The Limiting Departure Flow in an Infinite Series of Queues

An infinite series $S^0,S^1,\dots,$ of single servers is considered, the input of $S^N$ being identified with the output of $S^{N-1}$. The servers work on a “first come-first served” basis. The input of $S^0$ is given by a general stationary ergodic marked flow $\xi^0$ that forms a $G/G/1/\infty$ queue. The service time of a given customer is preserved in the course of passing from one server to another (the telegraph rule). If the service time distribution $\sigma^0$ in flow $\xi^0$ is supported on a finite number of values or, more generally, has a bounded support and an atom at the point $l^\ast=\rm{sup}[l:l\in\rm{supp}\sigma^0]$, we prove that the departure flow $\xi^n$ xN from a server $S^N$ converges as $N\to\infty$ to a limiting stationary flow $\bar\xi$ and specify $\bar\xi$. In the case where flow-$\xi^0$ service time distribution support $\rm{supp}\sigma^0$ is unbounded or bounded, but the point $l^\ast$ does not carry an atom, the flow $\xi^N$ converges (in some specific sense) to a tightly packed flow, in which the interval between arrival of two successive customers equals the service time of the first of them.