We study a system of infinitely many queues with Poisson arrivals and exponential service times. Let the net output process be the difference between the departure process and the arrival process. We impose certain ergodicity conditions on the underlying Markov chain governing the customer path. These conditions imply the existence of an invariant measure under which the average net output process is positive and proportional to the time. Starting the system with that measure we prove that the net output process is a Poisson process plus a perturbation of order 1. This generalizes a classical theorem (Burke (1956), Kelly (1979)) asserting that the departure process is a Poisson process. An analogous result is proven for the net input process.