Research on Stochastic Processes and Optimization

对于具有固定服务/路由策略的随机网络，通常很难在标准大数定律和扩散近似标度下唯一地表征极限。相关的困难出现在其他分析方法中，例如大偏差近似。另一种方法是允许路由/服务决策成为控制变量。当适当制定，类似的近似这些控制随机网络往往拥有更好的定性性质比他们的固定政策。此外，许多近似模型足够简单，可以得到封闭形式（或近似封闭形式）的解。研究人员将对几个密切相关的领域进行研究，这些领域可以利用这些功能：风险敏感控制和网络中罕见事件的控制;大数近似法（也称为流体模型）的鲁棒最优控制;流体模型控制的高阶校正。在每个这些主题的核心是一个变分问题的过程约束动力学（变分法或最优控制问题的大偏差和控制的流体模型，微分游戏的问题的鲁棒控制的流体模型或控制的罕见事件）。调查人员最近表明，在某些情况下，如何可以转换成一个变分问题，涉及约束和控制的动态和一个相对简单的成本结构到一个等价的问题，涉及无约束的动态和不同的成本。后一个问题，然后明确解决。建议的研究包括扩展这种技术，包括缓冲区溢出的大偏差和约束微分游戏的问题。今天应用概率的主要关注点之一是随机网络的易处理近似的发展。随机网络在现代计算机、通信和制造系统中普遍存在，但由于其复杂性和细节性，很难对其进行分析。因此，人们正在投入大量精力开发数学模型，这些模型要足够忠实于“真实的生活”，以便从中得出的结论可以放心地使用，而且可以通过分析或数值方法求解。该项目的目的是开发这种近似方法及其解决方案的技术。一个新的特征是允许关于路由和服务的决定（例如，在通信网络中应当服务哪个数据类以及应当将处理后的数据发送到哪里）是可以被优化的控制变量。两类特殊的网络问题将给予特别关注。一是罕见事件管控。在许多网络中，有一些事件并不经常发生，但仍然是主要关注的问题。一个例子是通信网络中的数据丢失。第二类是网络的鲁棒控制，这意味着对网络的某些方面建模不佳或不完全已知的网络的控制。

For stochastic networks with a fixed service/routing policy it is often difficult to uniquely characterize limits under standard law of large numbers and diffusion approximation scalings. Related difficulties appear in other methods of analysis, such as large deviation approximations. An alternative approach is to allow the routing/service decisions to be control variables. When properly formulated, the analogous approximations to these controlled stochastic networks frequently possess better qualitative properties than their fixed policy counterparts. In addition, many approximate models are simple enough that closed form (or nearly closed form) solutions are possible. The investigator will carry out research on several closely related areas that can take advantage of these features: risk-sensitive control and the control of rare events in queueing networks; robust optimal control of law of large number approximations (also known as fluid models); higher order corrections to the control of fluid models. At the heart of each of these topics is a variational problem for processes with constrained dynamics (calculus of variations or optimal control problems for large deviations and control of fluid models, differential games for the problems of robust control of fluid models or control of rare events). The investigator has recently shown how in certain cases one can convert a variational problem involving constrained and controlled dynamics and a relatively simple cost structure into an equivalent problem involving unconstrained dynamics and a different cost. The latter problem is then solved explicitly. The proposed research includes extending this technique to include problems of buffer overflow in large deviations and constrained differential games. One of the main concerns of applied probability today is the development of tractable approximations for stochastic networks. Stochastic networks are ubiquitous in modern computer, communication and manufacturing systems, but owing to their complexity and detail are very difficult to analyze. As a consequence, much effort is being put into the development of mathematical models that are faithful enough to "real life" that conclusions drawn from them can be used with confidence, and yet which can be solved by either analytical or numerical means. The purpose of this project is to develop such methods of approximation and also the techniques for their solution. A new feature is to allow decisions on routing and service (e.g., which data class should be served in a communication network and where the processed data should be sent) to be control variables that can be optimized. Two particular classes of network problems will be given special attention. The first is the control of rare events. In many networks there are events that do not occur very often, and yet which are nonetheless the main concern. An example is data loss in a communication network. The second class is the robust control of networks, which means the control of a network in which some aspects of the network are poorly modeled or otherwise imperfectly known.