负顾客、带启动期和备用服务员的M/M/1休假排队系统

考虑一类有正、负顾客, 带启动期和有备用服务员的M/M/1休假排队系统. 负顾客一对一抵消队尾的正顾客(若有), 若系统中无正顾客, 到达的负顾客自动消失, 负顾客不接受服务.系统中两个服务员, 其中一个在岗工作时另外一个备用.上岗服务员若因为某种原因休假, 备用服务员立即替换上岗.当系统变空时, 系统关闭.用拟生灭过程和矩阵几何解方法, 得到了稳态队长的分布, 此外, 证明了稳态条件下队长的条件随机分解并得到了附加队长的分布. 最后, 通过两个数值例子说明该模型可以较好的模拟一些实际问题.

The M/M/1 vacation queueing with negative customers,set-up times and spare servers

We consider an M/M/1 vacation queueing system with set-up period and spare servers,in which customers are either "positive" or "negative".Negative customers remove positive customers one by one only at the end(if present).When a negative customer arrives,if there isn’t positive customer in system,it will disappear.Negative customers don’t accept service.In the system,there are two servers, one goes on duty,the other keep on standby.If the first server is on vacation for some reason,the spare one replaces immediately.When system is empty,the system turn off.Using QBD(quasi-birth-and death) process and matrix-geometric solution method,we obtain the steady-state distribution for queue length. Furthermore,we prove the conditional stochastic decomposition of queue length process in the stationary state and gain the distributions for additional queue length.Using two numerical examples,we verify that our model can represent some practical problems reasonably well finally.