# Default Detection Rate-Dependent Software Reliability Model with Imperfect Debugging
## 5 Numerical Example
### Performance Verification
#### 5.3.1 Performance Analysis of the FDR-Related Imperfect Debugging Model  
To verify the difference in FDRF, the experiment will be verified and observed in the type I model that only considers incomplete debugging. In the type I imperfect debugging model, the parameters are \(a(t)\), \(p\) and FDRF, so it is easy to analyze the performance of FDR. The type I imperfect debugging model, namely, the IDI model, is tested on three data sets. Figure 2 shows the fitting curve of the IDI model. \begin{table}
\begin{tabular}{l l} \hline \hline
**Model** & **Cumulative Fault Detection Quantity \(m(t)\)** \\ \hline
**M-1:** Y-Exp [38] & \(m(t)=\frac{ab\left(e^{it}-e^{-it}\right)}{4+b};b(t)=b\) \\ \hline
**M-2:** Y-Lin [38] & \(m(t)=a(1-e^{-bt})(1-\frac{\epsilon}{b})+sat;b(t)=b\). \\ \hline
**M-3:** Pham Zhang IFD [14] & \(m(t)=a\left[1-e^{-bt}\left(1+(b+d)t+bdt^{2}\right)\right];b(t)=b\). \\ \hline
**M-4:** P-Z [35] & \(m(t)=a(1+\beta t)\left[1-e^{-\frac{p^{2}t^{2}}{2(1+t)}}\right];b(t)=b\). \\ \hline
**M-5:** [PERSON] [39] & \(m(t)=\frac{a}{1-\tau}\left[1-e^{-(1-r)bt}\right];b(t)=b\). \\ \hline
**IDI Frame Series Model: IDI\({}_{\textbf{1--IDI}}\)** & \(m(t)=e^{-\int_{0}^{t}k(u)(u)du}\int_{0}^{t}k(v)p(v)e^{\int_{0}^{t}k(s)(p)ds}dv\) \\ \hline
**IDII Frame Series Model: IDII\({}_{\textbf{1--IDI}}\)** & \(m(t)=a\int_{0}^{t}k(u)\cdot e^{-\int_{0}^{t}p(1-(\gamma(t))\cdot k(\tau)t)dt}du\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Participating models. The first model is a unified incomplete debugging model. As shown directly from Figure 2, these models have the same curve shape on three data sets, which reflects that five different FDRs have no substantial impact on the essential trend of the curve. To distinguish the performance of different models, especially to determine the performance differences of different FDRs, Table 2 gives the metrics of the model. From Table 2, it can be seen that IDI_2 and IDI_5 are almost the same in five standards, which is obviously superior to the other three models. This shows that the FDR (i.e., \(b2(t)\) and \(b5(t)\)) adopted by IDI_2 and IDI_5 are more in line with the real fault detection, and its performance is better than the other three FDR models. In fact, the two \(b(t)\) functions in IDI_2 and IDI_5 have the same structure in form. They all show the S-shaped trend of change and can adapt to various changes in software testing. Therefore, the cumulative fault detection function \(m(t)\) corresponding to IDI_2 and IDI_5 has similar performance. Then, Figure 3 shows the fitting results of the type II imperfect debugging model on three failure data sets. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Model** & **DS** & _MSE_ & _R-Square_ & _Variation_ & _RMS-PE_ & _BMMRE_ \\ \hline IDI\_1 & & 1.511895 & 1.045318 & 1.335027 & 1.357991 & 0.382202 \\ IDI\_2 & & 1.194023 & 1.001295 & 1.121184 & 1.121655 & 0.092767 \\ IDI\_3 & DS1 & 1.860862 & 1.071312 & 1.546524 & 1.590827 & 0.524525 \\ IDI\_4 & & 1.431813 & 1.035174 & 1.274901 & 1.289997 & 0.329218 \\ IDI\_5 & & 1.194023 & 1.001209 & 1.121203 & 1.12168 & 0.092741 \\ \hline IDI\_1 & & 25.64698 & 1.211092 & 5.779541 & 5.952458 & 0.325066 \\ IDI\_2 & & 9.980156 & 1.047866 & 3.264629 & 3.272011 & 0.078299 \\ IDI\_3 & DS2 & 39.52952 & 1.301613 & 7.372471 & 7.641251 & 0.550163 \\ IDI\_4 & & 51.54921 & 1.528432 & 7.635519 & 7.718815 & 0.491672 \\ IDI\_5 & & 9.980156 & 1.047869 & 3.264629 & 3.272011 & 0.078299 \\ \hline IDI\_1 & & 0.968407 & 0.99874 & 1.011346 & 1.011442 & 0.057902 \\ IDI\_2 & & 0.934999 & 0.980322 & 0.998502 & 1.000092 & 0.095439 \\ IDI\_3 & DS3 & 1.425435 & 1.060463 & 1.299476 & 1.321645 & 0.127577 \\ IDI\_4 & & 0.9799 & 0.995025 & 1.017026 & 1.017026 & 0.057254 \\ IDI\_5 & & 0.934999 & 0.98032 & 0.998502 & 1.000092 & 0.09544 \\ \hline \end{tabular}
\end{table} 
Table 2: Performance indicators of IDI-type models. Figure 2: IDI Model Fitting Curve. (**a**) IDI Model Fitting Curve on DS1; (**b**) IDI Model Fitting Curve on DS2; (**c**) IDI Model Fitting Curve on DS3.