Assume we have a $M/M/\infty$ queue with arrival rate $\lambda$ and a service rate $\mu$. From Burke's theorem, the departure process of the queue is a Poisson process with rate $\lambda$.
However, what is the relationship between the arrival and departure processes? Are they independent? If not, is it possible to characterize their relationship?
The arrival and departure processes are obviously not independent: suppose that, with some very bad luck, no customers arrived to the system up to now; then (after the completion of the service of those that were there initially) nobody will departure as well.
For a single-server queue, one can, for instance, calculate the covariance of arrival/departure counts: see The input and output processes in a single-server queue; for the $M/M/\infty$ that might be even easier, but I didn't check.
