The Basic Cyclic Scheduling Problem with Linear Precedence Constraints

In this paper, we consider a nite set of generic tasks that we execute a large number of times. We introduce the linear constraints between two generic tasks : the di erence of iteration indices between the execution of two tasks subject to a precedence constraint is a linear function of these indices. These constraints are modeled by a linear graph G. Moreover, we consider no resource constraints. We aim at characterizing the optimal frequencies of the generic tasks. Since the executions of the generic tasks are only subject to precedence constraints, the maximum frequencies are achieved in the earliest schedule. The study of its asymptotic behavior allows us to develop an algorithm to compute the maximum frequencies of the generic tasks : it is based on a particular decomposition of the linear graph and on the expansion of its components. We construct a valued graph by contracting these components and we show that the longest paths of that graph provide the maximum values of the frequencies.