Analyzing {M}arkov chains using {K}ronecker products : theory and applications

Abstract

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