Analysis of the Mx/G/1 queue with a random setup time

We consider an M-x/G/1 queueing system with a random setup time, where the service of the first unit at the commencement of each busy period is preceded by a random setup time, on completion of which service starts. For this model, the queue size distributions at a random point of time as well as at a departure epoch and some important performance measures are known [see Choudhury, G. An M-x/G/1 queueing system with setup period and a vacation period. Queueing Sys. 2000, 36, 23-38]. In this paper, we derive the busy period distribution and the distribution of unfinished work at a random point of time. Further, we obtain the queue size distribution at a departure epoch as a simple alternative approach to Choudhury([4]). Finally, we present a transform free method to obtain the mean waiting time of this model.