The numerical solution of stochastic automata networks

Stewart, William J.; Atif, Karim; Plateau, Brigitte

doi:10.1016/0377-2217(94)00075-n

Cited by 84 publications

(71 citation statements)

References 15 publications

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“…For example, the web ranking and information retrieval [1][2][3], queuing systems [4][5][6][7], stochastic automata networks [8,9], manufacturing systems and inventory control [10] and communication systems [11,12] and so on. In order to analyze their performance measures, it is required to find their stationary probability distributions π by solving the linear system 0, 0, 1, For a finite irreducible and aperiodic Markov chain, there exists a unique stationary probability distribution π whose elements are strictly greater than zero; see, e.g., [13,14].…”

Section: Introductionmentioning

confidence: 99%

An Accelerated Two-level Multigrid Method for Markov Chains

2015

J Appl Computat Math

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Based on the quadratic extrapolation method and its generalization, this paper presents an accelerated two-level multigrid method for speeding up the numerical computation of the stationary probability vector of an irreducible Markov chain. It shows how to combine these vector extrapolation methods with the two-level multigrid method on the coarse level in detail. Numerical results on two Markov chain problems are provided to illustrate the effectiveness of our proposed method in terms of reducing the iteration counts and computing time.

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Section: Introductionmentioning

confidence: 99%

An Accelerated Two-level Multigrid Method for Markov Chains

2015

J Appl Computat Math

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“…These methods include the power method and projection methods such as GMRES and Arnoldi. The GMRES and Arnoldi methods have been found to outperform the power method when applied to SANs [12]. The previously considered projection methods, GMRES and Arnoldi, both use long recurrences.…”

Section: Introductionmentioning

confidence: 99%

“…Preconditioning for Markov models is reviewed in References [4,7,14]. Preconditioners based on incomplete LU factorizations, which have been successfully employed in Markov models, cannot be readily used for SANs for the same reasons that direct methods based on L, U factors cannot be used [12]. A SAN preconditioner based on the Neumann series was proposed in Reference [12].…”

Section: Introductionmentioning

confidence: 99%

“…Preconditioners based on incomplete LU factorizations, which have been successfully employed in Markov models, cannot be readily used for SANs for the same reasons that direct methods based on L, U factors cannot be used [12]. A SAN preconditioner based on the Neumann series was proposed in Reference [12]. Yet such a preconditioner is computationally too expensive to compute, as experiments showed.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Second, the space required to store the description of components is in general much smaller than the explicit list of transitions, even in a sparse representation. However, using this representation instead of the usual sparse matrix form increases the time required for numerical analysis of the chains [6,15,37,33]. Note that we are interested in performance indices R defined as reward functions on the steady-state distribution (i.e.…”

Section: Introductionmentioning

confidence: 99%

An Algorithmic Approach to Stochastic Bounds

Fourneau

Pekergin

2002

Performance Evaluation of Complex Systems: Techniques and Tools

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Abstract. We present a new methodology based on the stochastic ordering, algorithmic derivation of simpler Markov chains and numerical analysis of these chains. The performance indices defined by reward functions are stochastically bounded by reward functions computed on much simpler or smaller Markov chains. This leads to an important reduction on numerical complexity. Stochastic bounds are a promising method to analyze QoS requirements. Indeed it is sufficient to prove that a bound of the real performance satisfies the guarantee.

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The numerical solution of stochastic automata networks

Cited by 84 publications

References 15 publications

An Accelerated Two-level Multigrid Method for Markov Chains

An Accelerated Two-level Multigrid Method for Markov Chains

A Kronecker product approximate preconditioner for SANs

An Algorithmic Approach to Stochastic Bounds

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