An iterative bounding method for stochastic automata networks

A method to bound stationary distributions of large Markov chains resulting from networks of stochastic automata is presented. It combines the concepts for bounding the stationary distribution using eigenvector polyhedra with the exploitation of the specific structure of Markov chains resulting from stochastic automata networks. The quality of the bounds depends on the coupling between automata. Three consecutive steps of the method are presented. In the first step bounds are computed using information about single automata in isolation. Bounds for single automata are refined in a second step by considering the environment of an automaton given by the other automata in the network. In a third step, bounds are further improved using a disaggregation step. By means of two small examples it is shown that the method yields tight bounds for loosely coupled automata and that the approach is extremely efficient compared to other bounding methods, let alone compared to an exact numerical analysis.