带启动时间和可修服务台的M/M/1/N工作休假排队系统

Analysis of M/M/1/N working vacation queuing system with setup times and repairable service station

分析带有启动时间、服务台可故障的M/M/1/N单重工作休假排队系统.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.假定服务台允许出现故障且当出现故障时,服务台停止为顾客服务且立即进行修理.服务台的失效时间和修理时间均服从指数分布,且工作休假期和正规忙期具有不同的取值;同时,从关闭期到正规忙期有服从指数分布的启动时间.建立此工作休假排队系统的有限状态拟生灭过程(QBD),使用矩阵几何方法得到QBD的各稳态概率相互依赖的率阵,从而求得稳态概率向量.通过有限状态QBD的最小生成元和稳态概率向量得到系统的基本阵和协方差矩阵,求解出系统方差、系统稳态可用度、系统吞吐率、系统稳态队长、系统稳态故障频度等系统性能.数值分析体现了所提出方法的有效性和实用性,通过敏感性分析将各参数对系统性能的影响进行了初探,为此模型的实际应用提供了很好的理论依据.

The paper analyzes the M/M/1/N single working vacation queueing system with setup time and server breakdowns.In this system,the server works at a lower service rate instead of stoping working completely during the vacation period.It is assumed that the server is subject to breakdown.The server stops services for customers and is repaired immediately when a breakdown occurs.Both the breakdown time and the repair time for the server follow exponential distributions,and they have different values in the working vacation period and regular busy period respectively.Meanwhile,the setup time from shut down period to regular busy period follows exponential distribution too.We establish the finite quasi birth and death(QBD)process of the system.Matrix-geometric approach is utilized to develop the interdependent rate matrix of the QBD,which helps to get the steady state probability vector.The fundamental matrix and covariance matrix of the system are obtained through the infinitesimal generator of the finite QBD and the steady state probability vector.With the fundamental matrix and covariance matrix,the steady state performances of the system,such as the output variance,availability,throughput,the queue length and fault frequency of the system,are obtained.Numerical analysis shows the effectiveness and feasibility of the proposed approach.Meanwhile,sensitivity analysis studies the influence of the parameters on the performances of the system,which provides a good theoretical basis for the practical application of the proposed model.