For  and , this is due to the fact that $(C^+ - C^-) x = C x = 0$.

We now complete the description of the continuous dynamics by additional equations which arise from the specification of routing rules. Such rules occur in the following two situations:

Given $p \in {\mathcal{P}_\textsf{conflict}}$, we suppose that tokens are routed according to a stationary distribution specified by weights $\mu_{qp} > 0$ associated with each output transition $q$. Therefore, $$\label{eq:conflict_routing}
  \forall p \in {\mathcal{P}_\textsf{conflict}}, \; \forall q \in p{^{\textrm{out}}}, \quad a^-_{qp}f_q(t) = \mu_{qp} \dfrac{m_p(t)}{\tau_p} \, .$$

Let $p \in {\mathcal{P}_\textsf{priority}}$, and $q_+$ and $q_-$ be the two output transitions, as illustrated in (c). In order to specify that the flow is routed in priority to transition $q_+$, we require that: $$\begin{aligned}
f_{q_+}(t) &= \min_{r \in q_+{^{\textrm{in}}}, \, w_r(t) = 0} \dfrac{m_r(t)}{a^-_{q_+{r}}\tau_r} \,,
\label{eq:p1}\\
f_{q_-}(t) &= \begin{cases}
    \min_{r \in q_-{^{\textrm{in}}}\setminus \{{p}\},  w_r(t) = 0} \dfrac{m_r(t)}{a^-_{q_-r}\tau_r} & \text{if}\  w_{{p}}(t) \neq 0 \, ,\\
\min \Bigl(\dfrac{m_{{p}}(t)}{a^-_{q{p}}\tau_{{p}}} - \dfrac{a^-_{q_+{p}}}{a^-_{q{p}}}f_{q_+}(t) \, , \min_{r \in q_-{^{\textrm{in}}}\setminus \{{p}\}, w_r(t) = 0} \dfrac{m_r(t)}{a^-_{q_-r}\tau_r}
\Big) 
& \text{if}\  w_{{p}}(t) = 0 \, .
    \end{cases} 
\label{eq:p2}\end{aligned}$$

The expression of $f_{q_-}(t)$ in , when $w_{p}=0$, indicates that only the outgoing flow from ${p}$ that is not 