## 5 Methodology for Reliability and Availability Evaluation
### Minimal Cut Set Generation  
After defining both the network failure condition and finding all the paths between the gateway and field devices that belong to the network failure condition, it becomes possible to generate a fault tree describing the network failure process. In this faulty tree, the top event represents the network failure condition and the basic events represent the device failure events. It could have been possible to feed this model directly into an evaluation tool in order to compute the dependability measures. However we  
Figure 9: Output flow of Algorithm 1 for the device \(Fd_{0}\) of Figure 5.  
choose to reduce this tree to a _minimal_ one. The goal is to obtain a less complex fault tree in order to enable a faster computation of the dependability measures. Therefore, we compute the minimal cut sets of the fault tree and use this data to build a _minimal_ fault tree. The algorithm used to compute the minimal cut sets is similar to the one proposed in [47]. The major difference is that in the original proposal the authors consider reliable devices and unreliable links, whereas in our proposal we consider the opposite (_i.e._, unreliable devices and reliable links). An inversion technique to generate the minimal cut sets from the _minimal path sets_ (MPS) was proposed in [48]. A MPS represents a set of components such that, if all components are properly working then the system is operational. However, if a component of the MPS fails, then the system also fails. It was proved in [48] that the application of [PERSON]'s law to an appropriate boolean polynomial of the minimal path set is related to a boolean polynomial of the minimal cut set. An efficient algorithm for path inversion was extended in [47] based on the procedure proposed in [48]. To clarify the minimal cut set generation, consider the following example that assumes the topology described in Figure 10. In this scenario, the field device \(Fd_{2}\) is the source whereas the gateway is the sink. \(L_{i}\) indicates if a link between two network devices is operational, whereas \(\overline{L_{i}}\) indicates the link is down. Based on [47], the minimal path set for this example is described by Equation (12). \[L_{1}L_{4}+L_{2}L_{5}+L_{2}L_{3}L_{4}+L_{1}L_{3}L_{5} \tag{12}\]  
As described in [48], applying the [PERSON]'s laws to the Equation (12) we can find a boolean polynomial related to the minimal cut set according to Equation (13). By applying the distributive law \(((A+B)C=AC+BC)\) and the absorption law \((A+AB=A)\) to Equation (13), we obtain the result presented in Equation (14). In other words, the communication between \(Fd_{2}\) and the gateway is broken if either the links \(L_{1}\) and \(L_{2}\) fail or if the links \(L_{1}\), \(L_{3}\) and \(L_{5}\) fail or if the links \(L_{2}\), \(L_{3}\) and \(L_{4}\) fail or if the links \(L_{4}\) and \(L_{5}\) fail. \[(\overline{L_{1}}+\overline{L_{4}})(\overline{L_{2}}+\overline{L_ {5}})(\overline{L_{2}}+\overline{L_{3}}+\overline{L_{4}})(\overline{L_{1}}+ \overline{L_{3}}+\overline{L_{5}}) \tag{13}\] \[\overline{L_{1}}\overline{L_{2}}+\overline{L_{1}}\overline{L_{3} }\overline{L_{5}}+\overline{L_{2}}\overline{L_{3}}\overline{L_{4}}+\overline{ L_{4}}\overline{L_{5}} \tag{14}\]  
On the other hand, if we assume unreliable devices and reliable links the minimal path sets of Figure 10 would be different from the represented in Equation (13). In this case, the minimal path sets  
Figure 10: Topology used to exemplify the minimal cut set generation. are described by Equation (15). In the same way, if we apply the DeMorgan's law and use the distributive and the absorption laws, the minimal cut sets for the example of Figure 10 can be obtained according to Equation (16). In other words, the communication between \(Fd_{2}\) and the gateway is interrupted if either the gateway fails or if \(Fd_{2}\) fails or if \(Fd_{0}\) and \(Fd_{1}\) fail.