The Busy Period of a Queue with Batch Service

In this paper, we study the distribution of the busy period for a queue with Poisson input, in which the customers are served m at the time if there are m or more present and all at once if there are less than m present. We show that the busy period is equal to the time between successive visits to the state 0 in an imbedded semi-Markov process, associated with the queuing process. Extending an argument of L. Takács for the M/G/1 queue, we obtain the transform of the distribution of the busy period. Explicit expressions in real time may in principle be obtained, using Lagrange's expansion.