A methodology for solving markov models of parallel systems

Plateau, Brigitte; Fourneau, Jean-Michel

doi:10.1016/0743-7315(91)90007-v

Cited by 100 publications

(75 citation statements)

References 4 publications

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“…This is the form of the Kronecker representation in hierarchical Markovian models [3], where rectangularity of the smaller matrices is possible due to the product state space of the modelled system being larger than its reachable state space [9]. When the product state space is equal to the reachable state space, the smaller matrices turn out to be square as in stochastic automata networks [19,20].…”

Section: Introductionmentioning

confidence: 99%

Compact Representation of Solution Vectors in Kronecker-Based Markovian Analysis

Buchholz

Dayar

Kriege

et al. 2016

Quantitative Evaluation of Systems

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Abstract.It is well known that the infinitesimal generator underlying a multi-dimensional Markov chain with a relatively large reachable state space can be represented compactly on a computer in the form of a block matrix in which each nonzero block is expressed as a sum of Kronecker products of smaller matrices. Nevertheless, solution vectors used in the analysis of such Kronecker-based Markovian representations still require memory proportional to the size of the reachable state space, and this becomes a bigger problem as the number of dimensions increases. The current paper shows that it is possible to use the hierarchical Tucker decomposition (HTD) to store the solution vectors during Kroneckerbased Markovian analysis relatively compactly and still carry out the basic operation of vector-matrix multiplication in Kronecker form relatively efficiently. Numerical experiments on two different problems of varying sizes indicate that larger memory savings are obtained with the HTD approach as the number of dimensions increases.

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Section: Introductionmentioning

confidence: 99%

Compact Representation of Solution Vectors in Kronecker-Based Markovian Analysis

Buchholz

Dayar

Kriege

et al. 2016

Quantitative Evaluation of Systems

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“…Given that a 1 k corresponds to the matrix element A 1i; j , then R k is deÿned similarly as the rearrangement of the block matrix R i; j . The ÿnal equality comes from the fact that a third-order tensor 3 (R) is actually m 1 n 1 second-order tensors and thus the sum of squares of the third-order tensor is the same as the sum of the m 1 n 1 sums of squares of the second-order tensors.…”

Section: Finding Higher-order Nkp For a General Matrixmentioning

confidence: 99%

“…This information regarding the automata and their types of transitions provides all the information needed to formally deÿne a SAN, as Atif and Plateau have done [2]. While this infrequent interaction (synchronizing transitions and functional transition rates) does complicate SANs, Plateau and her co-workers have shown that the SAN can still be represented in compact form as a sum of Kronecker products, known as the SAN descriptor [1,3,4]. Of course, too much interaction may complicate the SAN model to the point that all savings have been lost.…”

Section: Introductionmentioning

confidence: 99%

A Kronecker product approximate preconditioner for SANs

Langville

Stewart

2004

Numerical Linear Algebra App

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SUMMARYMany very large Markov chains can be modelled e ciently as stochastic automata networks (SANs). A SAN is composed of individual automata which, for the most part, act independently, requiring only infrequent interaction. SANs represent the generator matrix Q of the underlying Markov chain compactly as the sum of Kronecker products of smaller matrices. Thus, storage savings are immediate. The beneÿt of a SAN's compact representation, known as the descriptor, is often outweighed by its tendency to make analysis of the underlying Markov chain tough. While iterative or projections methods have been used to solve the system Q = 0, the time until these methods converge to the stationary solution is still unsatisfactory. SAN's compact representation has made the next logical research step of preconditioning thorny. Several preconditioners for SANs have been proposed and tested, yet each has enjoyed little or no success. Encouraged by the recent success of approximate inverses as preconditioners, we have explored their potential as SAN preconditioners. One particularly relevant ÿnding on approximate inverse preconditioning is the nearest Kronecker product approximation discovered by Pitsianis and Van Loan. In this paper, we extend the nearest Kronecker product technique to approximate the Q matrix for an SAN with a Kronecker product, A 1 ⊗ A 2 ⊗ · · · ⊗ A N . Then, we take M = A

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“…However, such computations rely on the underlying Markov chain of the Petri net model, so that Petri nets by themselves do not simplify the computation of statistics. In [10,11], however, particular structures of the transition matrix associated to certain Markov chains are used to decompose the statistical analysis of the system under consideration. Clearly, many real applications have been tackled by employing approaches developed within the Petri net community, and software products are available.…”

Section: Such Applications Require the Following Functionsmentioning

confidence: 99%

A Calculus of Stochastic Systems for the specification, simulation, and hidden state estimation of hybrid stochastic/non-stochastic systems

Benveniste

Levy

Fabre

et al. 1995

Lecture Notes in Computer Science

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Abstract.In this paper, we consider hybrid systems containing both stochastic and non-stochastic 3 components. To compose such systems, we introduce a general combinator which allows the specification of an arbitrary hybrid system in terms of elementary primitives of only two types. Thus, systems are obtained hierarchically, by composing subsystems, where each subsystem can be viewed as an "increment" in the decomposition of the full system. The resulting hybrid stochastic system specifications are generally not "executable", since they do not necessarily permit the incremental simulation of the system variables. Such a simulation requires compiling the dependency relations existing between the system variables. Another issue involves finding the most likely internal states of a stochastic system from a set of observations. We provide a small set of primitives for transforming hybrid systems, which allows the solution of the two problems of incremental simulation and estimation of stochastic systems within a common framework. The complete model is called CSS (a Calculus of Stochastic Systems),and is implemented by the SIG language, derived from the SIGNAL synchronous language. Our results are applicable to pattern recognition problems formulated in terms of Markov random fields or hidden Markov models (HMMs), and to the automatic generation of diagnostic systems for industrial plants starting from their risk analysis. A full version of this paper is available [1], omitted proofs can be found in this reference.

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A methodology for solving markov models of parallel systems

Cited by 100 publications

References 4 publications

Compact Representation of Solution Vectors in Kronecker-Based Markovian Analysis

Compact Representation of Solution Vectors in Kronecker-Based Markovian Analysis

A Kronecker product approximate preconditioner for SANs

A Calculus of Stochastic Systems for the specification, simulation, and hidden state estimation of hybrid stochastic/non-stochastic systems

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