Busy periods for queues alternating between two modes

We study the busy period of a single server queueing systemoperating in two alternating modes - working and vacation . In the two modes the systems run as an M X /G /1 queue with disasters, but with different parameters. The vacation mode starts once the number of customers drops to zero. It is terminated randomly (when it is not empty) with a transition to the working mode. At such a transition moment all the customers are transferred to the working mode; the service of the customer being served is lost and it starts from scratch in the working mode. Every busy period starts with a batch arrival into an empty system and terminates at the first time that the number of customers drops to zero. The working and the vacation periods are analyzed too. Finally, we apply the results to obtain the probability generating functions of the number of customers in the working, as well as in the vacation periods.