A Decomposition Property for an $M^{X}/G/1$ Queue with Vacations

We introduce a queueing system that alternates between two modes, so-called {\it working mode} and {\it vacation mode}. During the working mode the system runs as an $M^{X}/G/1$ queue. Once the number of customers in the working mode drops to zero the vacation mode begins. %Then working system becomes empty the vacation phase begins. During the vacation mode the system runs as a general queueing system (a service might be included) which is different from the one in the working mode. The vacation period ends in accordance with a given stopping rule, and then a random number of customers are transferred to the working mode. For this model we show that the conditional probability generating function of the number of customers given that the system is in the working mode is a product of three terms. This decomposition result puts under the same umbrella some models that have already been introduced in the past as well as some new models.