a specification of the downstream token flow matrix of the Petri net $C^-$ (introduced in Section \[sec:notations\]), associated with the policy $\pi$.

With this notation, the continuous dynamics of Petri nets with free choice and priority routing reads:

\[eq:matrix\_dyn\_system\] $$\begin{aligned}
  f(t) &= \inf_{\pi \text{ s.t. } S_{\pi} w (t) = 0} ({C^-_{\pi}})^{-1} S_{\pi} \left( \dfrac{m(t)}{\tau} \right) \label{eq:matrix_dyn_f} \,,\\
  \dot{m}(t) &= C^+ f(t) - \dfrac{m(t)}{\tau} \label{eq:matrix_dyn_dot_m} \,,\\
  \dot{w}(t) &= \dfrac{m(t)}{\tau} - C^- f(t) \label{eq:matrix_dyn_dot_w} \,,
  \end{aligned}$$

where the infimum must be understood for the partial order over ${\mathbb{R}}^{\mathcal{Q}}$ induced by $\leq$. Note that there is at least one policy $\pi^*$ (depending on $t$) attaining the infimum. It suffices to choose the policy $\pi^*$ introduced earlier ([*i.e.*]{}, to choose a policy that attains the minimum componentwise).

By choosing an upstream place for each transition, a policy actually defines a candidate “bottleneck net” of the Petri net, that is, a subnet with all the transitions of the original Petri net, and such that each transition has a unique upstream place. On each of these subnets, the dynamics is linear and yields a unique trajectory for a given initial condition. The trajectory is solved on a subnet of the original Petri net, but one 