## 5 Conclusions and Remarks  
In this study, we proposed a new SRM based on a Rayleigh fault detection rate function subject to the uncertainty of operating environments. Tables 5-7 summarize the estimated parameters for all eight models in Table 1 using the LSE technique and values of the 12 common criteria (RMSE, PRR, PP, AIC, BIC, PRV, RMSPE, TS, PC, PIC, R\({}^{2}\), and Adj R\({}^{2}\)). As shown in Table 5, the proposed model has the lowest RMSE, PRR, PP, AIC, BIC, PRV, RMSPE, TS, PC, and PIC values among all the models. In contrast, the proposed model has R\({}^{2}\) and Adj R\({}^{2}\) values that are the highest among all the models. Table 6 shows that the proposed model has the lowest RMSE, PRV, RMSPE, TS, and PIC values among all the models. Its PRR and PP values are the second lowest, and the R\({}^{2}\) and Adj R\({}^{2}\) values are the highest. Table 7 shows that the proposed model has the lowest RMSE, PRR, PP, AIC, BIC, PRV, RMSPE, TS, PC, and PIC values among all the models, whereas its R\({}^{2}\) and Adj R\({}^{2}\) values are the highest among all the models. The results show that the differences between the actual and predicted values of the proposed model are smaller than those of the other models. However, the AIC and BIC values, which measure the goodness-of-fit of an estimated statistical model, are much smaller for the proposed model than for other models. The R\({}^{2}\) and Adj R\({}^{2}\) values of the proposed model are closer to one than those of the other models. Furthermore, we established a cost model to address the risk level and time required to remove errors and to determine the optimal testing release policy for an NHPP SRM to minimize the expected total software cost. We showed the impact of the operating time, x, by changing it and comparing the ORTimes and minimum ETCs for all eight models. The ORTimes and minimum ETCs using the existing well-known models showed minimal changes after x = 10. Specifically, the testing coverage and proposed model are affected by the operating time, x, in real applications. However, the existing well-known  
Figure 8: ORTime versus function of the minimum ETC for the proposed model in Cases #2–#5 on real data. models are unaffected by the operating time x, after a certain period. By applying the proposed model, we can determine the ORTime in different operating environments. By predicting the fault detection rate and failure frequency of software through the software reliability model, it is possible to understand how reliably the system can be operated and to predict and optimize the costs that may occur in development and maintenance through the software cost model. In addition, it can be utilized in fields that require high reliability, such as aviation, space, military, medical devices, and financial systems related to software. In future studies, we suggest that extended research should be performed considering different fault detection rate functions. Additionally, it is necessary to estimate the parameters and confidence intervals of the proposed model using the maximum likelihood estimation method and the Bayesian estimation method to compare the differences between the limitations of the data format and the accuracy of model performance. There is also a need to further analyze the release time issue using existing and newly proposed models with various data sizes. In addition, further research on SRMs that propose integrated models [43; 44] using machine learning/deep learning algorithms [45; 46; 47; 48] seems necessary. Conceptualization, K.Y.S. and I.H.C. ; software, K.Y.S. ; writing--original draft preparation, K.Y.S. ; writing--review and editing, I.H.C. ; visualization, K.Y.S. ; funding acquisition, K.Y.S. and I.H.C. All authors have read and agreed to the published version of the manuscript. This research was supported by the Global-Learning & Academic research institution for Master's, PhD students, and Postdocs (LAMP) Program of the National Research Foundation of Korea (NRF) grant funded by the Ministry of Education (No.