Representations for the rate of convergence of birth-death processes. (English) Zbl 1026.60100

van Doorn, Erik A.

Teor. Jmovirn. Mat. Stat. 65, 33-38 (2001) and Theory Probab. Math. Stat. 65, 37-44 (2002).

Let \(\{X(t)\), \(t\geq 0\}\) be a birth-death process taking values in \(N=\{0,1,\dots\}\) with birth rates \(\{\lambda_n,n\in N\}\) and death rates \(\{\mu_n,n\in N\}\), all strictly positive except \(\mu_0\geq 0\). One of the main theorems proved by the author is as follows:
If \(\mu_0=0\), then the rate of convergence \(\alpha\) to stationarity of the birth-death process \(\{X(t)\}\) can be represented as
(i) \(\alpha=\max_{v}\{\inf_{n\geq 0} \{\lambda_n+\mu_{n+1}-\lambda_n\mu_n/v_n-v_{n+1}\}\}\), where \(v=(v_0,v_1,\dots)\) is any sequence of positive numbers;
(ii) \(\alpha=\max_{v}\{\inf_{n\geq 1}\frac{1}{2} \{\lambda_{n-1}+\mu_n+\lambda_n+\mu_{n+1}- ((\lambda_n+\mu_{n+1}-\lambda_{n-1}-\mu_n)^2+ 4/v_n^{-1}\lambda_n\mu_n)^{1/2}\}\}\), where \(v=(v_1,v_2,\dots)\) is any chain sequence;
(iii) \(\alpha=\max_{v}\{\lim_{n\to\infty}\inf\sum_{i=0}^n (v_i(\lambda_i+\mu_{i+1})-2\sqrt{v_{i-1}v_i\lambda_i\mu_i})\}\), where \(v_{-1}=0,v=(v_0,v_1,\dots)\) is any sequence of nonnegative numbers satisfying \(\sum v_i=1\).
A similar theorem is proved in the case \(\mu_0>0\). The obtained estimates are close to estimates proposed by M. V. Kartashov [Theory Probab. Math. Stat. 57, 53-60 (1998); translation from Teor. Jmovirn. Mat. Stat. 57, 51-59 (1997; Zbl 0948.60079)] who proved that the rate of convergence is bounded below by the quality \(\sup_{u}\{\inf_{n\geq 0} \{\lambda_n+\mu_{n+1}-\lambda_n/u_n-\mu_{n+1}v_{n+1}\}\}\), where \(u=(u_0,u_1,\dots)\) is any sequence of positive numbers.

Reviewer: Mikhail Moklyachuk (Kyïv)

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60K20	Applications of Markov renewal processes (reliability, queueing networks, etc.)
60K25	Queueing theory (aspects of probability theory)

Keywords:

birth-death process; decay parameter; duality; orthogonal polynomials; rate of convergence; spectral measure

Citations:

Zbl 0948.60079