W : F \to \mathbb{N^+} is a set of arc weights, which assigns to each arc f \in F some n \in \mathbb{N^+} denoting how many tokens are consumed from a place by a transition, or alternatively, how many tokens are produced by a transition and put into each place.
... until it hits the side of the Petri dish ...
... It's just like bacteria in a Petri dish. ...