## 4 Numerical Examples
### Estimating Model Parameters  
LSE was used in this paper to estimate the model's parameter values. In general, MLE can be used to estimate the parameter values of a model in software reliability modeling. The error utilizing LSE and MLE to estimate model parameter values is quite tiny, because the fault data sets have a small sample size. Furthermore, the value of the maximum likelihood function may not exist in some scenarios [57]. The LSE method can be presented as follows:  
\[\xi=\sum_{i=1}^{n}\left(\psi(t_{i})-\Lambda(t_{i})\right)^{2} \tag{17}\]  
where \(\psi(t_{i})\) represents the estimated number of faults detected by time \(t_{i}\). \(\Lambda(t_{i})\) denotes that the number of faults observed by time \(t_{i}\). Equation (17) takes partial differential equations on both sides.  
\[\frac{\partial\xi}{\partial a}=\sum_{i=1}^{n}\left[(\psi(t_{i})-\Lambda(t_{i}) )\left(\frac{\exp(1-\exp(-\theta t_{i}^{d})+\frac{1}{2}\sigma^{2}t_{i})}{1+ \beta\exp(-bt_{i})}-\frac{\sigma^{2}\exp(-bt_{i})(\exp((b+\frac{1}{2}\sigma^{ 2})t_{i})+\frac{2b+\sigma^{2}}{\sigma^{2}}-1)}{(1+\beta\exp(-bt_{i}))(2b+ \sigma^{2})}\right)\right]=0 \tag{18}\]  
\[\frac{\partial\xi}{\partial b}=\frac{\partial\left(\sum\limits_{i=1}^{n}\left[ (\psi(t_{i})-\Lambda(t_{i}))\left(\frac{a\exp(1-\exp(-\theta t_{i}^{d})+\frac{1 }{2}\sigma^{2}t_{i})}{1+\beta\exp(-bt_{i})}-\frac{a\sigma^{2}\exp(-bt_{i})( \exp((b+\frac{1}{2}\sigma^{2})t_{i})+\frac{2b+\sigma^{2}}{\sigma^{2}}-1)}{(1+ \beta\exp(-bt_{i}))(2b+\sigma^{2})}\right)\right]\right)}{\partial b}=0 \tag{19}\]  
\[\frac{\partial\xi}{\partial\beta}=\sum_{i=1}^{n}\left[(\psi(t_{i})-\Lambda(t_ {i}))\frac{a\theta dt_{i}^{d-1}\exp(1-\exp(-\theta t_{i}^{d})+\frac{1}{2} \sigma^{2}t_{i}-\theta t_{i}^{d})}{1+\beta\exp(-bt_{i})}\right]=0 \tag{20}\]  
\[\frac{\partial\xi}{\partial\sigma}=\frac{\partial\left(\sum\limits_{i=1}^{n} \left[(\psi(t_{i})-\Lambda(t_{i}))\left(\frac{a\exp(1-\exp(-\theta t_{i}^{d})+ \frac{1}{2}\sigma^{2}t_{i})}{1+\beta\exp(-bt_{i})}-\frac{a\sigma^{2}\exp(-bt_{ i})(\exp((b+\frac{1}{2}\sigma^{2})t_{i})+\frac{2b+\sigma^{2}}{\sigma^{2}}-1)}{(1+ \beta\exp(-bt_{i}))(2b+\sigma^{2})}\right)\right]\right)} \tag{21}\]  
Solving Equations (18)-(23), we can calculate the estimated values of the model parameters (\(a\), \(b\), \(\beta\), \(\theta\), \(d\), and \(\sigma\)). Additionally, the MLE function of the PM can be denoted as follows:  
\[\zeta=\Pr\{N(t_{1})=\kappa_{1},N(t_{2})=\kappa_{2},\ldots,N(t_{n})=\kappa_{n} \}=\prod_{j=1}^{n}\frac{\left[\psi(t_{j})-\psi(t_{j-1})\right]^{(\kappa_{j}- \kappa_{j-1})}\exp[-(\psi(t_{j})-\psi(t_{j-1}))]}{(\kappa_{j}-\kappa_{j-1})!} \tag{22}\]  
Take the logarithm on both sides of Equation (22).  
\[L=\log_{e}(\zeta)\ =\sum_{j=1}^{n}\left\{(\kappa_{j}-\kappa_{j-1})\log_{e}\left[ \psi(t_{j})-\psi(t_{j-1})\right]\right\}\ -\psi(t_{n})-\sum_{j=1}^{n}\log_{e}(\kappa_{j}-\kappa_{j-1})! \tag{23}\]  
The partial differential equation for Equation (23) can be obtained as follows (Equation (24)):  
We can obtain the estimated parameter values (\(a\), \(b\), \(\beta\),\(\theta\), \(d\), and \(\sigma\)) of the PM by solving Equation (24). Notably, the values of the maximum likelihood function may not exist when MLE is used to estimate the model parameter values [57].  
\[\frac{\partial L}{\partial a}=\frac{\partial L}{\partial b}=\frac{\partial L} {\partial\beta}=\frac{\partial L}{\partial\theta}=\frac{\partial L}{\partial d }=\frac{\partial L}{\partial\sigma}=0 \tag{24}\]