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In this paper we are concerned with several random processes that occur in M/G/1 queues with instantaneous feedback in which the feedback decision process is a Bernoulli process. Queue length processes embedded at various times are studied. It is shown that these do not all have the same asymptotic distribution, and that in general none of the output, input, or feedback processes is renewal. These results have implications in the application of certain decomposition results to queueing networks.

Burke [Burke, P. J. 1956. The output of a queueing system. Oper. Res. 4 699–704.] showed that the departure process from an M/M/1 queue with infinite capacity was in fact a Poisson process. Using methods from semi-Markov process theory, this paper extends this result by determining that the departure process from an M/G/1 queue is a renewal process if and only if the queue is in steady state and one of the following four conditions holds: (1) the queue is the null queue—the service times are all 0; (2) the queue has capacity (excluding the server) 0; (3) the queue has capacity 1 and the service times are constant (deterministic); or (4) the queue has infinite capacity and the service times are negatively exponentially distributed (M/M/1/∞ queue).

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