Mathematical Study of Queueing Theory

在本课题中，我们对排队网络的效率进行了理论分析。(1)给出了复杂到达顾客的排队网络模型，并给出了基于随机测量的数学表示。(2)将广义半马尔可夫过程在数学上进行了扩展，并将其应用于成组服务和顾客在每个节点上的服务时间相互依赖的排队网络中。(3)针对顾客服务的停止和路径的选择依赖于网络的情况，提出了一种新的排队网络模型，并对该网络模型的解析类进行了数学刻画。(4)通过改变原模型的条件，给出了不易找到网络中平稳分布的解析网络模型，并研究了它们之间的关系。(5)建立了同时发生变化的排队网络的分析方法。

In this project we gave a theoretical analysis on efficiencies of queueing networks.(1) A model to represent a queueing network for aomplicated arrivals of customers was given and its mathematical representation was shown based on random measures. (2) Generalized semi-Markov Processes was extended mathematically and applied to the queueing networks in case that group service and service time for customer on each node are depended each other. (3) A new model of a queueing network was presented for the case that the stop of the service for customers and choice of path depended on the network was investigated and the analytical class of network models was characterized mathematically. (4) The analytical network models was given by the change of the conditions of the original model which is not so easy to find the stationary distribution in the network and its relation was investigated. (5) The analytic method for a queueing network which happens some change at the same time was developed.