T-SPH排队队长平稳分布的尾部分析

Tail Analysis of the Stationary Queue Length Distributions for Two T-SPH Queues

在复杂随机模型的研究中,经常会出现水平和位相都是无限的拟生灭过程,对这类过程,平稳分布的计算仍然是一个未很好解决的难题.然而对一类比较特殊三对角无限位相拟生灭过程,简记为T-QBD过程,文献[1]指出,在一定条件下可以估计其平稳分布的尾部特征.本文对文献[1]中提出的方法在某一环节上作了改进,使之更适合于实际计算,并用此方法分析了两个具有实际应用背景的排队模型,即T-SPH/M/1排队和M/T-SPH/1排队,分析结果表明,在一定条件下,这两类排队系统的队长分布的尾部都具有几何衰减的特性.

In the research of complex stochastic models, Quasi-Birth-and-Death processes （QBDs） whose level and phase are all infinite will take place frequently. The computation of the stationary distributions for such processes is still an open problem. For some special T-QBD processes where the blocks of the density matrix have a tri-diagonal structure, Ref. [1] showed out that under certain conditions, the tail of the stationary distribution can be estimated. In this paper, we improve the method in Ref. [1] to make it more applicable in practice. We analyze the T-SPH/M/1 queue and the M/T-SPH/1 queue with this method and show that the tails of the stationary distributions of queue length for such two queues all have the characteristic of geometric decay.