# Fluid Net Models: From Behavioral Properties to Structural Objects
## 4 Discontinuities and Non-Monotonicities
### The Firing Rate Variation Case
Now, the equilibrium throughput \(\mathbf{f}_{e}^{\prime}\) is computed for the above TCPN as a function of \(\mathbf{f}_{e}\) when \(\lambda^{\prime}=k\lambda\) and \(k=1/\lambda_{2}\). Let us use the previous calculations of the TCPN system in Figure 8b. Consider point \(A^{\prime}\) in Figure 9; it is in the line that represents vector \(\mathbf{\lambda}= \begin{bmatrix}1&\lambda_{2}\end{bmatrix} ^{T}\) for interval \(0<\lambda_{2}<0.5\); the equilibrium throughput \(\mathbf{f}_{e}=\alpha_{A^{\prime}}\mathbf{x}\) for this interval is described by \(\alpha_{A^{\prime}}=\dfrac{\lambda_{2}}{1-\lambda_{2}}\). Now consider point \(A\) is on the line that represents vector \(\mathbf{\lambda^{\prime}}= \begin{bmatrix}\lambda_{1}^{\prime}&\lambda_{2}^{ \prime}\end{bmatrix} ^{T}= \begin{bmatrix}1&1\end{bmatrix} ^{T}=k\mathbf{\lambda}\)  
Figure 8: An MTS net, its equilibrium throughput when \(\lambda_{2}\) varies (Figure 8b), and its equilibrium throughput when \(\lambda_{1}\) varies (Figure 8c). Note that the first one is monotone and the last one is non-monotone. (**a**) An MTS net. The firing of \(t_{1}\) reduces the tokens in \(p_{1}\), while the firing of \(t_{2}\) increases its value. (**b**) Equilibrium throughput with respect to \(\lambda_{2}\) of the net system in Figure 8a. (**c**) Equilibrium throughput with respect to \(\lambda_{1}^{\prime}\) constructed using the time-scale property. A non-monotonicity is discovered from a discontinuity in the equilibrium throughput. \begin{table}
\begin{tabular}{l l l l} \hline \hline  & \multicolumn{2}{c}{**Equilibrium**} & \multicolumn{1}{c}{**Throughput w.r.t. \(\mathbf{\lambda_{2}}\)**} \\ \hline \multirow{2}{*}{\(\mathbf{\lambda_{2}}\)**Interval**} & **Equilibrium** & \(\mathbf{m_{e}}\)**Activates** & \(\mathbf{m_{e}}\)**Belongs to** \\  & **Throughput \(\mathbf{f_{e}}\)** & **Configuration:** & **Region:** \\ \hline \(0<\lambda_{2}<0.5\) & \multirow{2}{*}{\(\alpha=\dfrac{\lambda_{2}}{1-\lambda_{2}}\)} & \multirow{2}{*}{\(\mathcal{C}_{21}=(p_{2},t_{1}),(p_{1},t_{2})\)} & \multirow{2}{*}{\(\mathcal{R}_{21}\)} \\ \multirow{2}{*}{\(\lambda_{2}=0.5\)} & \multirow{2}{*}{\(\alpha=2.5\)} & \multirow{2}{*}{\(\mathcal{C}_{11}=(p_{1},t_{1}),(p_{1},t_{2})\) and \(\mathcal{C}_{13}=(p_{1},t_{1}),(p_{3},t_{2})\)} & \multirow{2}{*}{\(\mathcal{R}_{11}\cap\mathcal{R}_{13}\)} \\ \multirow{2}{*}{\(\lambda_{2}>0.5\)} & \multirow{2}{*}{\(\alpha=\dfrac{10\lambda_{2}}{1+2\lambda_{2}}\)} & \multirow{2}{*}{\(\mathcal{C}_{13}=(p_{1},t_{1}),(p_{3},t_{2})\)} & \multirow{2}{*}{\(\mathcal{R}_{13}\)} \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Equilibrium throughput w.r.t. \(\lambda_{2}\) and its characteristics of the MTS net depicted in Figure 8a. where \(k=\frac{1}{\lambda_{2}}\), for the interval \(\lambda_{1}^{\prime}>2\). Hence, now the equilibrium throughput is described by \(\alpha_{A}=k\alpha_{A^{\prime}}=\frac{1}{1-\lambda_{2}}=\frac{1}{1-\frac{1}{ \lambda_{1}^{T}}}=\frac{\lambda_{1}^{\prime}}{\lambda_{1}^{\prime}-1}\). The other two intervals can be determined in the same way. For \(\lambda_{2}=0.5\) it is obvious, and for \(0.5<\lambda_{2}\), points \(B\) and \(B^{\prime}\) are used. The equilibrium throughput of the system under variation of \(\lambda_{1}^{\prime}\), \(\boldsymbol{f^{\prime}}_{e}=\alpha^{\prime}\boldsymbol{x}\), is represented in Table 4.  
This is depicted in Figure 8c, and a _cached_ non-monotonicity (or decreasing) of the equilibrium throughput is now observed (there is a jump from a higher value \(\boldsymbol{f^{\prime}}_{e}=5\boldsymbol{x}\) to a lower one \(\boldsymbol{f^{\prime}}_{e}=2\boldsymbol{x}\) in the discontinuity value). Notice that the reasoning used in previous example can be generalized to any TCPN, when a discontinuity in its equilibrium throughput appears due to the variation of a firing rate \(\lambda_{i}\): the system throughput is non-monotonic with respect to \(\lambda\). This is formalized in the following theorem (written in the contrapositive form). **Theorem 2** ([14]).