A minimization problem in systems characterized by acyclic signal flow graphs

It is well known that some stochastic systems (e.g., semiMarkov systems) can be modeled by signal flow graphs. For such systems time-domain, transform, and simulation methods are available for analysis. The time-domain method has not been popular because of the requirement for numerous convolutions and deconvolutions, which are very timeconsuming and subject to error propagation. In this paper a method of minimizing the number of multiplications required to evaluate the symbolic gain expression for an acyclic signal flow graph is presented. For stochastic systems these multiplications correspond to time-domain convolutions. The method requires an N! search, where N is the number of internal nodes of the signal flow graph. To avoid this N! search a heuristic algorithm is implemented in a computer program (Symbolic Analysis-Minimal Multiplications (SAMM)) and examples of its application to semi-Markov systems are discussed.