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Stochastic monotonicity of birth–death processes

Published online by Cambridge University Press:  01 July 2016

Erik A. Van Doorn
Erik A. Van Doorn*
Affiliation:
Twente University of Technology
*
∗Postal address: Department of Applied Mathematics, Twente University of Technology, P.O. Box 217, 7500 AE Enschede, The Netherlands.
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Abstract

A birth–death process {x(t): t ≥ 0} with state space the set of non-negative integers is said to be stochastically increasing (decreasing) on the interval (t1, t2) if Pr {x(t) > i} is increasing (decreasing) with t on (t1, t2) for all i = 0, 1, 2, ···. We study the problem of finding a necessary and sufficient condition for a birth–death process with general initial state probabilities to be stochastically monotone on an interval. Concrete results are obtained when the initial distribution vector of the process is a unit vector. Fundamental in the analysis, and of independent interest, is the concept of dual birth–death processes.

Keywords

BIRTH-DEATH PROCESSESBIRTH-DEATH POLYNOMIALSDUAL BIRTH-DEATH PROCESSESKARLIN-MCGREGOR REPRESENTATIONSTOCHASTIC MONOTONICITY
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 1 , March 1980 , pp. 59 - 80
Copyright
Copyright © Applied Probability Trust 1980 

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References

Callaert, H. (1971) Exponential Ergodicity for Birth–Death Processes (in Dutch). Ph.D. Dissertation, University of Louvain.Google Scholar
Callaert, H. and Keilson, J. (1973) On exponential ergodicity and spectral structure for birth–death processes. Stoch. Proc. Appl. 1, 187–235.CrossRefGoogle Scholar
De Smit, J. H. A. (1972) The time dependent behaviour of the queue length process in the system M/M/s. CORE discussion paper no. 7217, University of Louvain.Google Scholar
Feller, W. (1959) The birth and death processes as diffusion processes. J. Math. Pures Appl. 38, 301–345.Google Scholar
Karlin, S. (1968) Total Positivity. Stanford University Press, Stanford, CA.Google Scholar
Karlin, S. and McGregor, J. L. (1957a) 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. (1957b) The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.Google Scholar
Karlin, S. and McGregor, J. L. (1958a) 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. (1958b) Linear growth, birth and death processes. J. Math. Mech. 7, 643–662.Google Scholar
Karlin, S. and McGregor, J. L. (1959) A characterization of birth and death processes. Proc. Nat. Acad. Sci. U.S.A. 45, 375–379.CrossRefGoogle ScholarPubMed
Keilson, J. and Kester, A. (1977) Monotone matrices and monotone Markov processes. Stoch. Proc. Appl. 5, 231–241.Google Scholar
Kemperman, J. H. B. (1962) An analytical approach to the differential equations of the birth-and-death process. Michigan Math. J. 9, 321–361.Google Scholar
Knopp, K. (1964) Theorie und Anwendung der Unendlichen Reihen. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Ledermann, W. and Reuter, G. E. H. (1954) Spectral theory for the differential equations of simple birth and death processes. Phil. Trans. R. Soc. London A 246, 321–369.Google Scholar
Reich, E. (1957) Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768–773.CrossRefGoogle Scholar
Shohat, J. A. and Tamarkin, J. D. (1950) The Problem of Moments. Mathematical Surveys Number I. American Mathematical Society, Providence, RI.Google Scholar
Stoyan, D. (1977) Qualitative Eigenschaften und Abschätzungen stochastischer Modelle. Akademie-Verlag, Berlin.Google Scholar
Szegö, S. (1959) Orthogonal Polynomials. American Mathematical Society Colloquium Publications. Vol. XXIII. American Mathematical Society, Providence, RI.Google Scholar
Van Doorn, E. A. (1978) The spectral measure of the M/M/s queue. Memorandum No. 223, Dept. of Applied Math., Twente University of Technology, Enschede.Google Scholar