## 4 Simulations and Results  
The last step of our methodology consists of running simulations of our timed CPN model. Since we are using the Erlang distribution, the Markovian hypothesis is not satisfied for our model, and we we choose the Monte Carlo simulation to assess predictive RAM indicators. Using Equation (4), we evaluate reliability and maintainability while we assess availability using Equation (8). The **CPN'Replications.nreplications** function is deployed to automatically run simulations. The [PERSON] and Griffiths equation is employed to determine the minimum number of simulations, n, needed to achieve a given estimation error with a confidence level (1-\(\alpha\)). It is expressed as follows [29]:  
\[n=\big{(}\frac{z_{\alpha/2}\sigma}{e}\big{)}^{2} \tag{10}\]  
where:  
* \(\sigma\): Standard deviation;
* \(\epsilon\): Standard error. If we admit a standard error equal to 0,02 \(\sigma\) and a confidence interval of 95% (\(\alpha\) = 0.05), the \(z_{\alpha/2}\) value for a 95% confidence level is 1.96. We obtain a number of needed iterations equal to 9604, and, thus, we choose 10,000 iterations. Table 2 below shows the different dysfunctional distributions defined for each type of component, we use the same parameter values mentioned in the work of [12; 13] extracted from available databases. We specify that the data are provided as an example, without being truly faithful to reality, the purpose of this section is to apply the methodology for evaluating predictive dependability attributes and assessing the RBS performance. We highlight Erlang distributions for repair transitions in the Table 3 below: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Component** & **Distribution** & **Average Parameters Values** \\ \hline Pedal & Weibull & \(\eta\) = 1000, \(\beta\) = 1.5 \\ Master cylinder & Weibull & \(\eta\) = 1000, \(\beta\) = 1.5 \\ Hydraulic group & Weibull & \(\eta\) = 1000, \(\beta\) = 1.5 \\ Reservoir & Weibull & \(\eta\) = 1000, \(\beta\) = 1.5 \\ Electrical machine & Weibull & \(\eta\) = 1000, \(\beta\) = 1.5 \\ Disc & Weibull & \(\eta\) = 1900, \(\beta\) = 1.5 \\ Caliper & Weibull & \(\eta\) = 1900, \(\beta\) = 1.5 \\ Brake pad & Weibull & \(\eta\) = 1900, \(\beta\) = 1.5 \\ Type & Weibull & \(\eta\) = 1000, \(\beta\) = 1.5 \\ Speed sensor & Exponential & \(\lambda\) = \(3\times 10^{-4}\) \\ DC-DC converter & Exponential & \(\lambda\) = \(7.5\times 10^{-3}\) \\ Battery & Exponential & \(\lambda\) = \(5\times 10^{-4}\) \\ ECU & Exponential & \(\lambda\) = \(50\times 3\times 10^{-6}\) \\ AData & Exponential-Musa & \(\lambda\) = 5.04 \(\times 10^{-4}\), N0 = 10 \\ IO & Exponential-Musa & \(\lambda\) = 5.04 \(\times 10^{-4}\), N0 = 10 \\ OS & Exponential-Musa & \(\lambda\) = 5.04 \(\times 10^{-4}\), N0 = 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average parameter values for reliability distributions. Information provided by selected CPN Tools monitors are recorded in separate text files and in a single folder for each simulation. CPN Tools software provides a report file in HTML with different statistical and performance analysis. This report offers an overview of the statistics calculated over the course of a simulation. We list, in Table 4, the statistical results; 95% CI is intended for the half length of the 95% confidence interval. \begin{table}
\begin{tabular}{c c c c c} \hline
**Component** & **MTTFF** & **MTBF** & **MTTR** & **Unavailability** \\ \hline Ped: Average & 1119.90 & 512.19 & 2.43 & 0.000050 \\  
hline \end{tabular}
\end{table} 
Table 4: RBS and component mean times (10,000 replications). \begin{table}
\begin{tabular}{c c} \hline
**Component** & **Average Parameters Values** \\ \hline Pedal & 1/2 \\ Master cylinder & 1/3 \\ Hydraulic group & 1/3 \\ Reservoir & 1/2,5 \\ Electrical machine & 1/3 \\ Disc & 1/1.2 \\ Caliper & 1/1.2 \\ Brake pad & 1/1.2 \\ Tyre & 1/1.5 \\ Speed sensor & 1/5 \\ DC-DC converter & 1/2 \\ Battery & 1/3 \\ ECU & 1/5 \\ AData: & 1/1.08 \\ IO & 1/2.17 \\ OS & 1/0.5 \\ \hline \end{tabular}
\end{table}