带工作休假和工作故障的M/M/1/N排队系统性能分析

Performance analysis of M/M/1/N queueing system with working vacation and working breakdown

本文在可修M/M/1/N排队系统中引入了启动时间、工作休假和工作故障策略.在该系统中,服务台在休假期间不是完全停止工作,而是处于低速服务状态.设定服务台在任何时候均可发生故障,当故障发生时立刻进行维修.且当服务台在正规忙期出现故障时,服务台仍以较低的服务速率为顾客服务.服务台的寿命时间和修理时间均服从指数分布,且在不同的时期有不同的取值.同时,从关闭期到正规忙期有服从指数分布的启动时间.本文建立此模型的有限状态拟生灭过程(QBD),使用矩阵几何方法得到系统的稳态概率向量,并应用基本阵和协方差矩阵理论,计算出系统稳态可用度、系统方差、系统吞吐率、系统稳态队长及各系统稳态概率等系统性能指标.同时,通过数值实验对各系统参数对系统性能的影响进行了初探.文中的敏感性分析体现了这种方法的有效性和可用性.实验表明,文中提出的模型,可有效改善仅带有工作休假或工作故障策略排队模型的系统性能.

In this paper,the strategies,such as the working vacation,working breakdown and setup time,are introduced into the M/M/1/N repairable queueing system.In the system,the server works at a lower service rate instead of stop working completely during the vacation period.The server is subject to breakdown at any time and is repaired immediately when a breakdown occurs.Furthermore,if breakdowns occur during the regular busy period,the server also works at a lower service rate for customers.Both the time to breakdown and the time to the end of repair for the server follow exponential distributions,and they have different values in different period respectively.Meanwhile,setup times from shut down period to regular busy period follow exponential distribution too.The paper establishes the finite quasi birth and death process(QBD)of the system.Matrix-geometric approach is utilized to develop the steady state probability vector of the system.Based on the fundamental matrix and covariance matrix thoery,the steady state performances of system,such as availability,the output variance,throughput,the queue length of the steady state and some probabilities of the steady state,are obtained.The influences of the parameters on the performances of the system are discussed preliminarily and the effectiveness and availability of the approach are fully shown in the sensitivity analysis.Experiments demonstrate that the proposed model can effectively improve the performance of the queueing system either with working vacation or working breakdown.