修理设备可更换且修理工多重休假的机器维修模型等待时间分布函数研究

doi:10.15960/j.cnki.issn.1007-6093.2025.04.015

运筹学学报（中英文） ›› 2025, Vol. 29 ›› Issue (4): 191-204.doi: 10.15960/j.cnki.issn.1007-6093.2025.04.015

• • 上一篇

修理设备可更换且修理工多重休假的机器维修模型等待时间分布函数研究

吴文青¹, 徐海文^1,†, 余玅妙², 郑克龙¹

1. 中国民用航空飞行学院理学院, 四川成都 641419;
2. 四川师范大学数学科学学院, 四川成都 610068

收稿日期:2022-11-16 发布日期:2025-12-11
通讯作者: 徐海文 E-mail:hwxu@cafuc.edu.cn
基金资助:
2025年度中央高校基本科研业务费资助项目(Nos.25CAFUC09011,TD2025CZ10),教育部人文社会科学研究规划基金项目(No.24YJA630121),四川省科技厅资助项目(No.2025ZHCG0007),河南省重点研发专项项目(No.251111242100)

Research on the waiting time distribution of a machine maintenance model with replaceable repair facility and repairman’s multiple vacations

WU Wenqing¹, XU Haiwen^1,†, YU Miaomiao², ZHENG Kelong¹

1. Faculty of Science, Civil Aviation Flight University of China, Chengdu 641419, Sichuan, China;
2. School of Mathematical Sciences, Sichuan Normal University, Chengdu 610068, Sichuan, China

Received:2022-11-16 Published:2025-12-11

摘要/Abstract

摘要： 本文研究修理设备可更换, 且修理工多重休假的机器维修模型中, 故障机器的等待时间分布函数, 其中机器的工作时间和修理时间均服从负指数分布。利用吸收时间的马氏链过程、位相型分布和拉普拉斯--斯蒂尔切斯变换, 推导出了任意故障机器的等待时间分布函数及其平均等待时间的表达式。在此基础上, 讨论了故障机器等待时间分布函数随时间的变化情况, 并给出了相应的数值计算结果。

关键词: 机器维修模型, 修理设备可更换, 多重休假, 位相型分布, 故障机器的等待时间分布函数

Abstract: This paper considers the waiting time distribution of an arbitrary failed machine in a machine repair system with a replaceable repair facility and repairman's multiple vacations where the operating time and the repair time of machines all follow exponential distributions. By employing the Markov Chain with absorbing time, the probabilistic properties of the phase-type distribution and the Laplace-Stieltjes transform, the specific expressions of the waiting time distribution of failed machines and its mean waiting time are all derived. Based on this, we discuss the time-dependent behavior of the system performance measures and provide numerical results of the waiting time.

Key words: machine repair system, replaceable repair facility, multiple vacations, phase type distribution, the waiting time distribution of failed machine

中图分类号:

O226
O213.2

吴文青, 徐海文, 余玅妙, 郑克龙. 修理设备可更换且修理工多重休假的机器维修模型等待时间分布函数研究[J]. 运筹学学报（中英文）, 2025, 29(4): 191-204.

WU Wenqing, XU Haiwen, YU Miaomiao, ZHENG Kelong. Research on the waiting time distribution of a machine maintenance model with replaceable repair facility and repairman’s multiple vacations[J]. Operations Research Transactions, 2025, 29(4): 191-204.

参考文献

[1] Palm C. The distribution of repairmen in serving automatic machines [J]. Industritidningen Norden, 1947, 75: 75-80.
[2] Gross D, Shortle J F, Thompson J M, et al. Fundamentals of Queueing Theory [M]. New York: Wiley & Sons, 2008.
[3] 高春燕. 兼职辅助业务的机器维修模型[D]. 秦皇岛: 燕山大学, 2006.
[4] 孟艳丽. 基于有限源排队的机器维修模型的优化研究[D]. 大连: 大连海事大学, 2019.
[5] 付永红, 余玅妙, 唐应辉, 等. 两水平修理策略下的M/(Mr,Gs)/1/N/N机器维修模型稳态概率算法与性能分析[J]. 山东大学学报(理学版), 2009, 4(44): 72-78.
[6] 张静. 带有休假策略的温贮备机器维修问题的性能分析[D]. 秦皇岛: 燕山大学, 2012.
[7] Ke J C, Wang K H. Vacation policies for machine repair problem with two type spares [J]. Applied Mathematical Modelling, 2007, 31(5): 880-894.
[8] Wang K H, Chen W L, Yang D Y. Optimal management of the machine repair problem with working vacation: Newton’s method [J]. Journal of Computational and Applied Mathematics, 2009, 233(2): 449-458.
[9] Ke J C, Wu C H. Multi-server machine repair model with standbys and synchronous multiple vacation [J]. Computers & Industrial Engineering, 2012, 62(1): 296-305.
[10] Wang K H, Su J H, Yang D Y. Analysis and optimization of an M/G/1 machine repair problem with multiple imperfect coverage [J]. Applied Mathematics and Computation, 2014, 242(1): 590-600.
[11] Chen W L, Wang K H. Reliability analysis of a retrial machine repair problem with warm standbys and a single server with N-policy [J]. Reliability Engineering & System Safety, 2018, 180(12): 476-486.
[12] 曹晋华. 服务设备可修的机器服务模型分析[J]. 数学研究与评论, 1985, 5(4): 89-96.
[13] Tang Y H. Revisiting the model of servicing machines with repairable service facility-a new analyzing idea and some new results [J]. Acta Mathematicae Sinica, 2010, 26(4): 557-566.
[14] Neuts M F. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach [M]. Baltimore: The Johns Hopkins University Press, 1981.

修理设备可更换且修理工多重休假的机器维修模型等待时间分布函数研究

Research on the waiting time distribution of a machine maintenance model with replaceable repair facility and repairman’s multiple vacations

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