In this paper, by considering the stochastic process of the busy period and the idle period, and introducing the unfinished work as a supplementary variable, a new vector Markov process was presented to study the M/G/1 queue again. Through establishing and solving the density evolution equations, the busy-period distribution, and the stationary distributions of waiting time and queue length were obtained. In addition, the stability condition of this queue system was given by means of an imbedded renewal process.