“enabling durations”, in which transitions preempt tokens. We refer to [@bowden2000brief] for a discussion on the meaning of time in Petri nets.

Policies and bottleneck places {#sec:policies}
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The analysis of the piecewise affine dynamical system – leads to introduce the notion of [*policy*]{}. Fixing a policy allows one to solve the dynamics on a region where it is linear. We shall see in that policies also arise in the characterization of stationary solutions.

Even if our continuous dynamics holds for more general classes of Petri nets, we focus in the following on strongly connected, autonomous Petri nets, so that each transition has at least one upstream place.

We observe that the dynamics of Petri nets with free choice and priority routing –, and – is linear on each region where the arguments of the minimum operators do not change. More precisely, at any time $t$, for any transition $q \in {\mathcal{Q}}$, there exists a place $p \in q{^{\textrm{in}}}$ such that $w_p(t)=0$ and $p$ is the unique upstream place of $q$ or $p$ realizes the minimum in the expression , or of $f_q(t)$. Place $p$ is then referred to as the *bottleneck place* of transition $q$ at time $t$.

We 