具有不耐烦顾客和K-重工作休假的M^X/M/1排队系统分析

Analysis of an M^X/M/1 Queueing System with Impatient Customers and K-Working Vacation Policy

研究了一个等待空间无限的具有不耐烦顾客和K-重工作休假M^X/M/1排队系统.当系统中没有顾客时服务员转入工作休假状态;服务员最多可进行K次休假,若K次之后系统中仍没有顾客,服务员进入闲期.顾客按Poisson过程批量到达,到达的批量服从一般离散分布.在工作休假期间,到达的顾客可能由于等待不耐烦而离开系统.文章建立了系统的稳态平衡方程,利用概率母函数的方法得到了稳态下正常忙期的平均队长和工作休假期的平均队长以及其他一些相关指标的解析表达式.最后,利用数值算例分析了系统参数以及参数K的变化对稳态指标的影响.

This paper presents an analysis for an infinite-buffer M^X/M/1 queue with impatient customers and K-working vacation policy. Whenever the system becomes empty, the server can take a working vacation. The server is allowed to take a maximum number K of consecutive vacations if the system remains empty after the end of a vacation. After the number K of consecutive vacations if the system is still empty, the server comes into the idle period. Customers arrive in batches according to a Poisson process. The batch size follows a general discrete distribution. During the working vacation, the customers may become impatient and leave the system.We obtain analytical expressions of the system sizes when the sever is in the normal period and in the working vacation respectively by using the probability generating function. We further derive analytical expressions of other performance measures.Finally, numerical results are presented to demonstrate effects of some parameters and the maximum number K on the performance measures of the system.