Regenerative cycle method in M/G/1 with many vacation rules

This paper gives a new method to derive the time average distribution of queue length in the model M/G/1 with many vacation rules. We extract many kinds of regenerative cycles of the well-known regenerative processes from the queue length process of the model by using the method shown by Fuhrmann and Cooper. The time average of the original model is represented by the combination of the time averages of such regenerative processes. We find this combination in various models, e.g., the model with setup time, the special decrementing service system, pure limited service system, feedback system and so on. We show a recursive calculation formula for moments.