The queue M/G/1 with a round-robin type queueing discipline is considered. The overall processing time of a job is assumed to consist of a sequence of “simple” processing times. There are a countably infinite number of waiting queues, 1,2,…, but a single processing facility. A job upon arrival enters the first, or highest priority waiting queue. When a job in the nth queue gains the processor, it receives its nth simple processing time in an uninterrupted fashion and then either departs or enters the n + 1st queue. The processor upon completion of a simple processing time selects the job to be next processed from the highest priority (lowest index number) non-empty queue. Laplace transforms and expressions for the moments of the time in system distributions are obtained.