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By applying the Takács’ technique about the busy period to the regenerative cycle method, this paper gives the strict proof for the time average distributions of the queue length in M/G/1 without depending on other methods. Moreover we extend its proof from the service time to the completion time(CT). That is, we choose the stochastic behavior on the completion time as the regenerative cycle and, by using its PGF, represent the queue length distributions in M/CT/1, M/CT/1 with N -policy, M/CT/1 with multiple vacation and their combinations. Our completion time is able to contain the additional service time, the vacation, the loss interval and the batch arrival. We can also consider some service disciplines on it like time-controlled service discipline. Thus the completion time realizes the wider application of the regenerative cycle method, unifies various variants of the fundamental models and derives their probability generating functions.

QUEUE LENGTH DISTRIBUTION IN M/G/1, M^x/G/1 AND THEIR VARIANTS WITH COMPLETION TIME