本文考虑一个具有修正$(p, N)$-策略和单重休假的$M/G/1$排队系统, 其中修正$(p, N)$-策略是指当服务员的休假结束回到系统时, 如果系统中有顾客但顾客数少于$N$, 则服务员以概率$p(0 \le p \le 1)$启动服务, 以概率$(1-p)$不启动服务直到系统中的顾客数累积到$N$个。运用更新过程理论、全概率分解技术和Laplace变换工具, 我们讨论了系统队长的瞬态分布, 得到队长瞬态分布关于时间$t$的L变换表达式。然后使用洛必达法则, 通过直接运算得到队长稳态分布的递推公式, 同时获得稳态队长分布的概率母函数和平均队长的显示表达式。最后, 应用更新报酬定理给出系统在长期运行单位时间内的期望费用的显示表达式, 并通过数值实例讨论了使得系统期望费用最小的最优控制策略$N^*$, 以及休假时间为定长$T(T\geqslant 0)$时的二维最优控制策略$(N^*, T^*)$。

This paper considers an $M/G/1$ queueing model with single vacation and modified $(p, N)$-policy. The modified $(p, N)$-policy means that when the vacation ends and the server returns to the system, if there are less than $N$ customers but at least one customer in the system, the server begins service with probability $p (0 \le p \le 1)$ or stays idle with probability $(1-p)$ until there are $N$ customers in the system and starts its service at once. By the renewal process theory, total probability decomposition technique and Laplace transform tool, we study the transient queue length distribution of the system, and obtain the expressions of the Laplace transform of the transient queue length distribution with respect to time $t$. Then, employing L'Hospital's rule and some algebraic manipulations, the recursive formulas of the steady-state queue length distribution are derived. Meanwhile, the explicit expressions for probability generating function of the steady-state queue length distribution and the expected queue size are presented. Finally, employing the renewal reward theorem, the explicit expression of the long-run expected cost per unit time is also presented. Numerical examples are provided to discuss the optimal control policy $N^*$ for economizing the system cost as well as the optimal two-dimensional control policy $(N^*, T^*)$ when the vacation time is a fixed length $T (T \ge 0)$.