本文以制造系统为背景, 提出一个在服务启动$N$-策略控制下具有检修策略和不同到达率的M/G/1排队模型。首先运用更新过程理论、全概率分解技术和拉普拉斯变换, 研究系统在任意时刻$t$队长的瞬态性质, 得到了瞬态队长分布关于时间$t$的拉普拉斯变换表达式。然后在瞬态分析的基础上, 使用洛必达法则得到队长稳态分布的递推表达式。最后, 在建立费用模型下, 应用更新报酬定理, 得到系统在长期运行下单位时间内的期望费用表达式, 并通过数值实例讨论了系统启动服务的最优控制策略和最优检修策略。

Based on the background of the manufacturing system, this paper proposes an M/G/1 queuing model with maintenance strategy and different arrival rates, in which the server begins its service under the control of $N$-policy. Firstly, using the renewal process theory, total probability decomposition technique and Laplace transform, we study the transient properties of the queue length at any time $t$, and obtain the expressions of the Laplace transform of the transient queue size distribution with respect to time $t$. Then, on the basis of the transient analysis, the recursive formulas of the steady-state queue-length distribution are presented by employing L'Hospital's rule. Finally, applying the renewal reward theory, the explicit expression of the long-run expected cost per unit time of the system is obtained under a given cost structure, and numerical examples are provided to discuss the optimal control policy that the server begins its service and optimal maintenance strategy.