Multivariate Discrete Phase-Type Distributions with Their Applications

A dynamic random environment usually induces statistical dependence among components of a system operating in such an environment thereby making explicit computation of the system performance measures very difficult and possibly even intractable. The multivariate phase-type distribution is a powerful tool for unifying a variety of stochastic models and for constructing new models that yield to algorithmic analysis in such applications. In this project, we introduce the multivariate discrete phasetype distribution and use it to study the dependence structures of multi-component systems operating in dynamic random environments. The advantages of multivariate discrete phase-type distributions stem from the facts that such distributions can be easily computed and they have close ties with multivariate continuous phase-type distributions. In this project, detailed properties of multivariate discrete phase-type distributions will be established, and computational algorithms, both matrix algorithms and simulation procedures, will be developed. Using stochastic comparison methods and computational experimentation, the dependence structures of multivariate discrete phase-type distributions will be throughly examined. The methods and results to be obtained in this project will also be applied to analyze various operating systems, including assemble-to-order systems, synchronized communication networks, and reliability systems with simultaneous subsystem failures. The objectives are to develop good computable bounds, efficient heuristics, and important characterizations of system performance measures whose explicit computations are either intractable or computationally intensive. The insights gained from the proposed research will be summarized in a set of design and control principles for the efficient and effective management of synchronized operating systems. ∗Supported in part by the NSF grant DMI 9812994