# Fluid Net Models: From Behavioral Properties to Structural Objects
## 2 Basic Concepts and Notations
### Timed Continuous Petri Net Systems  
If the firing of transitions is timed, the marking of the continuous system evolves deterministically along a trajectory within the set of reachable markings. In this sense, Equation (2) depends explicitly on time (that is, \(\mathbf{m}(\tau)=\mathbf{m}_{0}+\mathbf{C}\sigma(\tau)\)), and its time derivative gives the _state equation_:  
\[\dot{\mathbf{m}}(\tau)=\mathbf{C}\dot{\sigma}(\tau),\quad m(0)=m_{0}. \tag{3}\]  
The derivative of the firing count vector is known as the _firing flow_ or _throughput_ vector of the timed model \(\mathbf{f}(\tau)=\dot{\mathbf{\sigma}}(\tau)\). As already pointed out, in this work, the firing flow is defined using the _infinite servers semantics_[6; 7]:  
\[\mathbf{f}[t_{j}]=\mathbf{\lambda}[t_{j}]\min_{\forall p\in\mathbf{t}_{j}}\left\{\frac\mathbf{{m}[p]}{\mathbf{Pre}[p,t_{j}]}\right\}, \tag{4}\]  
where \(\mathbf{\lambda}:T\to\mathbb{Q}_{>0}^{[T]}\) is a mapping known as the _firing rate_ vector, which assigns to each transition of \(N\) a positive real value. The firing flow represents the number of transition firings per unit time, i.e., the throughput of the transition. Given a net structure, according to previous equation, it depends on two sets of parameters that may be tuned: the firing rate vector \(\mathbf{\lambda}\) and the current marking \(\mathbf{m}\). Defined by \(\mathbf{Pre}[p,t_{j}]\) and \(\mathbf{Post}[p,t_{j}]\), the structure of the net plays a key role in the possible evolution of the flow, being able to introduce undesirable behaviors (such as non-monotonicity in the equilibrium throughput). In Section 5, structural objects denoted as _problematic_ configurations are used to explain these undesirable behaviors. Notice that both \(\mathbf{\lambda}\) and \(\mathbf{m}\) directly affect the flow, so it suggests a \"kind of duality\". However, this duality is a _weak_ one, since \(\mathbf{\lambda}\) is a fixed timed net parameter, while \(\mathbf{m}\) is changing during the net system evolution. The following sections show the scope of such a duality. **Definition 4**. : _A timed continuous Petri net (TCPN) system is a CPN system together with a firing rate vector \(\mathbf{\lambda}\) and it is denoted as \(\langle N,\mathbf{\lambda},\mathbf{m}_{0}\rangle\)._  
Due to the min operator in Equation (4) and the token conservation laws, the marking evolution of a TCPN system can be represented by an _affine positive piecewise linear system_[6]. The following concepts are useful to study their structure and behaviors:1. A _configuration_\(\mathcal{C}\) of a net \(N\) is a set of \((p,t)\) arcs, only one per each transition, such that \(p\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bullet}t\). The _T-coverture_ of a configuration \(\mathcal{C}\) is \(\mathcal{T}_{\mathcal{C}}=\{p\mid\forall t\in T,(p,t)\in\mathcal{C}\}\), i.e., the places preceding the arcs of \(\mathcal{C}\). 2. A configuration \(\mathcal{C}\) has associated a _configuration matrix_\(\boldsymbol{\Pi}_{\mathcal{C}}\in\mathbb{Q}_{\geq 0}^{|\mathcal{T}|\times|P|}\) where: \[\boldsymbol{\Pi}_{\mathcal{C}}[t_{j},p_{i}]= \begin{cases}\dfrac{1}{\boldsymbol {Pre}[p_{i},t_{j}]}&\text{if }(p_{i},t_{j})\in\mathcal{C}\\ 0&\text{otherwise.}\end{cases} \] (5) A configuration \(\mathcal{C}\) is _active_ at marking \(\boldsymbol{m}\) if \(\boldsymbol{\Pi}_{\mathcal{C}}\boldsymbol{m}=\boldsymbol{enab}(\boldsymbol{m})\) (i.e., if their arcs are constraining the flow of transitions). 3. A _region_\(\mathcal{R}_{\mathcal{C}}\) is a (sub)state space in which a unique configuration \(\mathcal{C}\) is active. Regions constitute a partition--except on the borders--of the full polytope (possibly unbounded) of reachable markings. 4. The _associated net_ of a configuration \(\mathcal{C}\) is defined as the subnet \(N_{\mathcal{C}}=(\mathcal{T}_{\mathcal{C}},\mathcal{T}_{\mathcal{C}}^{\bullet },\boldsymbol{Pre}^{\prime},\boldsymbol{Post}^{\prime})\). 5.