A Whirl decomposition of stochastic systems based on the substitution property

A new method is proposed for realizing a stochastic system from interconnected smaller stochastic systems. Each smaller stochastic system considered is composed of two states. The basic tool is the partition with a substitution property which exists on the transition matrix of the interconnected stochastic system. Using the authors' method, a stochastic system with
n
states is decomposed into
m
(
m
≥
n
-1) component systems where each system has two states. The following results are derived: for a given outer input, each component system has
n
-1 state transition matrices; a relationship exists among some entries of transition matrices; and the whole system is composed of
m
component stochastic systems, but
m
-n+1 component systems are selected freely.