Let G = (V; E) be a directed graph with n vertices. For each e 2 E there is an associated reward r e ; the n \Theta n matrix A = [a ij ] is defined by a ij = r ij if ij 2 E and a ij = \Gamma1 if ij 62 E. The max-algebra system of equations y(k) = y(k \Gamma 1)\Omega A is a deterministic dynamic programming recursion for the maximum total reward y i (k) when the system is in state i at stage k and the one-stage transition reward matrix is A; when r ij represents the duration of activities that begin and end with the occurrence of events i and j, respectively, this is a discrete-event simulation. Max-algebra systems have also been used to simulate certain automated manufacturing systems. If G is strongly connected, the solution exhibits periodic behavior after an initial transient: if is the maximum cycle mean in G, then y(k+dA ) = y(k)+dA 1 for all k KA , where KA and dA are the max-algebra transient and period of the matrix A. For given initial conditions y(0), the transient K an..