# Prediction and Comparative Analysis of Software Reliability Model Based on NHPP and Deep Learning
## 4 Conclusions  
In this study, the software reliability models using NHPP and deep learning were compared. Regarding the software reliability model using deep learning, a model composed of deep neural networks and recurrent neural networks (RNN, LSTM, and GRU) that have been applied to time-series data characteristics was used. It was constructed by including the hidden layer of the neural network. Using Datasets 1 and 2, it was confirmed that this model showed better estimation and predictive power than existing software reliability models. Among them, the software reliability of the recurrent neural network series showed better results, and as a result of fitting with 80% and 90% datasets, the software reliability model with the deep neural network added to the GRU showed the best results. In addition, the software reliability model of the recurrent neural network showed good results when the predicted values were compared with the remaining datasets. The NHPP software reliability model showed results that relatively fit the data well with a small number of time points; however, it showed a significant increase as the number of time points increased. On the other hand, the software neural network using deep learning showed results that fit the data trend well, even when the time point was large. However, software reliability models using deep learning always face the problem of overfitting. Therefore, to address this, we plan to conduct research on software reliability models based on mathematical and statistical assumptions by grafting them together with the NHPP software reliability models. Through this, we aim to create a stable, data-dependent, and complementary model with a more mathematical basis. Conceptualization, K.Y.S. and I.H.C. ; funding acquisition, K.Y.S. and I.H.C. ; software, Y.S.K. ; writing--original draft, Y.S.K. ; writing--review and editing, 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 Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (NRF-2021R1F1A1048592, NRF-2021R11A1A01059842). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Data is available in a publicly accessible repository. **Acknowledgments:** This study was supported by the National Research Foundation of Korea (NRF). **Conflicts of Interest:** The authors declare no conflict of interest. **Appendix A**  
**Table A1. ** Cumulative number of software failures in Dataset 1. \begin{tabular}{c c c c c c c c c c} \hline \hline
**Index** & **Failures** & **Cumulative** & **Failures** & **Index** & **Failures** & **Cumulative** & **Index** & **Failures** & **Cumulative** \\ \hline  
& & \\ \hline \hline \end{tabular}   
**Table A2. ** Cumulative number of software failures in Dataset 2. \begin{tabular}{c c c c c c c c c c c} \hline \hline
**Index** & **Failures** & **Cumulative** & **Index** & **Failures** & **Index** & **Failures** & **Cumulative** & **Index** & **Failures** & **Cumulative** \\ \hline
1 & 1 & 34 & 0 & 25 & 67 & 0 & 85 & 100 & 3 & 197 \\
2 & 0 & 1 & 35 & 1 & 26 & 68 & 1 & 86 & 101 & 6 & 203 \\
3 & 1 & 2 & 36 & 0 & 26 & 69 & 2 & 88 & 102 & 3 & 206 \\
4 & 0 & 2 & 37 & 0