考虑基于~$\mathrm{M}\mathrm{i}\mathrm{n}(N,D,V)$- 策略和单重休假的~$M/G/1$ 排队系统, 在建立费用结构模型的基础上, 导出了系统长期单位时间内的期望费用的显示表达式. 通过数值计算实例, 讨论了使得系统在长期单位时间内的期望费用最小的联合控制策略~$(N^{\ast },D^{\ast }).$ 进一步, 当服务员休假时间为定长时间~$T$ 时, 给出了具有三维决策变量~$(N,D,T)$ 的系统长期单位时间内的期望费用的显示表达式, 而且通过数值实例确定了使得系统在长期单位时间内的期望费用最小的最优控制策略~$(N^{\ast },D^{\ast },T^{\ast })$, 以及在~$N\to \mathrm{\infty }$ 时的二维最优控制策略~$(D^{\ast },T^{\ast })$ 和在~$D\to \mathrm{\infty }$ 时的二维最优控制策略~$(N^{\ast },T^{\ast })$.

In this paper, we consider the $M/G/1$ queueing system based on the $\mathrm{M}\mathrm{i}\mathrm{n}(N,D,V)$-policy and single server vacation. Under a given cost structure, the explicit expression of the long-run expected cost per unit time is derived. And through numerical calculation, the combined control policy $(N^{\ast },D^{\ast })$ which minimizes the long-run expected cost per unit time is discussed. When the vacation time is fixed as $T$, we present the explicit expression of the long-run expected cost rate with three-dimensional decision variables $(N,D,T)$. Meanwhile, we numerically determine the combined control policy $(N^{\ast },D^{\ast },T^{\ast })$ for minimizing the long-run expected cost per unit time. Furthermore, by numerical examples, we also determine the two-dimensional optimal control policy $(D^{\ast },T^{\ast })$ when $N\to \mathrm{\infty }$ as well as the two-dimensional optimal control policy $(N^{\ast },T^{\ast })$ when $D\to \mathrm{\infty }$.