Technical Note—A Note on the M/G/1 Queue with Server Vacations

Consider an M/G/1 queuing system in which the server begins a vacation of random length each time that the system becomes empty. This model has been analyzed in several papers, which show that the number of customers present in the system at a random point in time is distributed as the sum of two independent random variables: (i) The number of Poisson arrivals during a time interval that is distributed as the forward recurrence time of a vacation, and (ii) the number of customers present in the corresponding standard M/G/1 queuing system (without vacations) at a random point in time. This note gives an intuitive explanation for this result, while simultaneously providing a more simple and elegant method of solution.