some general order linear recurrence relations such as arise in connection with the equilibrium distribution of state for some Markov processes whose natural state spaces are of dimension 2. Specific application is made to a multiserver version of a queueing problem studied by Neuts and Ramalhoto [13] and to a model proposed by Cohen [5] for repeated call attempts in teletraffic.

Keywords

SEARCH FOR CUSTOMERSREPEATED CALL ATTEMPTS
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 2 , June 1989 , pp. 357 - 375
Copyright
Copyright © Applied Probability Trust 1989 

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References

[1] Aleksandrov, A. M. (1974) A queueing system with repeated orders. Engineering Cybernetics 12, 1–4.Google Scholar
[2] Bernstein, L. (1971) The Jacobi–Perron Algorithm, Its Theory and Application. Lecture Notes in Mathematics 207, Springer-Verlag, New York.CrossRefGoogle Scholar
[3] Bodnarcuk, P. I., Pustomel'Nikov, I. P., Slon'Ovs'Kii, R. V. and Jarovii, S. S. (1972) On certain applications of branching continued fractions in the investigation of Markov processes (in Ukrainian). Dopovidi Akad. Nauk Ukrain. RSR, Ser., A 391–394, 475.Google Scholar
[4] Bodnarcuk, P. I. and Skorobogat'Ko, V. Ya. (1974) Branched Continued Fractions and Applications (in Ukrainian). Naukovaja Dumka, Kiev.Google Scholar
[5] Cohen, J. W. (1957) Basic problems of telephone traffic theory and the influence of repeated calls. Philips Telecomm. Rev. 18, 49–100.Google Scholar
[6] Erdélyi, A. (1953) Higher Transcendental Functions, Vol. 1. McGraw-Hill, San Francisco.Google Scholar
[7] Falin, G. I. (1979). A single-line system with secondary orders. Engineering Cybernetics 17, 76–83.Google Scholar
[8] Good, I. J. (1958) Random motion and analytic continued fractions. Proc. Camb. Phil. Soc. 54, 43–47.Google Scholar
[9] Jacobi, C. G. J. (1868) Allgemeine Theorie der kettenbruchähnlichen Algorithmen, in welchen jede Zahl aus drei vorhergehenden gebildet wird. J. Reine Angew. Math. 69, 29–64.Google Scholar
[10] Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
[11] Murphy, G. M. (1960) Ordinary Differential Equations and their Solutions. Van Nostrand Reinhold, New York.Google Scholar
[12] Naumova, E. O. (1983) The loss probability measurement accuracy for the subscriber line with repeated calls. 10th Int. Teletraffic Congress, Montréal, 4.4b-6 1-4.Google Scholar
[13] Neuts, M. F. and Ramalhoto, M. F. (1984) A service model in which the server is required to search for customers. J. Appl. Prob. 21, 157–166.Google Scholar
[14] Pearce, C. E. M. (1985) Holomorphic solutions about an irregular singular point of an ordinary linear differential equation. J. Austral. Math. Soc. A 39, 178–186.Google Scholar
[15] Pearce, C. E. M. (1987) On the problem of re-attempted calls in teletraffic. Stoch. Models 3, 393–407.CrossRefGoogle Scholar
[16] Perron, O. (1907) Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Annalen 64, 1–76.Google Scholar
[17] Schweiger, F. (1973) The Metrical Theory of Jacobi-Perron Algorithm. Lecture Notes in Mathematics 334, Springer-Verlag, Berlin.Google Scholar
[18] Skorobogat'Ko, V. Ya (1983) Continued Fractions and their Applications in Computational Mathematics (in Russian). Nauka, Moscow.Google Scholar