# Fluid Net Models: From Behavioral Properties to Structural Objects
## 5 Configurations, Continuity, and Monotonicity in MTS  
Section 4 illustrated how discontinuities imply non-monotonicities in the equilibrium throughput. Now, we will see that undesired behaviors only occur when the equilibrium markings belong to regions of the reachable marking in which the associated T-covertures do not contain the support of a P-semiflow. For instance, the discontinuity that appears in the equilibrium throughput of our running example (Figure 1a) occurs when the equilibrium marking belongs to configuration \(\mathcal{C}_{1,2,10,11,8,6,7}\) (see Figure 4). Its associated net has no P-Semiflows, but there is a P-flow \(\mathbf{y}^{T}=[-1\;-1\;0\;0\;0\;0\;-1\;1\;0\;1\;1\;0]\). Moreover, the deadlocks shown in Figure 2, either with respect to the initial marking or with respect to the firing rates, have an equilibrium marking belonging to configuration \(\mathcal{C}_{1,8,10,11,8,6,7}\) with associated subnet in Figure 5a; now, a subnet in which there are no P-Semiflows and no P-flows. Section 3 showed that with varying \(\lambda_{2}\) in the net depicted in Figure 1b, a discontinuity appears in the equilibrium throughput. This is possible because the subnet does not contain a P-flow; hence, no token conservation laws are present in the subnet even if the whole net is conservative. The fact that configurations without P-flows lead to a undesired equilibrium throughput can be generalized, as will be shown in Section 5. Now, configurations are classified into _suitable_ or _problematic_ to systematically study, from a structural point of view (parametrized by the equilibrium marking), the equilibrium throughput behavior in Mono-T-semiflow TCPN systems when the firing rates vary. **Definition 8**. : _A configuration \(\mathcal{C}_{i}\), of an MTS net \(N\), is said to be:_  
1. _Suitable_ if the support of a P-semiflow is contained in its T-coverture:_ \[\exists\mathbf{y}\geq\mathbf{0}\;|\;\mathbf{y}^{T}\mathbf{C}=\mathbf{0}\;\text{and}\;\|\mathbf{y}\| \subseteq\mathcal{T}(\mathcal{C}_{i}).\] _Otherwise,_ \(\mathcal{C}_{i}\) _is termed a_ problematic _configuration._
2. _nB-Problematic_ if its T-coverture does not contain the support of a P-semiflow and it contains the support of a P-flow._
3. _B-Problematic_ if its T-coverture does not contain the support of P-semiflows nor P-flows._  
B- and nB- refer to the existence or nonexistence of _bifurcations_ (jumps) in the equilibrium throughput, respectively, as already discussed in Section 3. Given a configuration of a net, deciding if it is suitable is quite simple. It can be checked using a linear algebraic problem: find if there exists a P-semiflow \(\mathbf{y}\), whose support belongs to the T-coverture of the configuration, \(\|\mathbf{y}\|\in\mathcal{T}_{C}\). If the configuration is not suitable, then it remains to be decided if it is B-Problematic or nB-Problematic. This can also be checked with a linear algebraic problem: find if there exists a P-flow \(\mathbf{y}\), whose support belongs to the T-coverture of the configuration, \(\|\mathbf{y}\|\in\mathcal{T}_{C}\). If it exists, the configuration is nB-Problematic. This is equivalent to check the rank of the token-flow matrix, if \(rank(\mathcal{C}_{\mathcal{T}_{C}})<|\mathcal{T}_{C}|\), then the configuration is nB-Problematic; otherwise, it is B-Problematic. Algorithm 3 performs this verification and runs in polynomial time. In Figure 4, the first two configurations \(\mathcal{C}_{1,2,3,4,5,6,7}\) and \(\mathcal{C}_{1,8,3,4,6,7}\) are suitable since they have a P-semiflow, \(\mathbf{y}^{T}=[1\;1\;1\;1\;1\;1\;0\;0\;0\;0\;0]\) and \(\mathbf{y}^{T}=[0\;0\;1\;0\;0\;1\;0\;1\;0\;0\;0]\), respectively. Configuration \(\mathcal{C}_{1,2,10,11,8,6,7}\) is nB-Problematic since it does not have any P-semiflow, but it has the P-flow \(\mathbf{y}^{T}=[-1\;-\;1\;0\;0\;0\;0\;-\;1\;1\;0\;1\;1\;0]\). Configuration \(\mathcal{C}_{9,8,10,11,8,6,7}\) is B-Problematic since it has no P-flows (and thus, no P-semiflows). Definition 9: Let \(\langle N,\lambda,\mathbf{m}_{0}\rangle\) be an MTS TCPN system.