带有止步和中途退出的成批到达的M^x/M/R/N同步休假排队系统的性能分析

Performance Analysis of M^x/M/R/N Queuing System with Balking Reneging Batch Arrival and Multiple Synchronous Vacations

研究了带有止步和中途退出的Mx/M/R/N同步休假排队系统.顾客成批到达.到达的顾客如果看到服务员正在休假或者全忙,他或者以概率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).系统根据一定的原则以概率nk在未止步的k个顾客中选择n个进入系统.在系统中排队等待服务的顾客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,R个服务员立即进行同步多重休假.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,在证明了相关矩阵可逆性的基础上,利用矩阵解法求出了系统稳态概率的明显表达式,并得到了系统的平均队长、平均等待队长及顾客的平均损失率等性能指标.

We consider an M^x/M/R/N queuing system with balking, reneging, batch arrival and multiple synchronous vacations. Customers arrive in batch. If the servers are all busy or on vacation, arriving customers either decide to enter the system with a probability b or balk （do not enter） with a probability 1 - b. The system can select n of k customers who do not balk to enter according to a certain plan. The impatient customer in the queue may leave the system after entering （renege） if it has not been served after a period of waiting time. The servers take multiple synchronous vacations when they are all idle at a service completion instant. First, by the Markfov process method, we obtain the steady-state probability equations. Second, by matrix solution method, we derive the explicit expressions of the steady-state probability. Some performance measures of the system such as the expected number of the customers in the system, the expected number of the customers in the queue and the average rate of the customer loss are also presented.