In this paper, an M/G/1 queue with exponentially working vacations is analyzed. This queueing system is modeled as a two-dimensional embedded Markov chain which has an M/G/1-type transition probability matrix. Using the matrix analytic method, we obtain the distribution for the stationary queue length at departure epochs. Then, based on the classical vacation decomposition in the M/G/1 queue, we derive a conditional stochastic decomposition result. The joint distribution for the stationary queue length and service status at the arbitrary epoch is also obtained by analyzing the semi-Markov process. Furthermore, we provide the stationary waiting time and busy period analysis. Finally, several special cases and numerical examples are presented.