Periodic trajectories in P-time event graphs and the non-positive circuit weight problem

TL;DR

This paper links periodic trajectories in P-time event graphs to the non-positive circuit weight problem, providing a polynomial algorithm to determine periodic behaviors in timed discrete-event systems.

Contribution

It formulates the periodic trajectory problem as an instance of the PIC-NCP and applies a polynomial algorithm, establishing a connection with max-plus algebra for P-TEGs.

Findings

The existence of a 1-periodic trajectory implies the existence of d-periodic trajectories for all d.

The polynomial algorithm efficiently solves the PIC-NCP in the context of P-TEGs.

A necessary and sufficient condition relates 1-periodic and d-periodic trajectories in P-TEGs.

Abstract

P-time event graphs (P-TEGs) are specific timed discrete-event systems, in which the timing of events is constrained by intervals. An important problem is to check, for all natural numbers $d$, the existence of consistent $d$-periodic trajectories for a given P-TEG. In graph theory, the Proportional-Inverse-Constant-Non-positive Circuit weight Problem (PIC-NCP) consists in finding all the values of a parameter such that a particular parametric weighted directed graph does not contain circuits with positive weight. In a related paper, we have proposed a strongly polynomial algorithm that solves the PIC-NCP in lower worst-case complexity compared to other algorithms reported in literature. In the present paper, we show that the first problem can be formulated as an instance of the second; consequently, we prove that the same algorithm can be used to find $d$-periodic trajectories in P-TEGs. Moreover, exploiting the connection between the PIC-NCP and max-plus algebra we prove that, given a P-TEG, the existence of a consistent 1-periodic trajectory of a certain period is a necessary and sufficient condition for the existence of a consistent $d$-periodic trajectory of the same period, for any value of $d$.