This paper considers the departure process of the $M/G/1$ queue with Min$(N,D)$-policy. Using the total probability decomposition technique and the renewal process theory, we discuss the expected number of departures occurring in finite time interval from an arbitrary initial state. Both the transient expression and the steady state expression of the expected departure number are obtained. The important relation among departure process, server state process and renewal process of service during server busy period is discovered. The relation displays the stochastic decomposition characteristic of the departure process, i.e., the expected departure number is decomposed into two parts: one is the server busy probability, and the other is the expected departure number during server busy period, which simplifies the discussion on the departure process. Furthermore, the approximate expansion for convenient calculation of the expected departure number is obtained. Finally, we derive the corresponding results for some special cases. Since the departure process also often corresponds to an arrival process in downstream queues in queueing network, it is expected that the results obtained in this paper would provide useful information for queueing network.