There has been a virtual explosion in the last several years of the use of Markovian models, in at least three different areas: development of algorithms for efficient computation of the stationary measures for Markov chains, fed for example by the use of MCMC (Markov chain Monte Carlo) approaches to statistical inference and spatial modeling; development of more complex stochastic models for networks of queues, especially in communication and manufacturing systems; and modeling of physical phenomena by non-linear time series models, which needs approaches different from those of traditional linear models. This proposal concerns the development and application of new methodologies motivated by problems in these diverse applications areas. The investigator will carry out research in five main aspects of the theory of stochastic models which can benefit all of these applications: (i) Methods of applying coupling theory and minorization theory to the development of perfect sampling strategies; (ii) Methods of computational evaluation of the rates at which models achieve a stable regime; (iii) Properties of stochastically ordered or partially stochastically ordered models, especially in continuous time, for both diffusions and jump-continuous processes; (iv) Robustness of model behavior against perturbations in assumptions; and (v) Use of non-Markovian and other general stationary processes in describing systems and developing algorithms.
Overall this project can be expected to deliver new methodologies which will provide a significant theoretical underpinning, and methods of speedup, to simulation and modeling methodologies currently being used more and more widely in solving problems in areas as diverse as manufacturing plant design, atmospheric and climatology modeling, and economic theory. In the analysis of complex network, queueing and storage systems, applications envisaged include the currently high-priority area of manufacturing systems, and more general queueing systems with feedback. High dimensional spatial models being used for the understanding of environmental issues should also be faster to analyze using some of the proposed methods. The development of simulation approaches for non-linear time-series models in this context can be expected to provide new results in models of physical, social and economic phenomena.
