A decomposition property for an MX/G/1 queue with vacations

We introduce a queueing system that alternates between two modes, so-called working mode and 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. 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 number of customers given that the system is in the working mode is distributed as the sum of two independent random variables, one of them is the number of customers in an $M^X/G/1$ queue given that the server is busy. This decomposition result puts under the same umbrella some models that have already been introduced in the past as well as some new models.