PDF(591 KB)
具有温储备失效和延迟修理的M/G/1可修排队系统的可靠性指标
刘金银, 唐应辉, 朱亚丽, 余玅妙
系统工程理论与实践 ›› 2015, Vol. 35 ›› Issue (2) : 413-423.
PDF(591 KB)
PDF(591 KB)
具有温储备失效和延迟修理的M/G/1可修排队系统的可靠性指标
Reliability indices of M/G/1 repairable queueing system with warm standby failure and delayed repair
本文考虑了具有温储备失效特征的M/G/1可修排队系统. 在该系统中, 服务台故障分为两类: 第一类是服务台在服务员的"广义忙期"中以故障率为α(0≤α<∞) 的泊松过程发生故障, 第二类是服务台在系统闲期中以分布函数为Y(t)的更新过程发生故障, 而且发生第二类故障时不能得到立即修理. 利用全概率分解技术和拉普拉斯变换工具, 分别讨论了在两类故障模式下服务台的瞬态不可用度和稳态不可用度, (0,t]时间内的平均故障次数和稳态故障频度等可靠性指标, 进一步还讨论了服务台由温储备失效引起等待修理的概率. 最后, 通过数值计算例子讨论了系统有关参数对服务台的第二类稳态不可用度和第二类稳态故障频度的影响.
This paper considers the M/G/1 repairable queuing system with warm standby failure. While the service station is in the course of "generalized busy period" it is subject to breakdowns (we call this the 1 type failure) according to a Poisson process with rate α(0≤α<∞), while the service station is in the system idle period it is subject to breakdowns (we call this the 2 type failure) according to a renewal process with distribution Y(t) and the broken service station don't be repaired at once. By using the total probability decomposition technique and the Laplace transform, some reliability indices of the service station, such as the transient-state and steady-state unavailability, the expected failure number during (0,t] and so on, are studied under two failed states. Further, a new index which is the waiting repair probability of the service station is achieved. At last, we present several numerical examples under some special cases to discuss the sensitivity of the steady-state unavailability and the failure frequency in the 2 type failure of the service station.
可修排队系统 / 温储备失效 / 不可用度 / 故障次数 / 全概率分解 {{custom_keyword}} /
repairable queueing system / warm standby failure / unavailability / failure number / total probability decomposition {{custom_keyword}} /
[1] 曹晋华, 程侃. 服务台可修的M/G/1排队系统分析[J]. 应用数学学报, 1982, 5(2): 113-127.Cao Jinhua, Cheng Kan. Analysis of M/G/1 queueing system with reparable service station[J]. Acta Mathematicae Applicatae Sinica, 1982, 5(2): 113-127.
[2] 唐应辉. 服务台可修的M/G/1排队系统的进一步分析[J]. 系统工程理论与实践, 1996, 16(4): 45-51.Tang Yinghui. Further analysis of M/G/1 queueing system with reparable service station[J]. Systems Engineering —— Theory & Practice, 1996, 16(4): 45-51.
[3] 唐应辉. 分析M/G/1排队系统队长分布的方法注记[J]. 系统工程理论与实践, 1996, 16(1): 46-50.Tang Yinghui. A note on the method of analyzing the queue of M/G/1 queueing system[J]. Systems Engineering —— Theory & Practice, 1996, 16(1): 46-50.
[4] 史定华, 张文国. 具有多重延误休假的Mx/G(M/G)/1(M/G)可修排队系统分析[J]. 应用数学学报, 1994, 17(2): 201-214.Shi Dinghua, Zhang Wenguo. Analysis of repairable queueing system Mx/G(M/G)/1(M/G) with multiple delay vacations[J]. Acta Mathematicae Applicatae Sinica, 1994, 17(2): 201-214.
[5] 唐应辉, 唐小我. 推广的Mx/G(M/G)/1(M/G)可修排队系统(I) ——- 一些排队指标[J]. 系统科学与数学, 2000, 20(4): 385-397.Tang Yinghui, Tang Xiaowo. The generalized Mx/G(M/G)/1(M/G) repairable queueing system(I) —— Some queueing indices[J]. Journal of Systems Science & Mathematical Sciences, 2000, 20(4): 385-397.
[6] 唐应辉, 唐小我. 推广的Mx/G(M/G)/1(M/G)可修排队系统(II) ——- 一些可靠性指标[J]. 系统工程理论与实践, 2000, 20(2): 84-91.Tang Yinghui, Tang Xiaowo. The generalized Mx/G(M/G)/1(M/G) repairable queueing system(II) —— Some reliability indices[J]. Systems Engineering —— Theory & Practice, 2000, 20(2): 84-91.
[7] Tang Y H. Some reliability problems arising in the Mx/G(M/G)/1(M/G) repairable queueing system single delay vacation[J]. Journal of Systems Science & Systems Engineering, 2001, 10(3): 306-314.
[8] 唐应辉, 赵玮. 可修排队系统可靠性的分解特性[J]. 运筹学学报, 2004, 8(4): 73-84.Tang Yinghui, Zhao Wei. The decomposition properties of reliability indices in repairable queueing systems[J]. OR Transactions, 2004, 8(4): 73-84.
[9] 王聚丰, 朱翼隽, 孙凤欣. 服务速度有变化的可修M/G/1排队系统[J]. 兰州大学学报, 2003, 39(5): 19-24.Wang Jufeng, Zhu Yijun, Sun Fengxin. M/G/1 repairable queueing system with changeable service speed[J]. Journal of Lanzhou University, 2003, 39(5): 19-24.
[10] 朱翼隽, 张峰. 具有两种服务速度的可修Mx/G/1排队系统[J]. 江苏大学学报, 2005, 26(5): 417-420.Zhu Yijun, Zhang Feng. Repairable Mx/G/1 queueing system with two service speeds[J]. Journal of Jiangsu University, 2005, 26(5): 417-420.
[11] 骆川义, 唐应辉, 曹保山. 启动-关闭型多级适应性休假Mx/G/1可修排队的可靠性分析[J]. 工程数学学报, 2007, 24(2): 222-228.Luo Chuanyi, Tang Yinghui, Cao Baoshan. Reliability of Mx/G/1 repairable queue with adaptive multistage vacations and set-up-close-down time[J]. Journal of Engineering Mathematics, 2007, 24(2): 222-228.
[12] 余#
[42] 妙, 唐应辉. 多级适应性延误休假Mx/G(M/G)/1可修排队系统的可靠性指标[J]. 运筹学学报, 2008, 12(3): 103-112.Yu Miaomiao, Tang Yinghui. Some reliability indices in Mx/G(M/G)/1(M/G) repairable queueing system with adaptive multistate delay vacation[J]. OR Transactions, 2008, 12(3): 103-112.
[13] Kuo H W, Shu L C, Wen L P. Maximum entropy analysis to the N policy M/G/1 queueing system with a removable server[J]. Applied Mathematical Modelling, 2002, 26: 1151-1162.
[14] 岳德权, 吕胜利, 李静铂. 一个修理工的M/M/N可修排队的可靠性分析[J]. 应用数学学报, 2004, 17(增): 5-9.Yue Dequan, Lü Shengli, Li Jingbo. Reliability analysis of M/M/N queueing system with repairable service stations and a single repairman[J]. Acta Mathematicae Applicatae Sinica, 2004, 17(supplement): 5-9.
[15] 岳德权, 李静铂, 吕胜利. 两个修理工的M/M/2可修排队系统[J]. 数学物理学报, 2008, 28A(6): 1109-1118.Yue Dequan, Li Jingbo, Lü Shengli. The M/M/2 repairable queue system with two repairmen[J]. Acta Mathematica Scientia, 2008, 28A(6): 1109-1118.
[16] 朱翼隽, 鲍媛媛. Mx/M/N可修排队系统[J]. 工程数学学报, 2009, 26(6): 1057-1061.Zhu Yijun, Bao Yuanyuan. Mx/M/N repairable queue system[J]. Journal of Engineering Mathematics, 2009, 26(6): 1057-1061.
[17] Kuo H W, Tsung Y W, Wen L P. Optimal control of the N policy M/G/1 queueing system with server breakdowns and general stratup times[J]. Applied Mathematical Modelling, 2007, 31: 2199-2212.
[18] Tang Y H, Yun X, Huang S J. Discrete-time Geox/G/1 queue with unreliable server and multiple adaptive delayed vacation[J]. Journal of Computational and Applied Mathematics, 2008, 220: 439-455.
[19] Ke J C, Huang K B. Analysis of an unreliable server Mx/G/1 system with a randomized vacation policy and delayed repair[J]. Stochastic Models, 2010, 26: 212-241.
[20] Tang Y H, Yu M M, Yun X, et al. Reliability indices of discrete-time Geox/G/1 queueing system with unreliable service station and multiple adaptive delayed vacations[J]. Journal of Systems Science & Complexity, 2012, 25(6): 1122-1135.
[21] Yu M M, Tang Y H, Fu Y H, et al. A geometric process model for M/PH(M/PH)/1/k queue with new service machine procurement lead time[J]. International Journal of Systems Science, 2013, 44(6): 1061-1075.
[22] 唐应辉, 冯慧侠, 吴文青. 修理设备可更换的M/G/1可修排队系统分析[J]. 系统工程学报, 2013, 28(6): 830-839.Tang Yinghui, Feng Huixia, Wu Wenqing. Analysis of M/G/1 repairable queueing system with a replaceable repair facility[J]. Journal of Systems Engineering, 2013, 28(6): 830-839.
[23] 唐应辉, 朱亚丽, 吴文青. 修理设备可更换的N-策略M/G/1可修排队系统分析[J]. 系统工程理论与实践, 2014, 34(3): 746-755.Tang Yinghui, Zhu Yali, Wu Wenqing. Analysis of M/G/1 repairable queueing system with N-policy and a replaceable repair facility[J]. Systems Engineering —— Theory & Practice, 2014, 34(3): 746-755.
[24] Tang Y H. A single-serve M/G/1 queueing system subject to breakdowns-some reliability and queueing problems[J]. Microelectronics Reliability, 1997, 37(2): 315-321.
[25] 牟永聪. 在闲期内可能发生故障的Mx/G/1可修排队系统分析[D]. 成都: 四川师范大学, 2011.Mou Yongcong. Analysis of Mx/G/1 repairable queueing system in which the service station may fail in the idle periods[D]. Chengdu: Sichuan Normal University, 2011.
[26] 李才良, 唐应辉, 牟永聪, 等. 在第二类故障期间以概率p进入的M/G/1可修排队系统 ——- 一些排队指标[J]. 数学物理学报, 2012, 32A(6): 1149-1157.Li Cailiang, Tang Yinghui, Mu Yongcong, et al. M/G/1 repairable queueing system with p-entering discipline during second type failure times —— Some queueing indices[J]. Acta Mathematica Scientia, 2012, 32A(6): 1149-1157.
[27] 唐应辉, 牟永聪, 余#
[42] 妙. 在第二类故障期间以概率p进入的M/G/1可修排队系统 ——- 一些可靠性指标[J]. 系统工程学报, 2012, 27(4): 559-567.Tang Yinghui, Mu Yongcong, Yu Miaomiao. Analysis of reliability on M/G/1 repairable queueing system with p-entering discipline during second type failure times —— Some reliability indices[J]. Journal of Systems Engineering, 2012, 27(4): 559-567.
[28] 朱亚丽, 唐应辉. 具有温储备失效的M/G/1可修排队系统[J]. 系统工程理论与实践, 2014, 34(4): 944-950.Zhu Yali, Tang Yinghui. M/G/1 repairable queuing system with failure in warm standby[J]. Systems Engineering —— Theory & Practice, 2014, 34(4): 944-950.
[29] 唐应辉, 唐小我. 排队论:基础与分析技术[M]. 北京: 科学出版社, 2006.Tang Yinghui, Tang Xiaowo. Queueing theory: Foundations and analysis techniques[M]. Beijing: Science Press, 2006.
[30] 曹晋华, 程侃. 可靠性数学引论[M]. 北京: 科学出版社, 1986.Cao Jinhua, Cheng Kan. Introduction to reliability mathematics[M]. Beijing: Science Press, 1986.
国家自然科学基金(71171138, 71301111)
/
| 〈 |
|
〉 |
