Optimal Control Policy for the M/G/1 Repairable Queueing System with Random Start-up Time and Bi-level Threshold(m,N)-Policy

doi:10.12005/orms.2021.0315

Abstract

Abstract: This paper is concerned with M/G/1 repairable queueing system with random start-up time and bi-level threshold (m,N)-strategy. Firstly, some related queueing indices of the system are analyzed. Then, we study the following reliability indices of the system caused by faults, for example, the distribution function of the first failure time, the probability of the system failure, and the expected failure number within a time interval(0,t]。 Finally, on the basis of the established cost model and combined with the actual situation of the company's testing samples, we study the optimal control strategy of double threshold(m^*,N^*), as well as compare the optimal control policy of the queueing system under which the service station doesn't fail.

Key words: M/G/1 repairable queue, bi-level threshold(m,N)-policy, queueing index, reliability index, optimal control policy

摘要： 本文研究带随机启动时间与双阈值(m,N)-策略的M/G/1可修排队系统,首先讨论系统有关的排队指标,接着研究因为故障而产生的系统的下列可靠性指标,如:服务台首次失效前的寿命分布、不可用度和(0,t]时间内的平均故障次数。最后,在建立费用模型的基础上,结合实际中检测公司检测样品的这一现实情况,研究了双阈值最优控制策略(m^*,N^*),并在同一组参数下与服务台不发生故障时系统的双阈值最优控制策略进行了比较。

关键词: M/G/1可修排队, 双阈值(m,N)-策略, 排队指标, 可靠性指标, 最优控制策略

CLC Number:

O226

KUANG Xin-yu, TANG Ying-hui. Optimal Control Policy for the M/G/1 Repairable Queueing System with Random Start-up Time and Bi-level Threshold(m,N)-Policy[J]. Operations Research and Management Science, 2021, 30(10): 64-70.

旷欣宇, 唐应辉. 带随机启动时间与双阈值(m,N)-策略的M/G/1可修排队系统的最优控制策略[J]. 运筹与管理, 2021, 30(10): 64-70.

References

[1] Heyman D P. T-policy for the M/G/1 queue[J]. Management Science, 1977, 23(7): 775-778.
[2] Zhu Y J, Pan X C, Hu B. Geom/Geom/1 queue with working vacations and N-policy and set-up time[J]. Communication on Applied Mathematics & Computation, 2010, 24(1): 25-34.
[3] 唐应辉,蒲会,余玅妙.带启动时间的N-策略M/G/1排队系统的队长[J].系统工程理论与实践,2011,31(1):131-137.
[4] Lan S J, Tang Y H. Analysis of D-policy discrete-time Geo/G/1 queue with second J-optional service and unreliable server[J]. RAIRO-Operations Research, 2017, 51(1): 101-122.
[5] 唐应辉,吴文青,刘云颇.基于多重休假的Min(N,V)-策略M/G/1排队系统的队长分布[J].系统工程理论与实践,2014,34(6):1533-1546.
[6] 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[J].系统科学与数学,2015,35(6):729-744.
[7] Gu J X, Wei Y Y, Tang Y H, et al. Queue size distribution of Geo/G/1 queue under the Min(N,D)-policy[J]. Journal of Systems Science and Complexity, 2016, 29(3): 752-771.
[8] 兰绍军,唐应辉.具有Bernoulli反馈和Min(N,D)-策略控制的离散时间可修排队的可靠性分析[J].系统科学与数学,2016,36(11):2070-2086.
[9] 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[J].数学物理学报,2017,37(2):352-365.
[10] 潘取玉,唐应辉.在延迟Min(N,D)-控制策略下修理设备可更换的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):184-201.
[11] Lee H W, Park J G. Optimal strategy in N-policy production system with early set-up[J]. The Journal of the Operational Research Society, 1997, 48(3): 306-313.
[12] Lee H W, Park J G, Kim B K. Queue length and waiting time analysis of a batch arrival queue with bi-level control[J]. Computers & Operations Research, 1998, 25(3): 191-205.
[13] Lee H W, Park N I, Jeon J. Queue length analysis of batch arrival queues under bi-level threshold control with early set-up[J]. International Journal of Systems Science, 2003, 34(3): 195-204.
[14] Ke J C . Bi-level control for batch arrival queues with an early startup and unreliable server[J]. Applied Mathematical Modelling, 2004, 8(5): 469-485.
[15] 刘名武,杨迎春,马永开.一类排队服务系统的最优控制策略研究[J].控制理论与应用,2012,29(3):323-330.
[16] Luo C Y, Tang Y H, Yu K Z, et al. Optimal (r,N)-policy for discrete-time Geo /G∕1 queue with different input rate and setup time[J]. Applied Stochastic Models in Business & Industry, 2015, 31(4): 405-423.
[17] 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.
[18] 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.
[19] Tang Y H. A single-server M/G/1 queueing system subject to breakdowns——some reliability and queueing problems[J]. Microelectronics & Reliability, 1997, 37(2): 315-321.
[20] 唐应辉,赵玮.可修排队系统可靠性指标的分解特性[J].运筹学学报,2004,8(4):73-84.
[21] Tang Y H. Revisiting the model of servicing machines with repairable service facility—a new analyzing idea and some new results[J]. Acta Mathematicae Applicatae Sinica, 2010, 26(4): 557-566.
[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.
[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.
[24] 蔡晓丽,唐应辉.具有温储备失效特征和单重休假Min(N,V)-控制策略的M/G/1可修排队系统[J].应用数学学报,2017,40(5):692-701.
[25] Lan S J, Tang Y H. Performance and reliability analysis of a repairable discrete-time Geo/G/1 queue with Bernoulli feedback and randomized policy[J]. Applied Stochastic Models in Business and Industry, 2017, 33(5): 522-543.
[26] 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.
[27] 曹晋华,程侃.可靠性数学引论(修订版)[M].北京:高等教育出版社,2006.