# Fluid Net Models: From Behavioral Properties to Structural Objects
## 4 Discontinuities and Non-Monotonicities
### The Firing Rate Variation Case  
In order to provide a formal definition of discontinuities and non-monotonicities, some definitions are given. They capture possible equilibrium throughput changes when the firing rates vary. In the following properties of the equilibrium throughput, the initial marking \(\mathbf{m}_{0}\) is fixed and the firing speed of transitions \(\mathbf{\lambda}\) varies. Some kind of \"dual\" properties, fixing \(\mathbf{\lambda}\) and varying \(\mathbf{m}_{0}\) can be established [15]. Definition 6: Let \(\langle N,\mathbf{\lambda},\mathbf{m}_{0}\rangle\) be a TCPN system with \(\mathbf{m}_{0}\) fixed. Assume that the firing rate vector \(\mathbf{\lambda}\) is arbitrary. Its equilibrium throughput is:  
* Monotonic with respect to the _firing rates_, denoted as \(M(\mathbf{\lambda})\), if \(\forall(\mathbf{\lambda},\mathbf{\lambda}^{\prime})\), \(\mathbf{\lambda}\leq\mathbf{\lambda}^{\prime}\implies\alpha(\mathbf{\lambda})\leq\alpha( \mathbf{\lambda}^{\prime})\);
* Deadlock-free monotonic with respect to the _firing rates_, denoted as \(DFM(\mathbf{\lambda})\), if \(\forall(\mathbf{\lambda},\mathbf{\lambda}^{\prime})\) with \(\mathbf{\lambda}\leq\mathbf{\lambda}^{\prime}\), \(\alpha(\mathbf{\lambda})>\mathbf{0}\implies\alpha(\mathbf{\lambda}^{\prime})>\mathbf{0}\),
* Continuous with respect to the _firing rates_, denoted as \(C(\mathbf{\lambda})\), if for every \(\mathbf{\lambda}\), given \(\varepsilon>0\), there is \(\delta>0\), such that \(\|\mathbf{\lambda}-\mathbf{\lambda}^{\prime}\|_{2}<\delta\implies\|\alpha(\mathbf{\lambda} )-\alpha(\mathbf{\lambda}^{\prime})\|_{2}<\epsilon\), where \(\|\cdot\|_{2}\) is the euclidean norm. The following example shows a firing rate variation in an MTS-TCPN, which leads to discontinuous changes in the equilibrium throughput, similar to the discontinuous change observed in Figure 1(b). Then, it is illustrated how a discontinuous change implies non-monotonicity. In order to simplify the explanation, the apparently simple TCPN system of Figure 8 with only four configurations is used. The configurations of this net are: \(\mathcal{C}_{1,1}\), \(\mathcal{C}_{2,1}\), \(\mathcal{C}_{1,3}\), and \(\mathcal{C}_{2,3}\). The same property can be observed for the net model in Figure 1(b); however, for the sake of simplicity, it is not used in this case. Consider the MTS net in Figure 8a with an initial marking \(\mathbf{m}_{0}= \begin{bmatrix}10&9&0\end{bmatrix} ^{T}\) and firing rates \(\mathbf{\lambda}= \begin{bmatrix}1&\lambda_{2}\end{bmatrix} ^{T}\), where \(\lambda_{2}\) is taken as a variation parameter. The net has two minimal P-semiflows \(\mathbf{y}_{1}= \begin{bmatrix}1&0&1\end{bmatrix} ^{T}\) and \(\mathbf{y}_{2}= \begin{bmatrix}0&1&1\end{bmatrix} ^{T}\) (hence, \(m_{1}+m_{3}=10\) and \(m_{2}+m_{3}=9\)) and only one minimal T-semiflow \(\mathbf{x}= \begin{bmatrix}1&1\end{bmatrix} ^{T}\). Since the net is an MTS-TCPN, the equilibrium throughput of the system is \(\mathbf{f}_{e}=\Lambda\Pi_{C}\mathbf{m}_{e}=\alpha\mathbf{x}\). Table 3 shows the value of the equilibrium throughput, for some intervals of \(\lambda_{2}\), in which a discontinuity of the equilibrium throughput is observed. In this case, the discontinuity can be explained due to the bifurcation that occurs when \(\lambda_{2}=0.5\). The linear system which determines the marking evolution when place \(p_{1}\) restricts the firing of \(t_{1}\) and \(t_{2}\) is: unstable if \(\lambda_{2}<0.5\); critically stable if \(\lambda_{2}=0.5\); and stable if \(\lambda_{2}>0.5\). The loss of hyperbolicity in this case (bifurcation) produces a discontinuity in the equilibrium throughput. Despite the discontinuity at \(\lambda_{2}=0.5\), Figure 8b shows that the equilibrium throughput is monotone with respect to \(\lambda_{2}\) variations. In order to show the non-monotonicity of this system, the equilibrium throughput, when \(\lambda_{1}\) varies, is obtained using \(\lambda_{2}\) as a parameter and the homothecy properties.