PDF《带启动时间和工作故障的M/M/1/N排队系统性能分析》

第36 卷第

4 期2019 年4月控制理论与应用Control Theory &

Applications Vol.

36 No.

4 Apr.

2019 带带带启 启 启动 动 动时 时 时间 间 间和 和 和工 工 工作 作 作故 故 故障 障 障的 的的M/M/1/N排排排队 队 队系 系 系统 统 统性 性 性能 能 能分 分 分析 析析杨喜娟1,3? , 李忠学2 , 黎锁平3 , 武福2 (1. 兰州交通大学 电子与信息工程学院, 甘肃 兰州 730070;

2. 兰州交通大学 机电学院, 甘肃 兰州 730070;

3. 兰州理工大学 电气工程与信息工程学院, 甘肃 兰州 730050) 摘要: 在M/M/1/N可修排队系统中引入了工作故障和启动时间. 服务台在忙期允许出现故障, 且在故障期间不是 完全停止服务而是以较低的服务速率为顾客服务. 同时, 从关闭期到正规忙期有服从指数分布的启动时间. 通过分 析此模型的二维连续时间 Markov过程, 求解出系统平稳方程, 建立此系统的有限状态拟生灭过程(QBD). 根据系统 参数, 求解出水平相依的子率阵, 从而得到系统稳态概率向量的矩阵几何表示形式. 在系统稳态概率向量的基础上, 求解出系统吞吐率、 系统稳态可用度、 系统稳态队长及系统处于各个状态的概率等性能指标的解析表达式. 文中的 敏感性分析体现了这种方法的有效性和可用性, 同时, 对系统各性能受系统参数的影响进行了探索. 实验表明, 文 中提出模型的稳定性较好, 且更贴近实际服务过程, 因此这种模型将被广泛应用于各种实际服务中. 关键词: 可修排队系统;

工作故障;

拟生灭过程;

矩阵几何方法;

性能分析 引用格式: 杨喜娟, 李忠学, 黎锁平, 等. 带启动时间和工作故障的M/M/1/N排队系统性能分析. 控制理论与应用, 2019, 36(4):

561 C

569 DOI: 10.7641/CTA.2018.80415 Performance analysis of M/M/1/N queue with setup time and working breakdown YANG Xi-juan1,3?, LI Zhong-xue2, LI Suo-ping3, WU Fu2 (1. School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou Gansu 730070, China;

2. School of Mechanical Engineering, Lanzhou Jiaotong University, Lanzhou Gansu 730070, China;

3. School of Electrical and Information Engineering, Lanzhou University of Technology, Lanzhou Gansu 730050, China) Abstract: In this paper, the working breakdown and setup time strategies are introduced into the M/M/1/N repairable queueing system. The server is subject to breakdown when it is busy, rather than completely stopping service, it will decrease its service rate. Meanwhile, setup time, following exponential distribution, exists from idle period to regular busy period. The steady state equations are obtained by analyzing the two dimensional continuous time Markov process of the system, and the ?nite quasi birth and death (QBD) process of the system is established. According to system parameters, the level dependent sub-rate matrices are solved and the matrix geometric representation of the steady state probability vector of the system is obtained. Based on the steady state probability vector, the analytic expression of the performances, such as the throughput of the system, the steady state availability, the steady state queueing length and probability of each states, are obtained. The effectiveness and availability of the approach are fully shown in the sensitivity analysis and the in?uences of the parameters on the performances of the system are explored preliminarily. Experiments demonstrate that the proposed model is more stable and closer to the actual service process. Therefore, the model will be widely used in various practical services. Key words: repairable queueing system;