# Fluid Net Models: From Behavioral Properties to Structural Objects
## 5 Configurations, Continuity, and Monotonicity in MTS
* The system is _suitable_, denoted by \(\mathbf{S}\), if every reachable equilibrium marking \(\mathbf{m}_{e}\) activates a suitable configuration,
* The system is nB-Problematic, denoted by \(\mathbf{nB}\), if every reachable equilibrium marking \(\mathbf{m}_{e}\) that does not activate a suitable configuration, activates only nB-Problematic configurations. * The system is B-Problematic, denoted as \(\mathbf{B}\), if there exists a reachable equilibrium marking \(\mathbf{m}_{e}\) that does not activate a suitable configuration, but activates a B-Problematic one. ```
0:\(N\) and configuration \(\mathcal{C}\). 0: The type of configuration
1: Compute the T-coverture of \(\mathcal{C}\): \(\mathcal{T}_{C}\). 2: Compute the token-flow matrix of the associated subnet: \(\mathcal{C}_{\mathcal{T}_{C}}\). 3:\(\mathcal{P}_{3}:\left\{\text{Find }\mathbf{y}\geq\mathbf{0}\;\text{s.t. }\;\mathbf{y}^{T} \mathcal{C}_{\mathcal{T}_{C}}=0\wedge\sum\limits_{p_{i}\in\mathcal{T}_{C}}\bm {y}(p_{i})\geq 1\wedge\forall p_{j}\
ot\in\mathcal{T}_{C},\mathbf{y}(p_{j})=0\right\}\)
4:if\(\mathcal{P}_{3}\) has a solution then
5:\(\mathcal{C}\) is suitable
6:else
7:if\(rank(\mathcal{C}_{\mathcal{T}_{C}})=|\mathcal{T}_{C}|\)then
8:\(\mathcal{C}\) is B-problematic
9:else
10:\(\mathcal{C}\) is nB-problematic
11:endif
```  
**Algorithm 3** Computing the type of a given configuration. Figure 10 shows a complete picture of the relationships that exist between the reachable configurations of a system and the equilibrium properties of the equilibrium marking (monotonicity, continuity, and deadlock-free monotonicity), with respect to the initial marking and the firing rates. The upper and lower part of that diagram were discussed in Section 4. Discontinuities of the equilibrium throughput, either with respect to the firing rates or with respect to the initial markings, imply the non-monotonicity of the system. In the case of the firing rate variation, the non-monotonicity implies that the equilibrium marking is reaching regions in which the associated configuration is problematic. Moreover, the converse is also true. Theorem 4 ([14]): _Let \((N,\lambda,\mathbf{m}_{0})\) be an MTS TCPN system. The following two statements are equivalent:_1. _Every reachable equilibrium marking activates a suitable configuration._
2. _The equilibrium throughput is monotonic with respect to \(\Lambda\)._  
For example, continuing with the manufacturing system depicted in Figure 1. Let the initial marking be fixed to \(\mathbf{m}_{0}=\left[13\ 0\ 0\ 0\ 0\ 0\ 0\ 5\ 7\ 4\ 4\ 7\right]^{T}\), and the firing rates \(\Lambda=\left[1\ 1\ 1\ 1\ \lambda_{5}\ 1\ 1\right]^{T}\). Increasing the firing rate \(\lambda_{5}\), the equilibrium throughput exhibits a discontinuity as can be seen in Figure 11a. Since this system is discontinuous, according to the relations shown in Figure 10, it is also non-monotonic. This behavior occurs particularly when the firing rate \(\lambda_{2}\) is increased, as is illustrated by Figure 11b. In this last case, the system reaches a deadlock for any \(\lambda_{2}\) greater than \(\lambda_{5}\). Now, if the initial marking is set to \(\mathbf{m}_{0}=\left[13\ 0\ 0\ 0\ 0\ 0\ 6\ 7\ 4\ 4\ 7\right]^{T}\), one more mark in place \(p_{8}\), then the system cannot reach a deadlock for any value of the firing rates. This can be explained thanks to the P-flow \(\mathbf{y}^{T}=\left[-1\ -1\ 0\ 0\ 0\ 0\ -1\ 1\ 0\ 1\ 1\ 0\right]\). The invariant law \(\mathbf{y}^{T}\mathbf{m}_{0}=\mathbf{y}^{T}\mathbf{m}\) gives that \(1=\mathbf{m}(p_{8})+\mathbf{m}(p_{10})+\mathbf{m}(p_{11})-\mathbf{m}(p_{1})-\mathbf{m}(p_{2})-\mathbf{ m}(p_{7})\), and mathematically implies that the deadlock marking (where all those places must be zero) cannot be reached. The physical meaning is that the number of resources, \(\mathbf{m}_{0}(p_{8})=6\), together with the capacity of the processing machine (\(\mathbf{m}_{0}(p_{10})=4\) and \(\mathbf{m}_{0}(p_{11})=4\)), are greater than the number of products that the system handles, \(\mathbf{m}_{0}(p_{1})=13\). This prevents the blocking situation.