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On a property of finite-state birth and death processes

Published online by Cambridge University Press:  24 August 2016

A. J. Branford
A. J. Branford*
Affiliation:
The Flinders University of South Australia
*
∗ Postal address: School of Mathematical Sciences, The Flinders University of South Australia, Bedford Park, SA 5042, Australia.
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Abstract

A simple proof is given of the result that the ‘overflow' from a finite-state birth and death process is a renewal stream characterized by hyperexponential inter-event times. Our structure is utilized to give a converse result that any hyperexponential renewal stream can be so produced as the overflow from a finite-state birth and death process.

Keywords

HYPEREXPONENTIALOVERFLOWORTHOGONAL POLYNOMIALS

Information

Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 4 , December 1986 , pp. 859 - 866
Copyright
Copyright © Applied Probability Trust 1986 

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References

Chihara, T. S. (1978) An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.Google Scholar
Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., (eds.) (1953) Higher Transcendental Functions, Volume II. McGraw-Hill, New York.Google Scholar
Favard, J. (1935) Sur les polynomes de Tchebicheff. C. R. Acad. Sci. Paris 200, 2052–2053.Google Scholar
Karlin, S. and Mcgregor, J. L. (1957) The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.Google Scholar
Karlin, S. and Mcgregor, J. L. (1958) Many server queueing processes with Poisson input and exponential service times. Pacific J. Math. 8, 87–118.Google Scholar
Karlin, S. and Mcgregor, J. L. (1959) A characterization of the birth and death process. Proc. Acad. Nat. Sci. U.S.A. 45, 375–379.CrossRefGoogle Scholar
Keilson, J. (1971) Log-concavity and log-convexity in passage time densities of diffusion and birth-death processes. J. Appl. Prob. 8, 391–398.Google Scholar
Keilson, J. (1979) Markov Chain Models — Rarity and Exponentiality., Applied Mathematical Sciences Series 28, Springer-Verlag, New York.Google Scholar
Khintchine, A. Y. (1969) Mathematical Methods in the Theory of Queueing. Griffin's Statistical Monographs and Courses No. 7, 2nd edn., Griffin, London.Google Scholar
Mitrinovic, D. S. (1970) Analytic Inequalities. Springer-Verlag, Berlin.Google Scholar
Potter, R. M. (1979) Overflow Traffic in Telephony. Ph.D. Dissertation, University of Adelaide, Australia.Google Scholar
Rosenlund, S. I. (1978) Passage times in the finite M/M/1 queue. In Point Processes and Queueing Problems, Colloq. Math. Soc. János Bolyai 24, North-Holland, Amsterdam, 337–341.Google Scholar
Syski, R. (1960) Introduction to Congestion Theory in Telephone Systems. Oliver and Boyd, London.Google Scholar
Szegö, G. (1939) Orthogonal Polynomials. Amer. Math. Soc. Colloquium Publications 23.Google Scholar
Wendroff, B. (1961) On orthogonal polynomials. Proc. Amer. Math. Soc. 12, 554–555.CrossRefGoogle Scholar