Computational Methods In Markov Chains

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The goal of this project is to study both practical and theoretical aspects surrounding the application and implementation of algorithms for computing stationary probabilities and analyzing transient behavior of discrete Markov chains. One facet of the research is dedicated to applications of discrete models for understanding and analyzing the dynamics of large systems comprised of collections of loosely coupled subsystems such as in the study of queuing models and networks, telecommunications, computer performance evaluation, stochastic automata networks, economic and biological forecasting, and flexible manufacturing systems. Special attention is devoted to making Markov chain technology widely available to researchers in diverse areas. Specific research objectives include the following aims. (1) The project seeks to relate properties of specific Markov chain models to characteristics of numerical algorithms with the goal of devising a sieve from which the most appropriate algorithm and implementation for a given application can be extracted. (2) Another objective is to develop both theoretical and numerical mechanisms to facilitate the determination or estimation of stationary probabilities of large-scale finite Markov chains by utilizing a variety of aggregation/disaggregation methods. A specific goal is to formulate a comprehensive theory of errors which can simplify and unify the analysis of a broad range of aggregation/disaggregation processes and algorithms. (3) The project seeks to develop practical preconditioning techniques for iterative algorithms which can be applied to Markov models which are formulated as stochastic automata networks. (4) A final objective is to develop a comprehensive software package which provides access to sophisticated computational tools for Markov modeling. The goal is to structure the package for researchers in applications areas who do not want to be diverted by or become immersed in technical numerical analysis details.