We consider an infinite tandem of first-come-first-served queues. The service times have unit mean, and are independent and identically distributed across queues and customers. Let $\bI$ be a stationary and ergodic interarrival sequence with marginals of mean $\tau>1$, and suppose it is independent of all service times. The process $\bI$ is said to be a fixed point for the first, and hence for each, queue if the corresponding interdeparture sequence is distributed as $\bI$. Assuming that such a fixed point exists, we show that it is the distributional limit of passing an arbitrary stationary and ergodic interarrival process of mean $\tau$ through the infinite queueing tandem.
"The attractiveness of the fixed points of a $\cdot/GI/1$ queue." Ann. Probab. 31 (4) 2237 - 2269, October 2003. https://doi.org/10.1214/aop/1068646384