高负荷下分支型轮询排队网络的极限性态

The asymptotic behavior of a branching-type polling network in heavy traffic

本文研究服务策略具有一般分支性的轮询排队网络.所有队列共享一个服务器,且服务器依次服务每个队列.队列i中的顾客服务完后,以概率pi,j转移到队列j,或以概率pi,0离开系统.利用带移民的多型分支过程的极限定理,本文给出高负荷下服务器进入队列服务时队长联合分布的极限性态;结合平均准则方法,还得到平均等待时间的极限定理.

We consider a polling network with a single server that serves the queues in a cyclic order. Upon the completion of his service at queue i, a customer is routed to queue j with probability pi,j, or leaves the system with probability pi,j. The service policy is branching-type. Using the limit theorem of a multi-type branching process with immigration, we derive the asymptotic behavior of joint queue lengths at polling epochs in the heavy traffic. With the help of heavy traffic averaging principle, the limit behavior of the mean waiting times is also presented.