## 4 Case Study and Hybrid Modelling
### Design of Experiment and Simulation Results  
The case study has been analysed according to the design experiment of Table 5. The simulation scenarios 0 and 1 model a queueing system without failures, assuming interarrival and service times according to a negative exponential and a normal distribution; they are needed in order to validate the coded MCS and compare the results with closed analytical equations. In scenarios 2 and 3, on the other hand, the restoration of the system and the evolution of the failure rate due to aging (scenario 2) and due to aging, wear and variations of the operating conditions (scenario 3) are modelled. Figure 7 shows the comparison between the simulative process and the analytic benchmark of scenarios 0. With \(10^{3}\) runs of the simulation, it is possible to observe that the two models converge. In particular, the simulated mean awaiting in queue time is \(W_{0}=10.12\) h, and the analytic mean awaiting queue time \(W_{n\!rf}=10.0\) h (mean error \(\Delta e=\frac{W_{0}-W_{n\!rf}}{W_{n\!rf}}=1.12\) %). Figure 7: Scenario 0—Comparison between simulated progressive mean waiting time (\(W_{0}\)) and analytic value (\(W_{n\!rf}\)). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{_Scenario_} & & & & & \\ \hline _Parameter Description_ & _Symbol_ & **Measuring Unit** & **0** & **1** & **2** & **3** \\ \hline pdf of interarrival time & I & - & neg exp & & & norm \\ mean interarrival time & \(\mathfrak{t}_{\text{in}}\) & h & 10 & 10 & 10 & 10 \\ std. deviation of interarrival time & \(\sigma_{\text{i}}\) & h & 10 & 2 & 2 & 2 \\ \hline pdf of service time & Z & - & neg exp & & norm & \\ mean service time & \(\mathfrak{t}_{\text{sm}}\) & h & 5 & 5 & 5 & 5 \\ std. deviation of service time & \(\sigma_{\text{u}}\) & h & 5 & 1 & 1 & 1 \\ \hline pdf of failure time & \(\Lambda\) & & & & & Weibull \\ service station expected life time & \(\mathfrak{a}_{0}\) & h & - & 100 & 100 \\ shape factor & \(\beta\) & & - & 3 & 3 \\ \hline pdf of repair time & \(\Phi\) & & & \#N/A & Uniform \\ maintenance strategy & & & & \#N/A & Corrective \\ max restoration period & \(\text{TTR}_{\text{max}}\) & h & - & 8 & 8 \\ min restoration period & \(\text{TTR}_{\text{min}}\) & h & - & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Design of experiments. The simulation results of scenario 1 show that the mean waiting time is 5 h. With regard to this scenario, inferential statistics enables one to define an upper bound for mean waiting time in queue according to the following equation [44]:  
\[E(W_{q})\leq\frac{\sigma_{ti}^{2}+\sigma_{ts}^{2}}{ti_{m}(1-\rho)}=7\,\mathrm{h} \tag{22}\]  
where \(p=ts_{m}/ti_{m}\) is the level of service utilization; \(\sigma_{ti}\) and \(\sigma_{ts}\) are the standard deviations, respectively, of interarrival and service time; \(ti_{m}\) is the mean interarrival time. Both the previous results confirm the goodness of the MCS coded. Figure 8 shows the Monte Carlo simulation process of the progressive mean waiting time of Scenarios 1-3. The reported trends allow a person to conclude that the average waiting time in queue increases with the increased complexity of the operative scenario. In fact, Scenario 1 which characterizes a system that cannot fail (max reliability of the system) shows the shortest mean waiting time in the queue. In Scenarios 2 and 3, the mean waiting time increases due to the increasing unavailability of the service station caused by the modelling of the failure rate. In particular for Scenario 2, this result is mainly given by the aging of the service station; whereas in Scenario 3, this effect is more emphasized by the contribution of the operating temperature. The results given by the simulation shows that the average waiting in queue time is, respectively, W1 = 5.0 h; W2 = 5.94 h; W3 = 6.12 h. These cannot be compared with an analytical benchmark because, as explained, the modelling and evaluation of such scenarios are not doable with analytical methods.