Circulant approximation for preconditioning in stochastic automata networks
Circulant approximation for preconditioning in stochastic automata networks
Stochastic Automata Networks (SANs) are widely used in modeling practical systems such as queueing systems, communication systems, and manufacturing systems. For the performance analysis purposes, one needs to calculate the steady-state distributions of SANs. Usually, the steady-state distributions have no close form solutions and cannot be obtained efficiently by direct methods such as LU decomposition due to the huge size of the generator matrices. An efficient numerical method should make use of the tensor structure of SANs' generator matrices. The generalized Conjugate Gradient (CG) methods are possible choices though their convergence rates are slow in general. To speed up the convergence rate, preconditioned CG methods are considered in this paper. In particular, circulant based preconditioners for the SANs are constructed. The preconditioners presented in this paper are easy to construct and can be inverted efficiently. Numerical examples of practical SANs are also given to illustrate the fast convergence rate of the method.
stochastic automata networks, steady-state distributions, circulant approximation, preconditioners, conjugate gradient methods
147-160
Ching, Wai Ki
cfee9d26-97e3-42ce-a4e9-b91e6d8aeef7
Zhou, Xun Yu
67677e8b-0641-4ba0-af26-508701d8fd33
2000
Ching, Wai Ki
cfee9d26-97e3-42ce-a4e9-b91e6d8aeef7
Zhou, Xun Yu
67677e8b-0641-4ba0-af26-508701d8fd33
Ching, Wai Ki and Zhou, Xun Yu
(2000)
Circulant approximation for preconditioning in stochastic automata networks.
Computers & Mathematics with Applications, 39 (3-4), .
(doi:10.1016/S0898-1221(99)00341-7).
Abstract
Stochastic Automata Networks (SANs) are widely used in modeling practical systems such as queueing systems, communication systems, and manufacturing systems. For the performance analysis purposes, one needs to calculate the steady-state distributions of SANs. Usually, the steady-state distributions have no close form solutions and cannot be obtained efficiently by direct methods such as LU decomposition due to the huge size of the generator matrices. An efficient numerical method should make use of the tensor structure of SANs' generator matrices. The generalized Conjugate Gradient (CG) methods are possible choices though their convergence rates are slow in general. To speed up the convergence rate, preconditioned CG methods are considered in this paper. In particular, circulant based preconditioners for the SANs are constructed. The preconditioners presented in this paper are easy to construct and can be inverted efficiently. Numerical examples of practical SANs are also given to illustrate the fast convergence rate of the method.
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Published date: 2000
Keywords:
stochastic automata networks, steady-state distributions, circulant approximation, preconditioners, conjugate gradient methods
Organisations:
Operational Research
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Local EPrints ID: 29748
URI: http://eprints.soton.ac.uk/id/eprint/29748
PURE UUID: f4186de1-3730-4720-8cb0-8e4695a703f7
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Date deposited: 20 Jul 2006
Last modified: 15 Mar 2024 07:34
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Author:
Wai Ki Ching
Author:
Xun Yu Zhou
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