This paper is concerned with the stationary distribution of a one-dimensional circuit-switched network. We show that if arrival rates decay geometrically with distance, then under the stationary distribution the number of circuits busy on successive links of the network at a fixed point in time is a Markov chain. When each link of the network has unit capacity we show that translation invariant arrival rates lead to a stationary distribution which can be described in terms of an alternating renewal process.
F. P. Kelly. "One-Dimensional Circuit-Switched Networks." Ann. Probab. 15 (3) 1166 - 1179, July, 1987. https://doi.org/10.1214/aop/1176992089