of the net: at any time $t$, there is a policy $\pi^*$ (we can note $\pi^*(t)$ if we want to emphasize the dependence on time) such that $$\forall q \in {\mathcal{Q}}\, , \quad w_{p_{\pi^*}(q)}(t) = 0 \,, $$ and $$\begin{aligned}
  &\forall q \in {\mathcal{Q}}\text{ s.t.\ } p_{\pi^*}(q) \not\in ({\mathcal{P}_\textsf{conflict}}\cup {\mathcal{P}_\textsf{priority}}) \,, \; \dfrac{m_{p_{\pi^*}(q)}(t)}{\tau_{p_{\pi^*}(q)}} = a^-_{qp_{\pi^*}(q)} f_q(t) \,,\\
  &\forall q \in {\mathcal{Q}}\text{ s.t.\ } p_{\pi^*}(q) \in {\mathcal{P}_\textsf{conflict}}\,, \quad \dfrac{m_{p_{\pi^*}(q)}(t)}{\tau_{p_{\pi^*}(q)}} = \dfrac{a^-_{qp_{\pi^*}(q)}}{\mu_{qp_{\pi^*}(q)}} f_q(t) \,,\\
  &\forall q_+ \in {\mathcal{Q}}\text{ s.t.\ } p_{\pi^*}(q_+) \in {\mathcal{P}_\textsf{priority}}\,, \; \dfrac{m_{p_{\pi^*}(q_+)}(t)}{\tau_{p_{\pi^*}(q_+)}} = a^-_{q_+p_{\pi^*}(q_+)} f_{q_+}(t) \,,\\
  &\forall q_- \in {\mathcal{Q}}\text{ s.t.\ } p_{\pi^*}(q_-) \in {\mathcal{P}_\textsf{priority}}\,, \; \dfrac{m_{p_{\pi^*}(q_-)}(t)}{\tau_{p_{\pi^*}(q_-)}}
   = a^-_{q_-p_{\pi^*}(q_-)} f_{q_-}(t)
   + a^-_{q_+p_{\pi^*}(q_+)} f_{q_+}(t) \,.
\end{aligned}$$

Now, for any policy $\pi$, we denote by ${C^-_{\pi}}$ the ${\mathcal{Q}}\times {\mathcal{Q}}$ matrix such that the right-hand side of this system of equations reads ${C^-_{\pi}}f(t)$ , where $f(t)$ is the vector of the $(f_q(t))_{q \in {\mathcal{Q}}}$. In particular, the above system writes $$S_{\pi^*} \left(\dfrac{m(t)}{\tau}\right) = {C^-_{\pi^*}}f(t) \,,$$ where $(m(t)/\tau)$ is the vector of the $(m_p(t)/\tau_p)_{p \in {\mathcal{P}}}$. The diagonal entries of ${C^-_{\pi}}$ are positive. Moreover, if we order each transition of type $q_+$ before its associated transition $q_-$, the matrix ${C^-_{\pi}}$ becomes lower triangular.[^2] Hence, ${C^-_{\pi}}$ is invertible. Matrix ${C^-_{\pi}}$ can be seen as 