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.
General
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