Research on Stochastic Processes and Optimization

对于具有固定服务/路由策略的随机网络，通常难以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。当适当地表述时，这些受控随机网络的类似近似通常比它们的固定策略对应具有更好的定性性质。此外，许多近似模型非常简单，因此可以采用封闭形式（或近似封闭形式）求解。研究者将在几个密切相关的领域开展研究，这些领域可以利用这些特征：风险敏感控制和队列网络中罕见事件的控制；大数近似律（也称为流体模型）的鲁棒最优控制；对流体模型控制的高阶修正。每个主题的核心都是约束动力学过程的变分问题（对于大偏差和流体模型控制的变分演算或最优控制问题，对于流体模型的鲁棒控制或罕见事件控制问题的微分对策）。研究者最近展示了在某些情况下如何将涉及约束和控制动力学以及相对简单的成本结构的变分问题转化为涉及无约束动力学和不同成本的等效问题。后一个问题就得到了明确的解决。建议的研究包括将该技术扩展到包括大偏差和约束微分对策中的缓冲区溢出问题。当今应用概率论的主要关注点之一是发展随机网络的可处理近似。随机网络在现代计算机、通信和制造系统中无处不在，但由于其复杂性和细节性，很难对其进行分析。因此，人们在开发数学模型上投入了大量的精力，这些模型要足够忠实于“现实生活”，以便从中得出的结论可以放心地使用，同时又可以通过分析或数值方法来解决。这个项目的目的是发展这样的近似方法和解决它们的技术。一个新特性是允许将路由和服务（例如，在通信网络中应该提供哪个数据类以及应将处理后的数据发送到何处）的决策作为可以优化的控制变量。我们将特别关注两类特殊的网络问题。首先是对罕见事件的控制。在许多网络中，有些事件并不经常发生，但仍然是主要关注的问题。通信网络中的数据丢失就是一个例子。第二类是网络的鲁棒控制，这意味着对网络某些方面建模不良或不完全了解的网络的控制。

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.