The relaxation time for truncated birth-death processes. (English) Zbl 1134.60332
Summary: 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 \(\{0,1,\dots, N\}\) is the quasi-stationary exit time from the set \(\{1,2,\dots, N\}\) for a dual absorbing birth-death process on \(\{0,1,\dots, N,N+1\}\) with two-sided absorption at states 0 and \(N+ 1\). The existence of such a dual process has been observed by D. Siegmund [Ann. Probab. 4, 914–924 (1976; Zbl 0364.60109)] for stochastically monotone Markov processes on the real line. Exit times for birthdeath 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.
MSC:
| 60G35 | Signal detection and filtering (aspects of stochastic processes) |
| 92D25 | Population dynamics (general) |
Citations:
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