带止步与中途退出策略的M^x/M/1/N单重工作休假排队系统

M^x/M/1/Nqueue with Balking,Reneging and Single Working Vacations

研究了带有止步和中途退出的M^x/M/1/N单重工作休假排队系统.顾客成批到达,到达后每批中的顾客,或者以概率b决定进入队列等待服务,或者以概率1-b止步(不进入系统).顾客进入系统后可能因为等待的不耐烦而在没有接受服务的情况下离开系统(中途退出).系统中一旦没有顾客,服务员立即进入单重工作休假.首先,利用马尔科夫过程理论建立了系统稳态概率满足的方程组.其次利用矩阵解法求出了稳态概率的矩阵解并得到了系统的平均队长、平均等待队长以及顾客的平均消失概率等性能指标.最后通过数值例子分析了工作休假时的低服务率η和休假率θ这两个参数对系统平均队长的影响.

In this paper, we consider an queue with balking, reneging and single working vacations. Customers arrive in batch and every arriving customer either decides to enter the queue with a probability b or balk （do not enter） with a probability 1 - b The impatient customer in the queue may leave the system （renege） if it has not been served after a period of waiting time. The server takes single working vacations immediately when it becomes idle at a service completion instant. First, we obtain the steady-state probability equations by the Markov process method. Second, by the matrix solution method we derive the matrix form solution of the steady-state probability. Some performance measures of the system such as the expected number of the customers in the system or in the queue and the average loss probability of customers are also presented. Finally the effects of the parameters of the system are investigated, such as the lower service rate and the vacation rate on the expected queue length of the system by numerical examples.