Suppose that we have a queueing system with slotted time-line, single server and Bernoulli distribution for the arrival and service processes. The probability of a packet arrival into the queue is $p$ and the probability of service availability is $q$, and $p<q$ so the queue is stable. I think it is easy to see that the departure process follows a Bernoulli distribution with the average of $p$, but I was unable to find a reference to cite for this property. I know that there is Burke's theorem for stable M/M/1 queues with arrival rate of $\lambda$, that states the departure process is also Poisson with average rate of $\lambda$, but I could not find any reference to mention its equivalent counterpart in the case of discrete-time queues. Do you know any good reference discussing that?
Thanks in advance.