Mathematical Analysis of Stochastic Networks

0406191Ramanan该项目的主要目标是扩展具有不连续动力学的确定性和随机约束过程的一般理论，并将其应用于更好地了解随机网络的性能和控制。该项目有四个组成部分。第一部分研究反映了作为近似的多类随机网络与合作服务器（即服务器，使用所有的过剩容量分配给一个类，以服务于其他类）的扩散。这些扩散的泛函的估计是复杂的事实，这些扩散往往不能是半鞅。目前的项目描述了这些扩散的路径分析的新技术。第二部分的研究旨在首先建立具有不连续漂移的微分方程唯一性的一般充分条件，然后使用这些条件来表征在其域内部表现出不连续过渡机制的离散网络的流体极限。第三部分是致力于分析相变和吉布斯措施与一类随机损失网络，出现在电信网络中的多播模型。最后一部分研究了一类非平稳混沌网络的最优控制设计问题。 在过去的十年中，电信，计算机和制造系统的日益复杂性强调了系统开发工具的必要性，用于分析，设计和控制。由于这些随机系统的精确分析往往是不可行的一个诉诸渐近分析，是适当的性能指标的兴趣。感兴趣的三个渐近状态是流体状态，其捕获系统的平均行为，扩散状态，其描述平均值周围的随机变化，以及大偏差状态，其分析罕见但对网络的操作至关重要的事件的概率（例如，计算机网络中的数据丢失）。该项目将开发工具，促进渐近分析（在三个制度的每一个）的重要类别的随机网络，自然出现在应用程序。特别是，开发的工具，预计将是有用的重要的网络性能指标，如稳定性和缓冲区溢出概率的估计，也为最佳控制的设计。

0406191Ramanan The main objective of this project is to extend the general theory of deterministic and stochastic constrained processes that have discontinuous dynamics and to apply it to gain better insight into the performance and control of stochastic networks. The project has four components. The first part studies reflected diffusions that arise as approximations to multi-class stochastic networks with cooperative servers (i.e. servers that use all the excess capacity allocated to one class in order to serve other classes). The estimation of functionals of these diffusions is complicated by the fact that these diffusions often fail to be semimartingales. The current project describes new techniques for the pathwise analysis of these diffusions. The second part of the research aims to first establish general sufficient conditions for uniqueness of differential equations that have discontinuous drifts and to then use these conditions to characterize fluid limits of queueing networks that exhibit discontinuous transition mechanisms in the interior of their domains. The third part is devoted to the analysis of phase transitions and Gibbs measures associated with a class of stochastic loss networks that appear as models of multicasting in telecommunication networks. The last part focuses on the design of optimal controls for a class of non-stationary queueing networks. Over the last decade the increasing complexity of telecommunication, computer and manufacturing systems has emphasized the need for the systematic development of tools for their analysis, design and control. Since an exact analysis of these stochastic systems is often infeasible one resorts to an asymptotic analysis that is appropriate for the performance measure of interest. Three asymptotic regimes of interest are the fluid regime, which captures the mean behavior of the system, the diffusion regime, which describes stochastic variations around the mean and the large deviations regime, which analyzes probabilities of events that are rare, but of crucial importance to the operation of the network (such as, for example, data loss in a computer network). This project will develop tools that facilitate the asymptotic analysis (in each of the three regimes) of important classes of stochastic networks that arise naturally in applications. In particular, the tools developed are expected to be useful for the estimation of important network performance measures such as stability and buffer overflow probabilities and also for the design of optimal controls.