An Ergodic Theorem for Markov Processes and Its Application to Telephone Systems with Refusals

Let $P(x,t,A)$ be the probability of transition from state $x \in \Omega $ to a measurable set A in time t of a homogeneous Markov process in the space $\Omega $. Let $P_t (A)$ be the probability that the system is in the measurable set A at time t if at $t = 0$ the distribution $P_0 $ is valid. The stationary distribution P is called ergodic if for each initial distribution $P_0 $ the variation $V(P_t - P) \to 0$ for $t \to \infty $.The following ergodic theorem is proved:Theorem 1.A Markov process homogeneous in time has a unique stationary probability distribution which is ergodic if for any$\varepsilon > \infty $there exists a measurable setC, a probability distributionRin$\Omega $, and values$t_1 > 0,k > 0,K > 0$such that 1) $kR(A) \leqq P(x,t_1 ,A)$for all points$x \in C$and measurable sets$A \subseteq C$; for any initial distribution$P_0 $there exists a$t_0 $such that for any$t \geqq t_0 $ 2) $P_t (C) \geqq 1 - \varepsilon $; 3) $P_t (A) \leqq KR(A) + \varepsilon $for any measurable set$A \subseteq...