具有两类优先权顾客的M/M/1排队的优化分析

Optimization Analysis of M/M/1 Queue with Two Types of Priority Customers

本文研究了一个带有强占优先权和非强占优先权的M/M/1排队模型,顾客分为三个优先等级,第一类顾客享有强占优先权,第二类顾客享有非强占优先权,第三类顾客无优先权.三类顾客具有不同的到达率,当第一类顾客到达后它将打断正在接受服务的第二类或第三类顾客立即接受服务;当第二类顾客到达时,若系统中只有第三类顾客,则此顾客必须等待当前服务完成,才能接受服务,否则排队等待.同一类顾客遵循FCFS的排队规则.利用补充变量法构造多维向量马尔可夫过程并对此排队系统的状态转移方程进行分析,得到三类顾客队长联合分布的概率母函数,进而得出了每类顾客各自的平均队长以及服务台被三类顾客占有和闲置的概率.利用Matlab进行数值计算,主要考察服务率的变化对系统中各类顾客平均队长的影响.最后构建不同成本费用函数进行优化分析.

This paper consideres an M/M/1 queue model with preemptive and non-preemptive priorities.Customers are divided into three priority levels.The first category of customers enjoy the priority of preemptive,the second category of customers enjoy the priority of non-preemptive,and the third category of customers have no priority.When the first kind of customers arrive,they will interrupt the second or third kind of customers who are receiving the service immediately;When the second type of customers arrive,if there are only the third type of customers in the system,the customer must wait for the current service to be completed before accepting the service.Customers of the same type follow the queuing rule of FCFS.A multi-dimensional vector Markov process is constructed here using the supplementary variable method,and the probability generating functions of three types of joint distribution of queue length are obtained.Then,each category of customer's average queue length and the probabilities in service are obtained.The influence of the change of service rate on the average queue length of all kinds of customers in the system is studied using MATLAB software.Finally,different cost functions are constructed for optimization analysis.