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

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.

### 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)

Zbl 0948.60079