Research in Large Deviations and Applications to Queueing Networks

小行星9700852 本研究将集中在两个广泛的领域：（1）弱收敛方法， （2）跳马氏过程的大偏差理论， 应用于网络。在领域（1），研究部分基于一本新书 与Paul Dupuis共同撰写。它包括以下内容：（a）派生表示 公式的新类别的过程，包括跳跃马尔可夫过程;（B）证明 扩散和其他具有不连续统计量的过程的拉普拉斯原理 各种几何形状;（c）证明具有状态依赖噪声的随机游走模型的拉普拉斯原理;（d）证明马尔可夫的经验测度的拉普拉斯原理 链和相关过程，包括具有状态依赖性的度量值过程; (e)推广了一个新的Legendre-Fenchel变换公式， 凸函数 领域（2）的研究包括以下内容：（a）使用新的 弱收敛的大偏差概率表示公式 方法获得一个显式公式的率函数的一些 （B）扩展最近一篇关于一般模型的大偏差分析的论文， 将系统映射到其他映射模型，包括依赖于状态的队列;（c） 计算精确的大偏差率并确定溢出路径 对应于多个网络中的某些溢出事件。 这项研究计划在大偏差的动机是应用到 网络，这是至关重要的通信和高性能 计算的 首席研究员过去几年的工作， 最终在一些重要的论文和研究水平的书，开发了一套 新的和强大的技术来研究这种网络的行为。例如是 高速网络技术设计中的一个关键问题是确定缓冲区的大小 在这些网络中的通信交换机上， 是适当的小。超大型多处理机设计中的另一个关键问题 系统的一个重要任务是估计系统在某个时间间隔内发生故障的概率。这 研究将适应于研究这些问题和相关问题。

9700852 Ellis This research will focus on two broad areas: (1) a weak convergence approach to the theory of large deviations and (2) large deviations for jump Markov processes and applications to queueing networks. In area (1) the research is based in part on a new book written jointly with Paul Dupuis. It consists of the following: (a) deriving representation formulas for new classes of processes, including jump Markov processes; (b) proving the Laplace principle for diffusions and other processes having discontinuous statistics with various geometries; (c) proving the Laplace principle for a random walk model with state- dependent noise; (d) proving the Laplace principle for empirical measures of Markov chains and related processes, including measure-valued processes with state dependencies; (e) generalizing a new formula on the Legendre-Fenchel transform of compositions of convex functions. Research in area (2) consists of the following: (a) using a new representation formula for large deviation probabilities together with weak convergence methods to obtain an explicit formula for the rate function for a number of queueing models of interest; (b) extending a recent paper on the large deviation analysis of general queueing systems to other queueing models, including state-dependent queues; (c) evaluating the exact large deviation rates and determining the overflow paths corresponding to certain overflow events in a number of queueing networks. This research program in large deviations is motivated by applications to queueing networks, which are of fundamental importance in communication and high performance computing. The work of the Principal Investigator over the past few years, which culminated in a number of important papers and a research-level book, develops a set of new and powerful techniques for studying the behavior of such networks. For example, a critical problem in the design of high speed networking technologies is to size the buffe rs at communications switches within these networks so that certain overflow probabilities are suitably small. Another critical problem in the design of very large multiprocessor systems is to estimate the probability that the system will fail in some time interval. This research will be adapted to studying these and related problems.