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The Relaxation time for truncated birth-death processes

Published online by Cambridge University Press:  27 July 2009

Julian Keilson  and
Ravi Ramaswamy
Julian Keilson
Affiliation:
University of Rochester, Rochester, New York 14627
Ravi Ramaswamy
Affiliation:
BGS Systems Inc. Waltham, Massachusetts 02254-9111
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Abstract

The relaxation time for an ergodic Markov process is a measure of the time until ergodicity is reached from its initial state. In this paper the relaxation time for an ergodic truncated birth-death process is studied. It is shown that the relaxation time for such a process on states {0,1, …, N} is the quasi-stationary exit time from the set {,2, …, N{0,1,…, N, N + 1} with two-sided absorption at states 0 and N + 1. The existence of such a dual process has been observed by Siegmund [15] for stochastically monotone Markov processes on the real line. Exit times for birth- death processes with two absorbing states are studied and an efficient algorithm for the numerical evaluation of mean exit times is presented. Simple analytical lower bounds for the relaxation times are obtained. These bounds are numerically accessible. Finally, the sensitivity of the relaxation time to variations in birth and death rates is studied.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 1 , Issue 4 , October 1987 , pp. 367 - 381
Copyright
Copyright © Cambridge University Press 1987

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