Performance of queueing networks

复杂的制造工厂或电信网络可以被建模为嵌入式网络。 订单或数据包从一个节点到另一个节点，并在队列中等待获得服务。网络的性能通常通过吞吐量或订单或数据包成功通过网络的速率来衡量。其他考虑因素包括网络过载的罕见性和通过网络的平均时间。我的第一个项目涉及一个建设性的大偏差理论，用于估计网络过载等罕见事件的概率。 稀有事件需要数学研究，因为模拟只能通过大量的计算工作来生成此类事件。 我感兴趣的是在超载的时刻，因为人们可以使用这个作为出发点，研究一个建议的过载恢复算法的网络状态。 我还对多类型网络和轮询网络中的罕见事件感兴趣，这些事件无视当前的大偏差分析。我的第二个项目涉及互联网协议的平均场分析。我们已经描述了通过瓶颈路由器传输的源的所有传输窗口大小的直方图的演变。 理论分布已通过网络测量得到证实。 我们希望使用我们的工具来研究一些建议的控制协议。 分析还提出了关于网络动荡的有趣问题。 可能同时存在HTTP源传输文件而没有丢失的流动状态以及具有周期性突发的数据包丢失的有损状态。 这些制度可以被看作是一个马尔可夫链的亚稳态。 由于测度（或直方图）值过程的大偏差理论是不完整的，我们有很多新的数学发现和有趣的应用范围。

Complex manufacturing plants or telecommunication networks may be modelled as queueing networks.  Orders or packets progress from node to node and wait in queues to obtain service. The performance of the network is usually measured by the throughput or the rate at which orders or packets successfully travers the network. Other considerations include the rarity of network overloads and the mean time to transit the network. My first project involves a constructive large deviation theory for estimating the probability of rare events like network overload.  Rare events need a mathematical investigation since simulation can only generate such events with a lot of computational effort.  I am interested in the state of the network at the moment of overload since one can use this as the starting point for investigating a proposed overload recovery algorithm.  I am also interested in rare events in multitype networks and polling networks which defy current large deviations analysis. My second project involves a mean field analysis of the Internet protocols. We have described the evolution of the histogram of all the transmission window sizes of the sources transmitting through a bottleneck router.  The theoretical distribution has been confirmed by network measurement.  We want to use our tools to investigate some proposed control protocols.  The analysis also raises interesting questions about network turbulence.  There may simultaneous exist a fluid regime where HTTP sources transmit files without loss as well as a lossy regime with periodic burst of packet losses.  These regimes may be viewed as metastable states of a Markov chain.  Since the theory of large deviations for measure (or histogram) valued processes is incomplete we have lots of scope for new mathematical discoveries as well as interesting applications.