Cycle velocities with a transient remainder

Let $Q$ be a finite strongly connected directed multigraph. Its vertices include schedule phase and register type. Edge $e$ has displacement ${\Delta }_e\in {\mathbb{R}}^d$ and duration $t_e\ge t_{\min }>0$. The translation lift permits the declared copies of these transitions. Waiting is permitted in a manner compatible with the phase model.

For each simple directed cycle $C$, define

This is the timed version of the periodic-graph velocity construction Fritz,CheungVIIJ.