排队论中之一问题——M/W/n

<正> 排队论问题中最常遇到的也是较为重要的问题之一,就是在随机输入,服务时间按负指数分布的假定之下,决定有关各种概率的问题.我们用符号 M/M/n 表示这样的一个服务系统:“顾客”输入服从参数λ的 Poisson 分布,先到来,先服务,服务时间的长短服从参数产的负指数分布,共有 n 个服务台.若“顾客”到来时发现服务台有空,则他可在空下的服务台中随意挑选一个而立刻受到服务,若无服务台空下,则他即依到来的次序列队等待,直到被服务完毕之后才离开.近年来,不少的作者(1],2],3],4],5]皆集中于

ON THE PROBLEM M/M/n IN THE THEORY OF QUEUES

As usual,we use M/M/n to denote such a queuing process:the arrival and servicetime are assumed negative exponentially distributed with means λ~(-1) and μ~(-1) respectively,the number of servers is n.Let pk(t) be the chance that at time t there are k customerspresent including those being served.In this paper we prove thatTheorem 1.For k≥n,we have(?)where a is the root ofnμx~2-(λ+u+nμ)x+λ=0with minimum absolutely value,and(?)Theorem 2.For n=1,2 and 3,the limits(?)p_k(t) k=0,1,2,…exist.