$z_q$ satisfying the relations – and which ultimately behave as affine functions, [*i.e.*]{}, $x_p(t) =  u_p + t \rho_p$ and $z_q(t) = u_q + t \rho_q$ for all $t$ large enough. In this case, $\rho_p$ (resp. $\rho_q$) represents the asymptotic throughput of place $p$ (resp. transition $q$). We have shown in [@allamigeon2015performance Theorem 3] that these stationary solutions are precisely given by following system (we generalize the equations to the case with valuations):

\[eq:fixpoint\_germ\_rho\] $$\begin{aligned}
\forall p \in {\mathcal{P}}\, , \qquad \rho_p& = \sum_{q \in p{^{\textrm{in}}}} a^+_{qp} \rho_q \label{eq:fixpoint_germ1_rho} \,, \\
\forall p \in {\mathcal{P}_\textsf{conflict}}\, , \forall q \in p{^{\textrm{out}}}\, , \qquad \rho_q & = \mu_{qp} \rho_p / a^-_{qp}
\label{eq:fixpoint_germ2_rho} \,, \\
\forall q \in {\mathcal{Q}_\textsf{sync}}\, , \qquad \rho_q & = \min_{p \in q{^{\textrm{in}}}} \rho_p/a^-_{qp} \label{eq:fixpoint_germ3_rho}\,, \\
\forall p \in {\mathcal{P}_\textsf{priority}}\, , \qquad
\rho_{q_+} &= \min_{r \in q_+{^{\textrm{in}}}} \rho_r / a^-_{q_+r} \label{eq:fixpoint_germ4_rho}\,, \\
\forall p \in {\mathcal{P}_\textsf{priority}}\, , \quad
\rho_{q_-} = \min \Big(
&{\big(\rho_p - a^-_{q_+p}\rho_{q_+}\big)}/{a^-_{q_-p} }, \min_{r \in q_-{^{\textrm{in}}}\setminus \{ p \}} {\rho_r}/{a^-_{q_-r}} \Big) \,,
\label{eq:fixpoint_germ5_rho} \end{aligned}$$

\[eq:fixpoint\_germ\_u\] $$\begin{aligned}
\forall p \in {\mathcal{P}}\, &, \quad u_p = M_p^0 + \sum_{q \in p{^{\textrm{in}}}} a^+_{qp} u_q \label{eq:fixpoint_germ1_u} \,, \\
\forall p \in {\mathcal{P}_\textsf{conflict}}\, , \forall q \in p{^{\textrm{out}}}&, \; u_q = 