## 4 Results
### Maintenance 5.0 Metrics: Resilient Physical Assets
#### 4.3.2 Resilience Using Historic Data  
This calculation can be used when an industrial machine completes its total period of useful life from its start-up (\(\beta<1\)) to the end of its remaining useful life (RUL) (\(\beta>1\)). The performance of a signal coming from a machine can be classified according to its transmission quality as P\({}_{1}\)(t), where normal operation is found in P\({}_{2}\)(t)\({}_{\rm A}\) and decreases until  
Figure 11: Resilience for real-time data. A is the _No_Failure Level_ plus B, where B is the cosine amplitude. Figure 12: MTBF, MTTF and MITTR maintenance metrics. there is a catastrophic fault in P\({}_{2}\)(t)\({}_{\text{D}}\). Inside the Maintenance 5.0 control loop, technological tools are used in each of the stages that an incident causes: normal operation, risk events, symptoms, detection, mitigation, and output (see Figure 13). In this work, resilience is defined as the area that includes the change in the performance of a signal coming from a machine and the identification time, which is the time in which the signal remains outside the desired levels. [PERSON] distribution bathtub curve (Equation (3)) can be applied for resilience analysis:  
\[P_{n}(t)=\frac{\beta}{\theta}\Big{(}\frac{x-\delta}{\theta}\Big{)}^{\beta-1} \tag{3}\]  
In which the three parts of the curve correspond to symptoms, detection, and mitigation. A description of the Weibull distribution applied to maintenance is presented in Table 7.  
Resilience is the area between the maximum performance value line and the performance in the three stages of the Weibull distribution bathtub. \(P_{L}(t)\) is the maximum performance value line, presented in Equation (4):  
\[P_{L}(t)=\frac{P(t_{3})-P(t_{0})}{t_{3}-t_{0}}(t-t_{3})+P(t_{3}) \tag{4}\]  
The resilience calculation formula for use cases with historic data is presented in Equation (5):  
\[ \begin{split} Resilience=\int_{t_{0}}^{t_{1}}&[P_{L}(t )-P_{1}(t)]dt+\int_{t_{1}}^{t_{2}}[P_{L}(t)-P_{2}(t)_{n}]dt\\ &+\int_{t_{2}}^{t_{3}}[P_{L}(t)-P_{3}(t)]dt\end{split}  \tag{5}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stage** & **Description** & **Time** & **\(n\) Value** & **Performance \(P_{n}(t)\)** & \(\beta\) \\ \hline \multirow{2}{*}{**Symptoms**} & The incident gives rise to a state & \multirow{2}{*}{\(t_{1}-t_{0}\)} & \multirow{2}{*}{\(n=1\)} & \multirow{2}{*}{\(P_{1}(t)\)} & \multirow{2}{*}{\(\beta<1\)} \\  & of deteriorating circumstances. & & & & \\ \hline \multirow{3}{*}{**Detection**} & Struggle to overcome the & \multirow{3}{*}{\(t_{2}-t_{1}\)} & \multirow{3}{*}{\(n=2\)} & \multirow{3}{*}{\(P_{2}(t)_{m}=A,B,C,D,E\)} & \multirow{3}{*}{\(\beta=1\)} \\  & degradation obtained by & & & & \\ \cline{1-1}  & the incident. & & & & \\ \hline \multirow{3}{*}{**Mitigation**} & Procedures for recovering from & \multirow{3}{*}{\(t_{3}-t_{2}\)} & \multirow{3}{*}{\(n=3\)} & \multirow{3}{*}{\(P_{3}(t)\)} & \multirow{3}{*}{\(\beta>1\)} \\  & and overcoming incidents. & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Weibull distribution bathtub curve for maintenance. Figure 13: Resilience configuration. In Equation (5), performance changes over different time intervals. \((t_{1}-t_{0})\) is the apparition of symptoms interval, \((t_{2}-t_{1})\) is the detection time, \((t_{3}-t_{2})\) is the mitigation time, \(t_{3}\) is the repairs completed time, and \(P_{o}(t)\) is the desired signal; finally, \(P_{2}(t)_{m}\) is the signal obtained after an incident occurs and it can adopt \(m=A,B,C,D,E\) value. The result of the Equation (5) represents the resilience; the smaller the area obtained, the greater the resilience of that signal. The foregoing implies that the maintenance system executes the control loop quickly and both the technologies and the human being are acting in symbiosis. Table 8 shows the impact that each of the performance levels observed in the system has on the performance metrics, as well as the enabling technologies that can be used within each of the maintenance stages.