A decomposition property for an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e54" altimg="si3.svg"><mml:mrow><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:mi>G</mml:mi><mml:mo>/</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> queue with vacations

We introduce a queueing system that alternates between two modes, so-called working mode and vacation mode. During the runs as an MX/G/1 queue. Once number of customers in drops to zero begins. general (a service might be included) which is different from one The period ends accordance with given stopping rule, then random are transferred For this model we show distributed sum independent variables, them queue server busy. This decomposition result puts under same umbrella some models have already been introduced past well new models.