denotes the number of firings of $q$ that occurred up to time $t$ included. Similarly, the counter variable of place $p$ is a function $x_p:{\mathbb{R}_{\geq 0}}\to {\mathbb{N}}$ which represents the number of tokens that have visited place $p$ up to time $t$ included (taking into account the initial marking). On top of being non-decreasing, the counter variables are *[[càdlàg]{}]{}* functions, which means that they are right continuous and have a left limit at any time.

In this setting, the parameter $\tau_p$ associated with the place $p$ represents a minimal holding time. It is shown in [@allamigeon2015performance] that, if tokens are supposed to be fired as early as possible, the counter variables satisfy the following equations (we generalize the equations to the case with valuations):

\[eq:discrete\] $$\begin{aligned}
 \forall p \in {\mathcal{P}}\,&, \quad x_p(t) =  M_p^0  + \sum_{q\in p{^{\textrm{in}}}} a^+_{qp} z_q(t) \, , \label{eq:pnpriority1} \\
 \forall p \in {\mathcal{P}_\textsf{conflict}}\,&, \quad \sum_{q \in p{^{\textrm{out}}}} a^-_{qp} z_q(t) = x_p(t - \tau_p) \, ,\label{eq:pnpriority2} \\
 \forall q \in {\mathcal{Q}_\textsf{sync}}\,&, \quad z_q(t) = \min_{p \in q{^{\textrm{in}}}} x_p(t - \tau_p) / a^-_{qp} \, ,\label{eq:pnpriority3}
 \end{aligned}$$ $$\begin{aligned}
\forall & p \in {\mathcal{P}_\textsf{priority}}\,, \nonumber\\
 &
 z_{q_+}(t) = \min\biggl( 
\Bigl(\dfrac{1}{a^-_{q_+p}}x_p(t-\tau_p) - \dfrac{a^-_{q_-p}}{a^-_{q_+p}} \lim_{s \uparrow t} z_{q_-}(s) \Bigr),\min_{{r \in q_+{^{\textrm{in}}}, r 