在延迟Min (N, D)-策略M/G/1可修排队系统及最优控制策略

摘要/Abstract

摘要：

该文考虑基于延迟Min（N，D）-策略M/G/1可修排队系统，其中修理设备在修理故障服务台期间可发生故障且可更换.使用全概率分解技术和拉普拉斯变换工具，分别讨论了服务台和修理设备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.最后在给定的费用结构下，用数值计算实例确定了使系统长期单位时间内期望费用最小的最优控制策略（N^*，D^*）.

关键词: Min(N, D)-策略, 修理设备, 延迟Min(N, D)-策略, 排队指标, 可靠性指标, 最优控制策略

Abstract:

This paper considers the M/G/1 repairable queueing system with delay Min(N, D)-policy, in which the repair facility subject to breakdowns and then replaced during the repair facility busy period. By using the total probability decomposition technique and employing the Laplace transform tool, some reliability indices of the service station and the repair facility, such as the transient-state and steady-state unavailability, the expected failure number during (0, t] are discussed. Finally, it is determined the optimal control policy (N^*, D^*) such that the long-run expected cost rate is minimum under a given cost structure.

Key words: Repair facility, Delay Min(N, D)-policy, Queueing index, Reliability index, Optimal control policy

中图分类号:

O121

潘取玉,唐应辉. 在延迟Min (N, D)-策略M/G/1可修排队系统及最优控制策略[J]. 数学物理学报, 2018, 38(5): 1014-1031.

Quyu Pan,Yinghui Tang. Analysis of M/G/1 Repairable Queueing System and Optimal Control Policy with a Replaceable Repair Facility Under Delay Min(N, D)-Policy[J]. Acta mathematica scientia,Series A, 2018, 38(5): 1014-1031.

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参考文献 25

1	曹晋华, 程侃. 服务台可修的M/G/1排队系统分析. 应用数学学报, 1982, 5 (2): 113- 127
	Cao J H , Cheng K . Analysis of M/G/1 queueing system with repairable service station. Acta Mathematicae Applicatae Sinica, 1982, 5 (2): 113- 127
2	曹晋华. 服务设备可修的机器服务模型分析. 数学研究与评论, 1985, 5 (4): 89- 96
	Cao J H . Analysis of machine service model with a repairable service equipment. Journal of Mathematical Research and Exposition, 1985, 5 (4): 89- 96
3	Cao J H , Wu Y H . Reliability analysis of a multistate system with a replaceable repair facility. Acta Mathematicae Applicatae Sinica, 1988, 4 (2): 113- 121 doi: 10.1007/BF02006059
4	Cao J H , Wu Y H . Reliability analysis of a two unit cold standby system with a replaceable repair facility. Microelectronics and Reliability, 1989, 29 (4): 145- 150
5	唐应辉. 服务台可修的M/G/1排队系统的进一步分析. 系统工程理论与实践, 1996, 16 (4): 45- 51
	Tang Y H . Further analysis of M/G/1 queueing system with repairable service station. Systems EngineeringTheory & Practice, 1996, 16 (4): 45- 51
6	Tang Y H . A single-server M/G/1 queuing system subject to breakdowns-some reliability and queueing problems. Microelectronics and Reliability, 1997, 37 (2): 315- 321 doi: 10.1016/S0026-2714(96)00018-2
7	Li W , Shi D H , Chao X . Reliability analysis of M/G/1 queuing systems with server breakdowns and vacations. Journal of Applied Probability, 1997, 34 (2): 546- 555 doi: 10.2307/3215393
8	唐应辉, 赵玮. 可修排队系统可靠性指标的分解特性. 运筹学学报, 2004, 8 (4): 73- 84 doi: 10.3969/j.issn.1007-6093.2004.04.010
	Tang Y H , Zhao W . The decomposition properties of reliability indices in repairable queueing systems. Operations Research Transactions, 2004, 8 (4): 73- 84 doi: 10.3969/j.issn.1007-6093.2004.04.010
9	余玅妙, 唐应辉. 多级适应性延误休假M^X/G(M/G)/1可修排队系统的可靠性指标. 运筹学学报, 2008, 12 (3): 103- 112 doi: 10.3969/j.issn.1007-6093.2008.03.011
	Yu M M , Tang Y H . Some reliability indices in M^X/G(M/G)/1 repairable queueing system with adaptive multistate delay vacation. Operations Research Transactions, 2008, 12 (3): 103- 112 doi: 10.3969/j.issn.1007-6093.2008.03.011
10	Lee H W , Seo W J . Control policies of an M/G/1 queue under the dyadic Min(N, D)-policy and its cost optimization. Performance Evaluation, 2008, 65 (10): 742- 758 doi: 10.1016/j.peva.2008.04.006
11	刘云颇, 唐应辉. 多重休假中以概率p进入的M/G/1可修排队系统. 系统工程学报, 2011, 26 (5): 718- 723
	Liu Y P , Tang Y H . M/G/1 repairable queueing system with p-entering discipline during server vacations. Journal of Systems Engineering, 2011, 26 (5): 718- 723
12	Tang Y H , Yu M M , Yun X , Huang S J . Reliability indices of discrete-time GeoX/G/1 queueing system with unreliable service station and multiple adaptive delayed vacations. Journal of Systems Science and Complexity, 2012, 25 (6): 1122- 1135 doi: 10.1007/s11424-012-1062-9
13	唐应辉, 俞国建, 李晓东. 修理设备可更换且修理延迟的两同型部件并联可修系统. 工程数学学报, 2005, 22 (1): 1- 8 doi: 10.3969/j.issn.1005-3085.2005.01.001
	Tang Y H , Yu G J , Li X D . Two-unit same paralleled repairable system with a replaceable facility and delay repair. Journal of Engineering Mathematics, 2005, 22 (1): 1- 8 doi: 10.3969/j.issn.1005-3085.2005.01.001
14	刘仁彬, 唐应辉, 骆川义. 修理设备可更换且修理延迟的N部件串联系统分析. 数学的实践与认识, 2007, 37 (1): 49- 54 doi: 10.3969/j.issn.1000-0984.2007.01.009
	Liu R B , Tang Y H , Luo C Y . Analysis of N-unit series system with a replaceable facility and delay repair. Mathematics in Practice and Theory, 2007, 37 (1): 49- 54 doi: 10.3969/j.issn.1000-0984.2007.01.009
15	唐应辉, 魏瑛源. 修理有延迟且修理设备可更换的多状态可修系统的可靠性分析. 数学的实践与认识, 2010, 40 (4): 100- 106
	Tang Y H , Wei Y Y . Reliability analysis of a multi-state repairable system with a replaceable repair facility and delay repair. Mathematics in Practice and Theory, 2010, 40 (4): 100- 106
16	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. 系统工程学报, 2013, 28 (6): 830- 838
	Tang Y H , Feng H X , Wu W Q . Analysis of M/G/1 repairable queueing system with a replaceable repair facility. Journal of Systems Engineering, 2013, 28 (6): 830- 838
17	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. 系统工程理论与实践, 2014, 34 (3): 746- 754
	Tang Y H , Zhu Y L , Wu W Q . Analysis of M/G/1 repairable queueing system with N-policy and a replaceable repair facility. Systems Engineering-Theory & Practice, 2014, 34 (3): 746- 754
18	Wu W Q , Tang Y H , Yu M M . Analysis of an M/G/1 queue with N-policy, single vacation, unreliable service station and replaceable repair facility. OPSEARCH, 2015, 52 (4): 670- 691 doi: 10.1007/s12597-015-0201-1
19	兰绍军, 唐应辉. 具有Bernoulli反馈和Min(N, D)策略控制的Geo^{λ₁, λ₂}/G/1离散时间可修排队的可靠性分析. 系统科学与数学, 2016, 36 (11): 2070- 2086
	Lan S J , Tang Y H . Reliability analysis of discrete-time Geo^{λ₁, λ₂}/G/1 repairable queue with bernoulli feedback and Min(N, D)-policy. Journal of Systems Science and Mathematical Sciences, 2016, 36 (11): 2070- 2086
20	唐应辉, 吴文青, 刘云颇, 刘晓云. 基于多重休假的Min(N, V) -策略M/G/1排队系统的队长分布. 系统工程理论与实践, 2014, 34 (6): 1533- 1546
	Tang Y H , Wu W Q , Liu Y P , Liu X Y . Queue length distribution of M/G/1 queueing system with Min(N, V)-policy based on multiple server vacations. Systems Engineering-Theory & Practice, 2014, 34 (6): 1533- 1546
21	魏瑛源, 唐应辉, 余玅妙. 基于Min(N, D) -策略的M/G/1排队系统的队长分布及最优策略. 系统科学与数学, 2015, 35 (6): 729- 744
	Wei Y Y , Tang Y H , Yu M M . Queue length distribution and optimal policy for M/G/1 queueing system under Min(N, D)-policy. Journal of Systems Science and Mathematical Sciences, 2015, 35 (6): 729- 744
22	魏瑛源, 唐应辉, 余玅妙. Min(N, D) -延迟Min(N, D) -策略的M/G/1排队系统的队长分布与数值计算. 运筹学学报, 2016, 20 (2): 23- 37
	Wei Y Y , Tang Y H , Yu M M . Queue length distribution and numerical calculation of M/G/1 queueing system with delay Min(N, D)-policy. Operations Research Transactions, 2016, 20 (2): 23- 37
23	高丽君, 唐应辉. 具有Min(N, D) -策略控制的M/G/1可修排队系统及最优控制策略. 数学物理学报, 2017, 37A (2): 352- 365 doi: 10.3969/j.issn.1003-3998.2017.02.014
	Gao L J , Tang Y H . M/G/1 Repairable Queueing System and Optimal Control Policy with Min(N, D)-policy. Acta Mathematica Scientia, 2017, 37A (2): 352- 365 doi: 10.3969/j.issn.1003-3998.2017.02.014
24	曹晋华, 程侃. 可靠性数学引论(修订版). 北京: 科学出版社, 1986
	Cao J H , Cheng K . Introduction to reliability mathematics (revised edition). Beijing: Science Press, 1986
25	唐应辉, 唐小我. 排队论-基础与分析技术. 北京: 科学出版社, 2006
	Tang Y H , Tang X W . Queueing Theory:Foundations and Analysis Techniques. Beijing: Science Press, 2006

	D
N	1	2	3	4	5	5.1381	6	7	8	9	10
1	72.7177	72.7177	72.7177	72.7177	72.7177	72.7177	72.7177	72.7177	72.7177	72.7177	72.7177
2	62.9313	62.5521	62.5335	62.5325	62.5325	62.5325	62.5325	62.5325	62.5325	62.5325	62.5325
3	58.9588	57.9496	57.8746	57.8695	57.8692	57.8692	57.8692	57.8692	57.8692	57.8692	57.8692
4	57.7637	56.3324	56.1923	56.1802	56.1793	56.1792	56.1792	56.1792	56.1792	56.1792	56.1792
5	57.7136	56.3969	56.2830	56.2739	56.2733	56.2732	56.2732	56.2732	56.2732	56.2732	56.2732
6	57.9797	57.3362	57.4734	57.5142	57.5207	57.5209	57.5215	57.5216	57.5216	57.5216	57.5216
7	58.2257	58.6056	59.3190	59.5182	59.5541	59.5556	59.5591	59.5597	59.5598	59.5598	59.5598
8	58.3738	59.8524	61.4852	62.0234	62.1400	62.1452	62.1592	62.1618	62.1621	62.1621	62.1621
9	58.4444	60.8845	63.7075	64.8239	65.1145	65.1292	65.1712	65.1801	65.1813	65.1814	65.1814
10	58.4728	61.6369	65.7848	67.7421	68.3483	68.3828	68.4877	68.5132	68.5171	68.5176	68.5177