\neq p}} \dfrac{1}{a^-_{q_+r}}x_r(t - \tau_r) \biggr)\, ,
 \label{eq:pnpriority4} \\
&
z_{q_-}(t) = \min \biggr(
\Bigl(\dfrac{1}{a^-_{q_-p}}x_p(t-\tau_p) - \dfrac{a^-_{q_+p}}{a^-_{q_-p}} z_{q_+}(t)\Bigr),
\min_{{r \in q_-{^{\textrm{in}}}, r \neq p}}\dfrac{1}{a^-_{q_-r}} x_r(t - \tau_r) \biggr) \, ,
\label{eq:pnpriority5}\end{aligned}$$

where $q_+$ ($q_-$) is the priority (non priority) output transition of $p \in {\mathcal{P}_\textsf{priority}}$.

Note that if all the holding times $\tau_p$ are integer multiples of a fixed time $\delta$, the left limit $\lim_{s \uparrow t}z_{q_-}(s)$ in  can be replaced by $z_{q_-}(t-\delta)$. This is helpful in particular to simulate these equations.

In the setting of [@allamigeon2015performance], all conflicts are solved by a stationary distribution routing. The equivalent of the routing rule introduced to solve conflicts in the continuous setting is obtained here by allowing the tokens to be shared in fractions, so that the counter functions take real values. This corresponds to a *fluid approximation* of the discrete dynamics. In this setting, for each $p \in {\mathcal{P}_\textsf{conflict}}$ and $q\in p{^{\textrm{out}}}$, we fix $\mu_{qp}>0$, giving the proportion of the tokens routed from $p$ to $q$. We have: $$\forall p \in {\mathcal{P}_\textsf{conflict}}\, , \, \forall q \in p{^{\textrm{out}}}\,, \quad z_q(t) = \dfrac{\mu_{qp}}{a^-_{qp}} x_p(t - \tau_p) \ . 
\label{eq:pnpriority6fluid}$$

The stationary solutions of the discrete dynamics are defined as functions $x_p$ and 