具有不耐烦顾客的M^X/M/1多重工作休假排队系统

Analysis of Customers’ Impatience in an M^X/M/1 Queue with Multiple Working Vacations

考虑带有不耐烦顾客的M^X/M/1工作休假排队系统.顾客成批到达系统;工作休假期服务员以较低速率进行服务,并且顾客(除正接受服务的顾客)会因为等待时间超过其期望等待时间而离开服务系统;该模型可以为服务系统决策者提供一些理论依据.首先,建立模型得到稳态平衡方程;然后,利用概率母函数的方法得到正规忙期和工作休假期平均队长的解析表达式;进一步得到了稳态下顾客平均逗留时间,忙期到达顾客的平均逗留时间以及中途退出率等性能指标的解析表达式.最后利用数值算例分析了模型参数对系统性能指标的相关影响.

We consider an M^X/M/1 queue with impatient customers and working vacations. Customs arrive in batches;during the vacation, the service is provided at a lower rate and the customer (except customers who are receiving service) may leave the system because the waiting time exceeds the expected waiting time. The model can provide some theoretical guidances for the decision maker of some service systems. Firstly, the steady-state equilibrium equation is established by building the model. Then, using the probability generating function method, the analytic expressions for the mean queue length in the normal and the working vacation are obtained. We further derive the analytic expressions for other performance measures such as: the mean sojourn time of customers;the mean sojourn time of customers arriving in the busy period;the rate of abandonment due to impatience. Finally, some numerical results are presented to demonstrate the effects of some parameters on the performance measures of the system.