out in Remark \[rk:comparison\] that our model can be reduced to a variant of the standard continuous model [@vazquez2013introduction] in which we allow immediate transitions and require non-trivial routings to occur only at these transitions. A benefit of our presentation is to allow a more transparent comparison between the continuous model and the discrete time piecewise affine models studied in [@cohen1995asymptotic; @gaujal2004optimal; @allamigeon2015performance].

The use of the term “policy” refers to the theory of Markov decision processes, owing to the analogy between the discrete time dynamics and the value function of a semi-Markovian decision process. Note that in the context of continuous Petri nets, policies are also known as “configurations”, see [@mahulea2006performance] for an example.

Continuous dynamics of Petri nets {#sec:semantics}
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General notation {#sec:notations}
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A Petri net consists of a set ${\mathcal{P}}$ of places, a set ${\mathcal{Q}}$ of transitions and a set of arcs $\mathcal{E} \subset ({\mathcal{P}}\times {\mathcal{Q}}) \cup ({\mathcal{Q}}\times {\mathcal{P}})$. Every arc is given a valuation in ${\mathbb{N}}$. Each place $p \in {\mathcal{P}}$ is given an initial marking $M^0_p \in {\mathbb{N}}$, which represents the number of tokens initially occurring in the place.

We denote by $a^+_{qp}$ the valuation of the arc from transition $q$ to place $p$, 