A new proof that the stationary probability of finding i customers at a random point in time in an M/G/1 queueing system equals the stationary probability of finding i customers in the system just after a random service completion epoch. In an M/G/1 queue there is a single channel with arrivals generated by a random process. Service times are independent of each other and of arrival spacings. The queue discipline is irrelevant as long as it does not use information related to processing times. Traffic intensity is assumed to be less than 1. This proof may serve as a model for relating the stationary measure of an imbedded Markov chain to the stationary measure of the original queueing process. As examples, it turns out that these measures are also equal for the G/M/1 queue but not for bulk queues. 7 pp. Refs.