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排队分析

doi:10.15960/j.cnki.issn.1007-6093.2025.02.009

运筹学学报（中英文） ›› 2025, Vol. 29 ›› Issue (2): 113-127.doi: 10.15960/j.cnki.issn.1007-6093.2025.02.009

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排队分析

李丰芮¹, 唐应辉¹^,*()

四川师范大学数学科学学院, 四川成都 610068

收稿日期:2022-04-14 出版日期:2025-06-15 发布日期:2025-06-12
通讯作者: 唐应辉 E-mail:tangyh@sicnu.edu.cn
基金资助:
国家自然科学基金(71571127);四川师范大学学科建设专项项目(XKZX2021-04)

Analysis of M/G/1 queue with maintenance strategy and different arrival rates under service startup N-policy control

Fengrui LI¹, Yinghui TANG¹^,*()

School of Mathematical Sciences, Sichuan Normal University, Chengdu 610068, Sichuan, China

Received:2022-04-14 Online:2025-06-15 Published:2025-06-12
Contact: Yinghui TANG E-mail:tangyh@sicnu.edu.cn

摘要/Abstract

摘要：

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。

关键词: M/G/1排队, 检修策略, 服务启动N-策略, 队长分布, 最优策略

Abstract:

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.

Key words: M/G/1 queue, maintenance strategy, service startup N-policy, queue length distribution, optimal policy

中图分类号:

O226

李丰芮, 唐应辉. 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排队分析[J]. 运筹学学报（中英文）, 2025, 29(2): 113-127.

Fengrui LI, Yinghui TANG. Analysis of M/G/1 queue with maintenance strategy and different arrival rates under service startup N-policy control[J]. Operations Research Transactions, 2025, 29(2): 113-127.

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$ N $	$ \rho=0.2 $	$ \rho=0.6 $	$ \rho =0.9 $
1	76.132 47	88.749 37	75.691 30
2	48.758 01	68.096 89	69.530 15
3	37.846 70	55.992 24	66.243 28
4	32.938 70	49.176 65	$ \boxed{{\bf{64.993\ 83}}}$
5	30.769 62	45.416 99	64.994 91
6	$ \boxed{{\bf{30.061\ 81}}}$	43.483 75	65.758 61
7	30.222 71	42.699 93	67.001 17
8	30.941 61	$ \boxed{{\bf{42.676\ 70}}}$	68.556 02
9	32.040 11	43.179 49	70.322 41
10	33.408 55	44.060 01	72.237 31

$ N $	$ T_N^* $	$ F(T_N^*, N) $	$ {\left. {\frac{{{\rm \partial} F(T, N)}}{{{\rm \partial} T}}} \right\|_{T = T_N^ * }} $	$ N $	$ T_N^* $	$ F(T_N^*, N) $	$ {\left. {\frac{{{\rm \partial} F(T, N)}}{{{\rm \partial} T}}} \right\|_{T = T_N^ * }} $
1	12.47	47.18	0	9	3.98	42.89	0
2	12.29	47.11	0	10	4.34	43.75	0
3	11.72	46.88	0	11	4.80	44.87	0
4	10.06	46.24	0	12	5.35	46.18	0
5	5.54	44.65	0	13	5.96	47.63	0
6	4.09	43.05	0	14	6.61	49.21	0
7	3.76	$ \boxed{{\bf{42.37}}}$	0	15	7.31	50.87	0
8	3.77	42.39	0	16	8.03	52.61	0

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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排队分析

Analysis of M/G/1 queue with maintenance strategy and different arrival rates under service startup N-policy control

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