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

Part of: Nontrigonometric harmonic analysis Special processes

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 µ0 = 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
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.

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