# Fluid Net Models: From Behavioral Properties to Structural Objects
## 5 Configurations, Continuity, and Monotonicity in MTS
Nonetheless, the system is non-monotonic, as can be seen in Figure 11c. In these cases, the equilibrium marking reaches regions in which the associated configuration is not suitable. In Figure 11a,b, a region with B-Problematic configuration \(\mathcal{C}_{1,8,10,11,8,6,7}\) is reached and that is why the discontinuity induced bifurcation and the deadlocks appear (the configuration reached is associated with the subnet in Figure 5a). This is formalized in the next theorem. Theorem 5 ([15]): _Let \(\langle N,\mathbf{\lambda},\mathbf{m}_{0}\rangle\) be an MTS TCPN system with \(\Lambda\) fixed and \(\mathbf{m}_{0}\) arbitrary. If the equilibrium throughput is not deadlock-free monotonic with respect to \(\mathbf{m}_{0}\), then there exists an equilibrium marking \(\mathbf{m}_{e}\) that activates only a B-Problematic configuration._  
For the case of Figure 11c, a region with nB-Problematic configuration \(\mathcal{C}_{1,2,10,11,8,6,7}\) is reached and that is why the system presents a decrement of the equilibrium throughput. It is the same subnet of Figure 5a with the additional place \(p_{2}\); this place induces a restriction that prevents the system of reaching the deadlock. This is formalized in the next proposition. Proposition 6 ([15]): _Let \(\langle N,\mathbf{\lambda},\mathbf{m}_{0}\rangle\) be an MTS TCPN system with \(\Lambda\) fixed and \(\mathbf{m}_{0}\) arbitrary. If every reachable equilibrium marking \(\mathbf{m}_{e}\), which does not activate a suitable configuration, activates only an nB-Problematic configuration, then the equilibrium throughput is non-monotonic with respect to \(\mathbf{m}_{0}\)._  
Figure 11: Equilibrium throughput properties of the MTS TCPN model in Figure 1b. (**a**) Discontinuity of the equilibrium throughput for parameter \(\lambda_{5}\). (**b**) Non-monotonicity with deadlock of the equilibrium throughput for parameter \(\lambda_{2}\). (**c**) Non-monotonicity without deadlock of the equilibrium throughput for parameter \(\lambda_{2}\). Now, let us set the firing rate to \(\lambda=\left[1\,3\,1\,1\,1\,1\,1\,1\right]^{T}\), and the initial marking to \(\mathbf{m}_{0}=\left[k_{1}\,0\,0\,0\,0\,0\,0\,6\,7\,4\,4\,7\right]^{T}\), where \(k_{1}\) is taken as a variation parameter. If this initial marking is increased, then the equilibrium throughput exhibits a non-monotonic behavior, as can be seen in Figure 12. This complements the above explanation that the number of resources, together with the capacity of the machine, should be greater than the products that the system can handle, so that no blocking situations occur. In this case, the equilibrium marking reaches the region with nB-Problematic configuration \(\mathcal{C}_{1,2,10,11,8,6,7}\) (decreasing part of the equilibrium marking), and finally reaches a region associated with the B-Problematic configuration \(\mathcal{C}_{1,8,10,11,8,6,7}\). As can be seen in Theorems 4 and 5 and Proposition 6, the study of the equilibrium throughput properties depends not only on the type of the configuration (suitable or problematic), but also on the possibility that this configuration becomes activated from the initial marking. As mentioned above, Algorithm 3 decides the type of a configuration; now, Algorithm 4 determines when the configuration can be activated. ```
0:\(\langle N,\mathbf{\lambda},\mathbf{m}_{0}\rangle\), configuration \(\mathcal{C}\). 0: If configuration \(\mathcal{C}\) may be activated by an equilibrium marking
1: Compute the token-flow matrix \(\mathbf{C}=\mathbf{Post-Pre}\)
2: Compute a basis of P-semiflows \(\mathbf{B}_{y}\)
3: Compute the existence of a reachable equilibrium marking activating \(\mathcal{C}\): \(\mathcal{P}_{4}\): \(\{\text{Find}\ \mathbf{m}\geq\mathbf{0}\text{ such that}\)
1. \(\mathbf{B}_{y}^{T}\mathbf{m}=\mathbf{B}_{y}^{T}\mathbf{m}_{0}\); \(\{\text{Reachable marking}\}\)
2. \(\mathcal{C}\mathbf{\Lambda}\mathbf{\Pi}_{\mathcal{C}}\mathbf{m}=\mathbf{0}\); \(\{\text{Equilibrium marking}\}\)
3.