\in p{^{\textrm{in}}}}a_{qp}^+f_q(t) - \dfrac{m_p(t)}{\tau_p} \, .$$ The evolution of the number of tokens waiting in place $p$ is described by the relation: $$\label{eq:wp}
  \dot{w}_p(t) = \dfrac{m_p(t)}{\tau_p} - \sum_{q\in p{^{\textrm{out}}}} a_{qp}^- f_{q}(t) \, .$$ Moreover, for all transition $q$, we require that $$\min_{p\in q{^{\textrm{in}}}, \, w_p(t)=0}\Big(
\dfrac{m_p(t)}{\tau_p} - \sum_{q'\in p{^{\textrm{out}}}} a^-_{q'p}f_{q'}(t)\Big)  =0 
\,.
\label{eq:earliest}$$ In particular, this implies that at least one place $p \in q{^{\textrm{in}}}$ verifies $w_p(t) = 0$. In this case, means that each of the upstream places $p$ that has a zero quantity of waiting tokens ($w_p(t)=0$) must satisfy $\dot{w}_p(t)\geq 0$, and that at least one of these places satisfies $\dot{w}_p(t)=0$. In other words, there is at least one *bottleneck* upstream place $p$ of $q$, which has no waiting tokens and whose outgoing flow $\sum_{q'\in p{^{\textrm{out}}}} a^-_{q'p}f_{q'}(t)$ coincides with its processing flow ${m_p(t)}/{\tau_p}$.

The relation provided in  can be simplified in the case of conflict and synchronization patterns. In more detail, if $q$ has a unique upstream place $p$, and this place is free choice ([conflict]{}), then  reduces to: $$\label{eq:conflict}
\dfrac{m_p(t)}{\tau_p} - \sum_{q'\in p{^{\textrm{out}}}} a^-_{q'p}f_{q'}(t) =0 \,.$$ Now, if $q$ has several upstream places, which are all free choice ([synchronization]{}), then reads as: $$\label{eq:sync}
f_{q}(t) = \min_{p\in q{^{\textrm{in}}}, 