define a *policy* $\pi$ as a function from ${\mathcal{Q}}$ to ${\mathcal{P}}$, which maps any transition $q$ to one of its upstream places $p_{\pi}(q) \in q{^{\textrm{in}}}$. A policy is meant to indicate the bottleneck place of each transition $q$. We denote by $S_{\pi}$ the [*selection matrix*]{} associated with $\pi$, that is, the ${\mathcal{Q}}\times {\mathcal{P}}$ matrix such that $(S_{\pi})_{qp} = 1$ if $p = p_{\pi}(q)$, and $0$ otherwise. In particular, $(S_{\pi})_{qp} = 1$ implies that $a_{qp} < 0$.

Note that, if $p$ realizes the minimum in one of the equations , or  for some transition, then $p$ also realizes the minimum in . The converse is not true if places are subject to priority. For $p$ denoting a priority place and $q_+$ its priority output transition, if $p$ realizes the minimum in  for transition $q_+$, then, $p$ does not necessarily realize the minimum in . In other words, our definition of a bottleneck place is dependent on the routing rules of the net.

We point out that notions comparable to policies are used in [@mahulea2006performance] in the context of continuous Petri nets with time attached to transitions.

The dynamics of a Petri net can be expressed in terms of the different policies 