## 2 New Software Reliability Model
### Proposed NHPP Software Reliability Model  
Differential equations were used to derive the traditional NHPP software reliability model. The functions required to compute the NHPP software reliability model comprise those for the mean value function \(m(t)\), number of failures \(a(t)\), and failure detection rate function \(b(t)\), which are derived using Equation (1), assuming that software failures occur independently [5]. \[\frac{dm(t)}{dt}=b(t)[a(t)-m(t)] \tag{1}\]  
However, as software is utilized in various ways and its structure has become more complex, software failures typically depend on other failures, leading to failures in otherwise functioning software. In addition, as software utilization diversifies, the differences between software development and operating environments can result in software failure. Therefore, we propose a software reliability model for predicting and integrating multiple failures in diverse software applications. Previously, many software reliability studies assumed that software was independent, which is insufficient for describing complex software. Thus, research has been conducted on models that incorporate various assumptions, such as operating environment uncertainty and dependent failures. In this paper, we propose an NHPP software reliability model that considers both the uncertainty of the operating environment and dependent failures by extending a previously developed NHPP software reliability model. The differential equation for assuming operating environment uncertainty is expressed in Equation (2) [19]. \[\frac{dm(t)}{dt}=\eta b(t)[a(t)-m(t)] \tag{2}\]  
where \(a(t)\) is the total expected number of failures \(N\), and \(\eta\) follows a probability density function \(g\) generalized gamma distribution with shape parameter \(\alpha\) and scale parameter \(\beta\). When the initial value \(m(0)=0\), the process derives the mean value function \(m(t)\) using Equation (3). \[m(t)=\int_{\eta}N\bigg{(}1-\exp\bigg{(}-\eta\int_{0}^{t}b(x)dx\bigg{)}\bigg{)} dg(\eta) \tag{3}\]Furthermore, by defining \(b(t)=\frac{\gamma^{2}t}{\gamma t+1}\) and reorganizing Equation (3) using the Laplace transform, we obtain Equation (4) [31]:  
\[m(t)=N\left(1-\frac{\beta}{\gamma+\int_{0}^{t}\frac{\gamma^{2}s}{\gamma s+1}ds} \right)^{\alpha} \tag{4}\]  
where \(\gamma\) is the failure detection rate and is included in the denominator of Equation (4) and in the expression of \(b(t)=\frac{\gamma^{2}t}{\gamma t+1}\). The assumption of operating environment uncertainty is also included in the failure detection rate \(\gamma\), which assumes that dependent failures occurring in the operating environment are also considered. Therefore, the newly proposed NHPP software reliability model assumes both the dependent failure occurrence and operating environment uncertainty, as expressed in Equation (5). \[m(t)=N\left(1-\frac{\beta}{\gamma+\gamma t-\ln(\gamma t+1)}\right)^{\alpha} \tag{5}\]  
Table 1 lists the 15 NHPP software reliability models used in the study. Models 1-14 are traditional software reliability models with different assumptions of software failure, such as independent occurrence, dependent occurrence, and operating environment uncertainty. Model 15 is a new type of NHPP software reliability model proposed in this study, which considers both dependent failures and operating environment uncertainty. \begin{table}
\begin{tabular}{c c c} \hline \hline
**No. ** & **Model** & **Mean Value Function** & **Note** \\ \hline
1 & [PERSON] (GO) [5] & \(\text{m}(t)=\text{a}\left(1-e^{-bt}\right)\) & Concave \\ \hline
2 & [PERSON] et al. (DS) [7] & \(m(t)=\text{a}\left(1-(1+\text{bt})e^{-bt}\right)\) & S-Shape \\ \hline
3 & [PERSON] (IS) [8] & \(m(t)=\frac{\text{a}\left(t-e^{-bt}\right)}{1+\beta e^{-bt}}\) & S-Shape \\ \hline
4 & [PERSON] et al. (YID 1) [9] & \(m(t)=\frac{\text{a}\left(t-e^{bt}\right)}{\text{a}+\text{b}}\left(e^{at}-e^{- bt}\right)\) & [PERSON] \\ \hline
5 & [PERSON] et al.