  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.
