Mathematical Analysis of Stochastic Networks

本项目的主要目标是扩展具有不连续动态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洞察随机网络的性能和控制。该项目有四个组成部分。第一部分研究了具有合作服务器的多类随机网络(即服务器使用分配给某一类的所有过剩容量以服务于其他类)的近似所反映的扩散。这些扩散的泛函的估计由于这些扩散往往不是半鞅的事实而变得复杂。目前的项目描述了对这些扩散进行路径分析的新技术。第二部分的研究目的是首先建立具有不连续漂移的微分方程解唯一性的一般充分条件，然后利用这些条件来刻画在其区域内部呈现不连续转移机制的排队网络的流动极限。第三部分分析了一类随机丢失网络的相变和Gibbs度量，这类随机丢失网络在电信网络中表现为多播模型。最后一部分研究了一类非平稳排队网络的最优控制问题。在过去十年中，电信、计算机和制造系统日益复杂，强调了系统地开发用于分析、设计和控制这些系统的工具的必要性。由于对这些随机系统的精确分析通常是不可行的，所以人们求助于适合于感兴趣的性能度量的渐近分析。感兴趣的三个渐近区域是流体区域、扩散区域和大偏差区域，流体区域捕获系统的平均行为，扩散区域描述平均值周围的随机变化，大偏差区域分析对网络运行至关重要的事件的概率(例如，计算机网络中的数据丢失)。这个项目将开发工具，以促进对应用中自然出现的重要随机网络类的渐近分析(在三个区域中的每一个)。特别是，预计开发的工具将有助于估计稳定性和缓冲区溢出概率等重要的网络性能衡量标准，并可用于最佳控制的设计。

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.