## 3 Software Cost Model Considering Operating Environment
### Optimal Software Release Policy  
In this section, we describe the expected cost model to determine the optimal software release time (\(\mathrm{T}^{\bullet}\)) that minimizes the ETC and present the proof of the proposed model for determining the optimal software release policy. For details, refer to Appendix B.  
\[\mathrm{m(T)}=\mathrm{N}\left(1-\frac{\beta}{\beta+\left(\mathrm{aT}\right)^{ 2}}\right)^{\alpha}\]  
\[\lambda(\mathrm{t})=\frac{2\mathrm{Na}^{2}\alpha\beta\mathrm{t}\left(1-\frac{ \beta}{\beta+\left(\mathrm{at}\right)^{2}}\right)^{\alpha-1}}{\left(\beta+ \left(\mathrm{at}\right)^{2}\right)^{2}}\] \begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **Model** & \(\mathbf{m(t)}\) & \(\mathbf{\lambda(t)}\) \\ \hline
1 & [PERSON] (GO) [30] & \(\mathrm{m(t)=a\Big{(}1-e^{-bt}\Big{)}}\) & \(\lambda(t)=\mathrm{abe^{-bt}}\) \\
2 & Yamada et al. (DS) [31] & \(\mathrm{m(t)=a\Big{(}1-(1+bt)e^{-bt}\Big{)}}\) & \(\lambda(t)=\mathrm{ab^{2}te^{-bt}}\) \\
3 & [PERSON]–Hatoyama (IS) [32] & \(\mathrm{m(t)=\frac{a\big{(}1-e^{-bt}\big{)}}{1+\beta e^{-bt}}}\) & \(\lambda(t)=\frac{\mathrm{ab}(1+\beta)e^{-bt}}{\big{(}1+\beta e^{-bt}\big{)}^{2}}\) \\
4 & [PERSON] et al. (YID) [33] & \(\mathrm{m(t)=a\Big{[}1-e^{-bt}\Big{]}\big{[}1-\frac{\alpha}{b}\big{]}+\alpha at}\) & \(\lambda(t)=\mathrm{abe^{-bt}\big{[}1-\frac{\alpha}{b}\big{]}+\alpha a}\) \\
5 & [PERSON] et al. (PNZ) [27] & \(\mathrm{m(t)=\frac{a\big{[}1-e^{-bt}\big{]}\big{[}1-\frac{\alpha}{b}\big{]}+ \alpha at}}{1+\beta e^{-bt}}\) & \(\lambda(t)=\frac{\mathrm{ab}[1-\frac{\alpha}{b}\big{]}e^{-bt+\alpha a}}{1+ \beta e^{-bt}}\) \\
6 & [PERSON] (PZ) [34] & \(\mathrm{m(t)=\frac{\big{(}(c+a)\big{[}1-e^{-bt}\big{]}-\big{[}\frac{\alpha b}{ \beta+(at)}\big{]}\big{)}}{1+\beta e^{-bt}}}\) & \(\lambda(t)=\frac{\mathrm{ab}\big{(}1-\frac{\alpha}{b}\big{]}e^{-bt-\alpha}\, \beta e^{-bt}}{\big{(}1+\beta e^{-bt}\big{)}^{2}}\) \\
7 & [PERSON] et al. (TC) [35] & \(\mathrm{m(t)=N\Big{[}1-\Big{(}\frac{\beta}{\beta+(at)}\Big{)}^{\alpha}\Big{]}}\) & \(\lambda(t)=\frac{\mathrm{Na^{b}\alpha\beta^{a+b-1}}}{\big{(}\beta+(at)^{b} \big{)}^{\alpha+b}}\) \\
8 & New Model (Ray) & \(\mathrm{m(t)=N\Big{(}1-\frac{\beta}{\beta+(at)^{2}}\Big{)}^{\alpha}}\) & \(\lambda(t)=\frac{2\mathrm{Na^{2}a\beta\Big{(}1-\frac{\alpha}{b\beta+(at)^{2}} \Big{)}^{\alpha-1}}}{\big{(}\beta+(at)^{2}\big{)}^{2}}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Existing proposed NHPP SRMs and a new NHPP SRM. (3) Predictive power (PP) [36]  
\[\text{PP}=\sum_{i=1}^{n}\biggl{(}\frac{\hat{m}(t_{i})-y_{i}}{y_{i}}\biggr{)}^{2},\]  
(4) Akaike's information criterion (AIC) [37]  
\[\text{AIC}=-2\log(L)+2m,\]  
\[L=\prod_{i=1}^{n}\frac{(m(t_{i})-m(t_{i-1}))^{y_{i}-y_{i-1}}}{\bigl{(}y_{i}-y_{ i-1}\bigr{)}!}e^{-(m(t_{i})-m(t_{i-1}))},\]  
\[\log L=\sum_{i=1}^{n}\bigl{\{}\bigl{(}y_{i}-y_{i-1}\bigr{)}\ln((m(t_{i})-m(t_{i- 1}))-(m(t_{i})-m(t_{i-1}))-\ln(\bigl{(}y_{i}-y_{i-1})!)\bigr{\}}.\]  
(5) Bayesian information criterion (BIC):  
\[\text{BIC}=-2\log(L)+\text{mlog}(n).\]  
(6) Predicted relative variation (PRV) [6]  
\[\text{PRV}\,=\sqrt{\frac{\sum_{i=1}^{n}(y_{i}-\hat{m}(t_{i})-\text{Bias})^{2} }{n-1}},\]  
where the bias is as follows: \(\text{Bias}\,=\,\frac{1}{n}\sum_{i=1}^{n}(\hat{m}(t_{i})-y_{i})\).