Research on Stochastic Processes and Optimization

Brown University, Providence RI

Investigators

Abstract

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.