# Fluid Net Models: From Behavioral Properties to Structural Objects
## 2 Basic Concepts and Notations
### Fluid or Continuous Petri Net System  
A _marking_ is a mapping \(\textbf{m}:P\rightarrow\mathbb{R}^{|P|}_{\geq 0}\) that assigns to each place of \(N\) a non-negative real value. **Definition 3**. : _A fluid or continuous Petri net (CPN) system is a net \(N\) together with an initial marking \(\textbf{m}_{0}\), and it is denoted as \(\langle N,\textbf{m}_{0}\rangle\)._  
One characteristic of continuous Petri net systems is their evolution rule. It allows the firing of transitions in positive real amounts, while the reachable markings of the continuous system must be non-negative real values. The _enabling degree_ of a transition \(t\) at a marking \(\textbf{m}\) is defined as:  
\[\textbf{enab}[t]=\min_{\forall p\in t}\bigg{\{}\frac{\textbf{m}[p]}{\textbf{ Pre}[p,t]}\bigg{\}}. \tag{1}\]  
A transition \(t\in T\) is _enabled_ if \(\textbf{enab}[t]>0\). An enabled transition can be _fired_ in any real amount inside the interval \(0\leq\delta\leq\textbf{enab}[t]\). Its firing leads to a new marking \(\textbf{m}^{\prime}=\textbf{m}+\delta\textbf{C}[P,t]\). A marking \(\textbf{m}\) is _lim-reachable_ from \(\textbf{m}_{0}\) if there exists a (finite or infinite) firing sequence \(\sigma\), such that \(\textbf{m}\) can be reached from \(\textbf{m}_{0}\), which is denoted as \(\textbf{m}_{0}[\sigma\rangle\textbf{m}\). A transition \(t\) is called _dead_ if its enabling degree is zero (\(\textbf{enab}[t]=0\)) for every reachable marking. The set of reachable markings, denoted as \(RS(N,\textbf{m}_{0})\), satisfy the following _fundamental_ equation:  
\[\textbf{m}=\textbf{m}_{0}+\textbf{C}\sigma, \tag{2}\]where \(\mathbf{\sigma}\in\mathbb{R}_{\geq 0}^{|T|}\) is the firing count vector of the sequence \(\sigma\) such that \(\mathbf{m}_{0}[\sigma)\mathbf{m}\). Regarding Equation (2) and the P-flows of a net \(N\), a _token conservation law_ associated with a P-flow \(\mathbf{y}\) is described by the equation \(\mathbf{y}^{T}\mathbf{m}=\mathbf{y}^{T}\mathbf{m}_{0}\). This last equation is obtained by pre-multiplying Equation (2) by \(\mathbf{y}^{T}\). Proposition 1 ([6,7]): _Let \(\langle N,\mathbf{m}_{0}\rangle\) be a CPN system. If there is no empty siphon at \(\mathbf{m}_{0}\) and the net is consistent (not a real constraint in practical systems), then the following are equivalent:_  
1. _The set of reachable markings;_
2. _The set of solutions of Equation (2), where_ \(\mathbf{\sigma}\geq\mathbf{0}\)_,_ \(\mathbf{m}\geq\mathbf{0}\)_:_
3. _The set of solutions of equation_ \(\mathbf{B}_{y}^{T}\mathbf{m}=\mathbf{B}_{y}^{T}\mathbf{m}_{0}\)_, where_ \(\mathbf{m}\geq\mathbf{0}\)_._  
In this work, it is assumed that every system \(\langle N,\mathbf{m}_{0}\rangle\) fulfils the previous condition: _there is no empty siphon at \(\mathbf{m}_{0}\) and the net is consistent_. If that is not the case, the places associated with the empty siphon, together with their output transitions, can be removed from the net because they never evolve. In the case of continuous Petri net systems, the condition that there is not empty siphon at \(\mathbf{m}_{0}\) is equivalent to not having dead transitions in the CPN system. Property 1: _Let \(\langle N,\mathbf{m}_{0}\rangle\) be a consistent CPN system. There are no empty siphons at \(\mathbf{m}_{0}\) iff there are no dead transitions at \(\mathbf{m}_{0}\)._  
Proof: (Sufficiency) If there are not dead transitions at \(\mathbf{m}_{0}\), then every transition in \(T\) can be fired at least once. Choose an arbitrary place \(p\) in an arbitrary siphon \(\Sigma\); if \(p\) is marked at \(\mathbf{m}_{0}\), then the siphon \(\Sigma\) is marked at \(\mathbf{m}_{0}\). If \(p\) is not marked at \(\mathbf{m}_{0}\), then there should be a transition \(t\in{}^{\bullet}p\) that can be fired, which means that all the places in \({}^{\bullet}t\) are marked; thus, the siphon \(\Sigma\) is marked at \(\mathbf{m}_{0}\). If that is not the case, this reasoning can be done upwards to find the place that marks the siphon, because every transition can be fired at least once.