Analyzing Markov Chains using Kronecker Products [electronic resource] : Theory and Applications / by Tuğrul Dayar.

By: Dayar, Tuğrul [author.]
Contributor(s): SpringerLink (Online service)
Material type: TextTextSeries: SpringerBriefs in Mathematics: Publisher: New York, NY : Springer New York : Imprint: Springer, 2012Description: IX, 86 p. 3 illus. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9781461441908Subject(s): Mathematics | Mathematical statistics | Numerical analysis | Probabilities | Mathematics | Probability Theory and Stochastic Processes | Numerical Analysis | Probability and Statistics in Computer ScienceAdditional physical formats: Printed edition:: No titleDDC classification: 519.2 LOC classification: QA273.A1-274.9QA274-274.9Online resources: Click here to access online
Contents:
Introduction -- Background -- Kronecker representation -- Preprocessing -- Block iterative methods for Kronecker products -- Preconditioned projection methods -- Multilevel methods -- Decompositional methods -- Matrix analytic methods.
In: Springer eBooksSummary: 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.
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Introduction -- Background -- Kronecker representation -- Preprocessing -- Block iterative methods for Kronecker products -- Preconditioned projection methods -- Multilevel methods -- Decompositional methods -- Matrix analytic methods.

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.

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