Flowgraph Models in Reliability and Finite Automata

We discuss the interrelationship of two seemingly unrelated subjects: the theory of finite automata, and reliability theory, finite automata, more generally known as generalized transition graphs, are dasiaconvertedpsila to regular expressions by manipulating their pictorial representation, a directed graph, by elimination of its states one-by-one until two states are left, connected by an edge whose label is a regular expression equivalent to the initially given finite automata or generalized transition graph. Flowgraphs are used to represent semi-Markov reliability models. They are directed graphs with edges labeled with expressions of the form pG(s), where p is the probability of transition from node i to node j, say; and G(s) is the transform (Laplace transform, moment generating function, or characteristic function) of the waiting time in i given that the next transition is to j. Usually, transforms of waiting time distributions (e.g. time to first failure) are obtained from these graph representations by applying Mason's Rule (e.g. Huzurbazar, Mason, and Osaki), or, by the Cofactor Rule. In this paper we are concerned with obtaining transforms of waiting times by direct manipulation of the flowgraphs along the lines in finite automata. The goal of the paper is to observe that identical patterns of reasoning are applicable in both fields. This interconnects two apparently unrelated fields of knowledge, an interesting observation for its own sake but also important from a tool & technique point of view.