A Fast Computation for the State Vector in a Max-Plus Algebraic System with an Adjacency Matrix of a Directed Acyclic Graph

We provide a useful method for calculating the state vector of a state equation efficiently in a max-plus algebraic system. For a discrete event system whose precedence relationships are represented by a directed acyclic graph, computing the transition matrix, which includes the Kleene star operation of a weighted adjacency matrix, is occasionally the bottleneck. On the other hand, the common objective is to compute the state equation, rather than the transition matrix itself. Since the state equation is essentially the multiplication of the transition matrix and vector, we propose algorithms for efficiently calculating the multiplication and left division of the Kleene star of an adjacency matrix and a vector.