Numerische Mathematik

, Volume 87, Issue 1, pp 35–57

Circulant preconditioners for stochastic automata networks

  • Raymond H. Chan
  • Wai Ki Ching
Original article

DOI: 10.1007/s002110000173

Cite this article as:
Chan, R. & Ching, W. Numer. Math. (2000) 87: 35. doi:10.1007/s002110000173

Summary.

Stochastic Automata Networks (SANs) are widely used in modeling communication systems, manufacturing systems and computer systems. The SAN approach gives a more compact and efficient representation of the network when compared to the stochastic Petri nets approach. To find the steady state distribution of SANs, it requires solutions of linear systems involving the generator matrices of the SANs. Very often, direct methods such as the LU decomposition are inefficient because of the huge size of the generator matrices. An efficient algorithm should make use of the structure of the matrices. Iterative methods such as the conjugate gradient methods are possible choices. However, their convergence rates are slow in general and preconditioning is required. We note that the MILU and MINV based preconditioners are not appropriate because of their expensive construction cost. In this paper, we consider preconditioners obtained by circulant approximations of SANs. They have low construction cost and can be inverted efficiently. We prove that if only one of the automata is large in size compared to the others, then the preconditioned system of the normal equations will converge very fast. Numerical results for three different SANs solved by CGS are given to illustrate the fast convergence of our method.

Mathematics Subject Classification (1991): 65F10

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Raymond H. Chan
    • 1
  • Wai Ki Ching
    • 2
  1. 1.Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong; e-mail: rchan@math.cuhk. edu.hkHK
  2. 2.Department of Census and Statistics, Government of the Hong Kong Special Admininstrative Region, Hong KongHK
  3. 3.Judge Institute of Management Studies, Cambridge University, Cambridge, UKGB