In this note we observe that the recent argument of Ekhaus and Gray combined with the approach of Liggett and Shiga shows that the limit from passing a stationary ergodic arrival process of rate $\alpha < 1$ through a sequence of independent, rate one, exponential server queues is a Poisson process of rate $\alpha$. This builds on work of Liggett and Shiga and Anantharam.
"On the Weak Convergence of Departures from an Infinite Series of $\cdot /M/ 1$ Queues." Ann. Appl. Probab. 5 (1) 121 - 127, February, 1995. https://doi.org/10.1214/aoap/1177004831