Infinite precedence graphs for consistency verification in P-time event graphs

Precedence constraints are inequalities used to model time dependencies. In 1958, Gallai proved that a finite system of precedence constraints admits solutions if and only if the corresponding precedence graph does not contain positive-weight circuits. We show that this result extends naturally to the case of infinitely many constraints. We then analyze two specific classes of infinite precedence graphs -- $\mathbb{N}$-periodic and ultimately periodic graphs -- and prove that the existence of solutions of their related constraints can be verified in strongly polynomial time. The obtained algorithms find applications in P-time event graphs, which are a subclass of P-time Petri nets able to model production systems under cyclic schedules where tasks need to be performed within given time windows.