Analyzing Markov Chains using Kronecker Products: Theory and - download pdf or read online

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By Tugrul Dayar

ISBN-10: 1461441897

ISBN-13: 9781461441892

Kronecker items are used to outline the underlying Markov chain (MC) in numerous modeling formalisms, together with compositional Markovian types, hierarchical Markovian versions, and stochastic technique algebras. the incentive at the back of utilizing a Kronecker established illustration instead of a flat one is to relieve the garage standards linked to the MC. With this process, structures which are an order of value higher will be analyzed at the similar platform. The advancements within the resolution of such MCs are reviewed from an algebraic standpoint and attainable parts for extra study are indicated with an emphasis on preprocessing utilizing reordering, grouping, and lumping and numerical research utilizing block iterative, preconditioned projection, multilevel, decompositional, and matrix analytic tools. Case stories from closed queueing networks and stochastic chemical kinetics are supplied to encourage decompositional and matrix analytic equipment, respectively.

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Example text

LC1/ . H / D f0g. m;0/ D Q for iteration m. l/ /j; 1/ at level l for iteration m. mC1;lC1/ e D 1. The level to end recursion depends on available memory since there must be space to store and factorize the aggregated CTMC at that level. m;l/ , at each level. 13) changes from iteration to iteration, and hence, the method is nonstationary. l/ . l/ . l/ were ordered antilexicographically. l/ . l/ j since it has one nonzero per column by definition. These PH 1 QH vectors amount to a total storage of lD0 hDlC1 nh floating-point values if the recursion terminates at level H .

Diagonal blocks having identical off-diagonal parts and diagonals that differ by a multiple of the identity are exploited in [32]. There, it is shown that such diagonal blocks can share and work with the factorization of only one diagonal block. This approach saves not only time spent for factorization of diagonal blocks at the outset but also space. The same paper also discusses a three-level version of BSOR for MCs based on Kronecker products in which diagonal blocks that are too large to be factorized are solved using BSOR.

Surprisingly, block iterative solvers, which are sometimes called two-level (or two-stage) iterative solvers [112], have still not been incorporated into most analysis packages based on Kronecker representations, although they have been shown to be more effective than point solvers on many test cases [32, 146]. Furthermore, in contrast to the block partitionings of sparse MCs considered in [59], block partitionings of Kronecker products are nested and recursive due to the lexicographical ordering of states.

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Analyzing Markov Chains using Kronecker Products: Theory and Applications by Tugrul Dayar


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