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Evaluation of the decay parameter for some specialized birth-death processes

Part of: Special processes Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  14 July 2016

Masaaki Kijima
Masaaki Kijima*
Affiliation:
The University of Tsukuba, Tokyo
*
∗ Postal address: Graduate School of Systems Management, The University of Tsukuba, Tokyo, 3–29–1 Otsuka, Bunkyo-ku, Tokyo 112, Japan.
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Abstract

Let N(t) be an exponentially ergodic birth-death process on the state space {0, 1, 2, ···} governed by the parameters {λn, μn }, where µ 0 = 0, such that λn = λ and μn = μ for all n ≧ N, N ≧ 1, with λ < μ. In this paper, we develop an algorithm to determine the decay parameter of such a specialized exponentially ergodic birth-death process, based on van Doorn's representation (1987) of eigenvalues of sign-symmetric tridiagonal matrices. The decay parameter is important since it is indicative of the speed of convergence to ergodicity. Some comparability results for the decay parameters are given, followed by the discussion for the decay parameter of a birth-death process governed by the parameters such that limn→∞λn = λ and limn→∞µn = μ. The algorithm is also shown to be a useful tool to determine the quasi-stationary distribution, i.e. the limiting distribution conditioned to stay in {1, 2, ···}, of such specialized birth-death processes.

Keywords

EXPONENTIAL DECAY PARAMETERQUASI-STATIONARY DISTRIBUTIONTRANSITION PROBABILITY

MSC classification

Primary: 60K99: None of the above, but in this section
Secondary: 42C05: Orthogonal functions and polynomials, general theory

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 4 , December 1992 , pp. 781 - 791
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

Research partially supported by Grant-in-Aid for Scientific Research (C) (02680017) of the Ministry of Education, Science and Culture.

References

[1] Chihara, T. S. (1978) An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.Google Scholar
[2] Keilson, J. (1979) Markov Chain Models - Rarity and Exponentiality. Springer-Verlag, New York.CrossRefGoogle Scholar
[3] Kijima, M. (1990) On the largest negative eigenvalue of the infinitesimal generator associated with M/M/n/n queues. Operat. Res. Lett. 9, 59–64.CrossRefGoogle Scholar
[4] Kijima, M. and Seneta, E. (1991) Some results for quasi-stationary distributions of birth-death processes. J. Appl. Prob. 28, 503–511.CrossRefGoogle Scholar
[5] Kingman, J. F. C. (1963) The exponential decay of Markov transition probabilities. Proc. London Math. Soc. 13, 337–358.CrossRefGoogle Scholar
[6] Kingman, J. F. C. (1963) Ergodic properties of continuous-time Markov processes and their discrete skeletons. Proc. London Math. Soc. 13, 593–604.CrossRefGoogle Scholar
[7] van Doorn, E. (1981) Stochastic Monotonicity and Queueing Applications of Birth-Death Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
[8] van Doorn, E. (1985) Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process. Adv. Appl. Prob. 17, 514–530.CrossRefGoogle Scholar
[9] van Doorn, E. (1987) Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices. J. Approx. Theory, 51, 254–266.CrossRefGoogle Scholar
[10] van Doorn, E. (1991) Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Prob. 23, 683–700.CrossRefGoogle Scholar