In this paper the ``set-up time'' is introduced in the $M/G/1$ queueing system with Min($N,V$)-policy based on multiple server vacations. By applying the method of the total probability decomposition technique and the Laplace transform tool, we discuss the transient queue-length distribution and steady state queue-length distribution from the beginning of the any initial state. The expressions of the Laplace -transformation of the transient queue-length distribution are obtained. Furthermore, the recursive expressions of the steady state queue-length distribution and the stochastic decomposition of the steady state queue-length are also obtained, which have important value in the system capacity design. Meanwhile, some special cases are discussed.