## 4 Model Analysis
### Mean Time to Failure (MTTF) Analysis  
Setting the total number of fixes to zero and taking the limit as \(s\) approaches zero in Equation (56) enables us to calculate the system's mean time to failure (MTTF). \[MTTF=\underset{s\to 0}{\overset{\bullet}{\lim}}\overset{\bullet}{P}_{up}(s)\]  
The formula for \(MTTF\) can be derived by defining the limit as \(s\) goes to zero and treating all repairs in Equation (56) as zero. \[MTTF=\frac{1}{3(\sigma_{1}+\sigma_{2}+\sigma_{3})+\sigma_{4}} \Big{[}1+3\sigma_{1}\Big{(}\frac{1}{2\sigma_{1}+\sigma_{4}}+\frac{2\sigma_{1}} {\sigma_{1}+\sigma_{4}}\Big{)}+3\sigma_{2}\Big{(}\frac{1}{2\sigma_{2}+\sigma_ {4}}+\frac{2\sigma_{2}}{\sigma_{2}+\sigma_{4}}\Big{)}\\ +3\sigma_{3}\Big{(}\frac{1}{2\sigma_{3}+\sigma_{4}}+\frac{2 \sigma_{3}}{\sigma_{3}+\sigma_{4}}\Big{)}\Big{)}\Big{]} \tag{63}\]  
One can examine the impact of various parameter values (\(\sigma_{1}=0.01,\sigma_{2}=0.02\), \(\sigma_{3}=0.03,\sigma_{4}=0.04\)) on the mean time to failure (MTTF) by systematically changing each parameter individually. Specifically, we are varying \(\sigma_{1},\sigma_{2},\sigma_{3}\), and \(\sigma_{4}\) across a range of values (0.01 to 0.10 in increments of 0.01) in Equation (63). The resulting variations in MTTF, in relation to the corresponding failure rates, are documented in Table 4 and visualized in Figure 5. \begin{table}
\begin{tabular}{c c c c c} \hline
**Failure Rates** & **MTTF (\(\sigma_{1}\))** & **MTTF (\(\sigma_{2}\))** & **MTTF (\(\sigma_{3}\))** & **MTTF (\(\sigma_{4}\))** \\ \hline
0.01 & 14.9052 & 15.7955 & 17.5875 & 14.9052 \\
0.02 & 14.2286 & 14.9052 & 16.2737 & 14.9052 \\
0.03 & 13.3724 & 13.8651 & 14.9052 & 14.9052 \\
0.04 & 12.5392 & 12.8898 & 13.688 & 14.9052 \\
0.05 & 11.7801 & 12.0233 & 12.6432 & 14.9052 \\
0.06 & 11.1031 & 11.2651 & 11.7516 & 14.9052 \\
0.07 & 10.5027 & 10.6029 & 10.9881 & 14.9052 \\
0.08 & 9.9701 & 10.0229 & 10.3297 & 14.9052 \\
0.09 & 9.49622 & 9.51224 & 9.7578 & 14.9052 \\
0.1 & 9.07289 & 9.06025 & 9.25714 & 14.9052 \\ \hline \end{tabular}
\end{table} 
Table 4: Variation of \(MTTF\) with respect to failure rates. Figure 4: Reliability analysis of two cases. Case II: Cost analysis with general distribution:  
Applying the same set of conditions outlined in Section 4.1, Case I, along with Equations (59) and (64), one can derive the following Equation (66), you can find the anticipated profit outcomes in Table 7, and a visual representation of the data is presented in Figure 8.  
\[E_{p}(t)=\left(0.0321583-0.0240503e^{-1.27165t}-0.00068673e^{-1.0910 4t}-0.00131265e^{-1.0701t}\right.\\ -0.00131345e^{-1.05447t}-0.000263134e^{-1.02754t}-0.000540803e^{-1. 01579t}-0.00399124e^{-1.00535t}\\ +3.68768(10^{-12})e^{-0.1t}-2.09926(10^{-11})e^{-0.08t}+3.85423(10 ^{-11})e^{-0.07t}-2.14406(10^{-11})e^{-0.06t}\\ \left.+5.69202(10^{-12})e^{-0.05t}+0.961052t\right)V_{1}-V_{2}t \tag{66}\]  
The anticipated profit is displayed in Table 7 and Figure 8 for the following values of  
identification of various decisions that can be made. These decisions are derived from the insights gained through the analysis conducted. Table 2 and Figure 3 present the system's availability variations under copula repair and general repair for different failure rates. The data clearly illustrate that, as time progresses, the system's availability decreases across all scenarios. Notably, the availability is comparatively better when copula repair is employed. When employing the reduction strategy in Case III, we decrease the unit failure rates of the system by a factor \(\rho\) (where \(0<\rho<1\)). This strategic reduction is aimed at enhancing the system's availability. The evaluation of the system's reliability mirrors the assessment of its availability in both scenarios. As time progresses, the reliability of the initial system diminishes when intentionally manipulating failure rates in scenarios I and II. Introducing a reduction strategy involving a factor \(\rho\) leads to a decrease in the system's failure rate per unit, resulting in an improvement in its reliability.