研究了具有多级适应性休假和$\mathrm{M}\mathrm{i}\mathrm{n}(N,V)$-策略控制的$M/G/1$排队系统, 运用全概率分解技术和拉普拉斯(L)变换工具, 研究了从任意时刻出发队长的瞬态分布, 得到了队长瞬态分布的拉普拉斯变换的表达式和稳态队长分布的递推表达式, 同时求出了稳态队长分布的概率母函数和平均队长的表达式. 进一步, 在一些特殊情况下, 例如当休假次数服从几何分布或休假次数为固定正整数值$M$时, 我们获得了相应稳态队长分布更简洁的显示表达式, 并通过数值实例阐述了稳态队长分布的显示表达式在系统容量的优化设计中的重要价值. 最后, 在建立系统费用结构模型基础上, 我们导出了系统长期单位时间的期望费用的显示表达式, 并通过数值实例不但确定了使得系统在长期单位时间内的期望费用最小的控制策略$N^{\ast }$, 而且还确定了当休假次数为固定正整数值$M$时的联合控制策略$(N^{\ast },M^{\ast })$.

In this paper, we study the $M/G/1$ queueing system with multiple adaptive vacations and $\mathrm{M}\mathrm{i}\mathrm{n}(N,V)$-policy. By using the total probability decomposition technique and the Laplace transformation tool, the transient distribution of the queue length from the beginning of the any initial is discussed. Both the expressions of the Laplace transformation of the transient queue length distribution and the recursive expressions of the steady-state queue length distribution are obtained. Meanwhile, we obtain the probability function of the steady-state queue length distribution and the expression of average queue length. Moreover, some more concise explicit expressions of the corresponding steady-state queue length distribution are obtained under some special cases such as the vacation number obeys the geometric distribution or the vacation number is a fixed positive integer $M$. And by the numerical example we illustrate the important value of the explicit expressions of the steady-state queue length distribution in the optimal design of the system capacity. Finally, the explicit expression of the long-run expected cost rate is derived under a given cost structure. Furthermore, through numerical calculation, we determine the optimal control policy $N^{\ast }$ for minimizing the long-run expected cost per unit time as well as the combined control strategy $(N^{\ast },M^{\ast })$ when the vacation number is a fixed positive integer $M$.