带启动时间和工作故障的M/M/1/N排队系统性能分析

Performance analysis of M/M/1/N queue with setup time and working breakdown

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.

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 finite 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 influences 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.