On the Weak Convergence of Departures from an Infinite Series of ./M/1 Queues

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 .<1 through a sequence of independent, rate one, exponential server queues is a Poisson process of rate .. This builds on work of Liggett and Shiga and Anantharam