## 5 Numerical Examples
### Model Analysis  
Tables 5-7 summarize the results of the estimated parameters of all 10 models in Table 1 using the LSE technique and the values of the six common criteria: MSE, AIC, PRR, PP, SAE, and R\({}^{2}\). We obtained the six common criteria at \(\mathrm{t}=1,\ 2,\cdots,25\) from DS1 (Table 2), at \(\mathrm{t}=1,2,\cdots,17\) from DS2 (Table 3), and at cumulative testing CPU hours from DS3 (Table 4). As can be seen in Table 5, when comparing all of the models, the MSE and AIC values are the lowest for the newly proposed model, and the PRR, PP, SAE, and R\({}^{2}\) values are the second best. The MSE and AIC values of the newly proposed model are 7.361, 114.982, respectively, which are significantly less than the values of the other models. In Table 6, when comparing all of the models, all criteria values for the newly proposed model are best. The MSE value of the newly proposed model is 60.623, which is significantly lower than the value of the other models. The AIC, PRR, PP, and SAE values of the newly proposed model are 151.156, 0.043, 0.041, and 98.705, respectively, which are also significantly lower than the other models. The value of R\({}^{2}\) is 0.960 and is the closest to 1 for all of the models. In Table 7, when comparing all of the models, all the criteria values for the newly proposed model are best. The MSE value of the newly proposed model is 6.336, which is significantly lower than the value of the other models. The PRR, PP, and SAE values of the newly proposed model are 0.086, 0.066, and 36.250, respectively, which are also significantly lower than the other models. The value of R\({}^{2}\) is 0.9940 and is the closest to 1 for all of the models. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Time Index** & **Cumulative** & **Time Index** & **Cumulative** & **Time Index** & **Cumulative** \\
**(CPU hours)** & **Failures** & **(CPU hours)** & **Failures** & **(CPU hours)** & **Failures** \\ \hline
519 & 16 & 4422 & 58 & 8205 & 96 \\
968 & 24 & 5218 & 69 & 8564 & 98 \\
1430 & 27 & 5823 & 75 & 8923 & 99 \\
1893 & 33 & 6539 & 81 & 9282 & 100 \\
2490 & 41 & 7083 & 86 & 9641 & 100 \\
3058 & 49 & 7487 & 90 & 10,000 & 100 \\
3625 & 54 & 7846 & 93 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: DS3: Tandem Computers (TDC) data set. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Week Index** & **Failures** & **Cumulative Failures** & **Week Index** & **Failures** & **Cumulative Failures** \\ \hline
1 & 90 & 90 & 10 & 0 & 190 \\
2 & 17 & 107 & 11 & 2 & 192 \\
3 & 19 & 126 & 12 & 0 & 192 \\
4 & 19 & 145 & 13 & 0 & 192 \\
5 & 26 & 171 & 14 & 0 & 192 \\
6 & 17 & 188 & 15 & 11 & 203 \\
7 & 1 & 189 & 16 & 0 & 203 \\
8 & 1 & 190 & 17 & 1 & 204 \\
9 & 0 & 190 & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: DS2: medical record system (MRS) data set. Figures 2-4 show the graphs of the mean value functions for all 10 models for DS1, DS2, and DS3, respectively. Figures 5-7 show the graphs of the 95% confidence limits of the newly proposed model for DS1, DS2, and DS3. Tables A1-A3 in Appendix A list the 95% confidence intervals of all 10 NHPP software reliability models for DS1, DS2, and DS3. In addition, the relative error value of the proposed software reliability model confirms its ability to provide more accurate predictions as it remains closer to zero when compared to the other models (Figures 8-10).