Analyzing Markov Chains using Kronecker Products

Kronecker products are used to define the underlying Markov chain (MC) in various modeling formalisms, including compositional Markovian models, hierarchical Markovian models, and stochastic process algebras. The motivation behind using a Kronecker structured representation rather than a flat one is to alleviate the storage requirements associated with the MC. With this approach, systems that are an order of magnitude larger can be analyzed on the same platform. The developments in the solution of such MCs are reviewed from an algebraic point of view and possible areas for further research are indicated with an emphasis on preprocessing using reordering, grouping, and lumping and numerical analysis using block iterative, preconditioned projection, multilevel, decompositional, and matrix analytic methods. Case studies from closed queueing networks and stochastic chemical kinetics are provided to motivate decompositional and matrix analytic methods, respectively.

First to provide a solely Kronecker product based treatment of Markov chain analysis The subject matter is interdisciplinary and at the intersection of applied mathematics, specifically numerical linear algebra and computational probability, and computer science The exposition is concise and rigorous, yet it tries to be complete and touches almost all relevant aspects without being too technical. Includes supplementary material: sn.pub/extras Includes supplementary material: sn.pub/extras

Klappentext

Kronecker products are used to define the underlying Markov chain (MC) invarious modeling formalisms, including compositional Markovian models, hierarchical Markovian models, and stochastic process algebras. The motivation behind using a Kronecker structured representation rather than a flat one is to alleviate the storage requirements associated with the MC. With this approach, systems that are an order of magnitude larger can be analyzed on the same platform. The developments in the solution of such MCs are reviewed from an algebraic point of view and possible areas for further research are indicated with an emphasis on preprocessing using reordering, grouping, and lumping and numerical analysis using block iterative, preconditioned projection, multilevel, decompositional, and matrix analytic methods. Case studies from closed queueing networks and stochastic chemical kinetics are provided to motivate decompositional and matrix analytic methods, respectively.

Inhalt
Introduction.- Background.- Kronecker representation.- Preprocessing.- Block iterative methods for Kronecker products.- Preconditioned projection methods.- Multilevel methods.- Decompositional methods.- Matrix analytic methods.