The ergodicity of service systems with an infinite number of servomechanisms

Existence, uniqueness, and ergodicity are proved for a stationary distribution for a service system having countably many servomechanisms with input flow rate $\lambda_k$ depending on the number $k$ of servomechanisms occupied, and with arbitrary (identical) distribution of the service time with finite mean $\mu$, under the condition $\mu\varlimsup\limits_{k\to\infty}\frac{\lambda_k}{k+1}<1$. For this system we have, in particular, Erlang's formula
$$ p_k(t)\underset{k\to\infty}\longrightarrow p_k=\frac{\lambda_0\dots\lambda_{k-1}\mu^k}{k!}p_0,\quad k=0,1,\dots,\quad p_0^{-1}=\sum_{k=0}^\infty\frac{\lambda_0\dots\lambda_{k-1}\mu^k}{k!},\quad\lambda_{-1}=1. $$