Circulant Approximation for Preconditioning in Stochastic Automata Networks

Stochastic Automata Networks (SANs) are widely used in modeling many 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 eeciently by direct methods such as LU decomposition due to the huge size of the generator matrices. An eecient numerical method should make use of the tensor structure of SANs' generator matrices. The generalized Conjugate Gradient (CG) methods are possible choices, although 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 eeciently. Numerical examples of SANs are also given to illustrate the fast convergence rate of the method.