## 3 Modeling Fault Introduction Process
In addition, we assume that the number of faults discovered instantaneously is related to the number of remaining faults in the software [47], and FD obeys an NHPP. We also consider learning phenomena during FD. Thus, the following differential equation can be derived:  
\[\frac{d\psi(t)}{dt}=b(t)(\phi(t)-\psi(t)) \tag{4}\]  
Substituting Equation (3) into Equation (4),  
\[\psi(t)=\frac{a\exp(1-\exp(-\theta t^{d})+\frac{1}{2}\sigma^{2}t)}{1+\beta\exp (-bt)}-\frac{a\sigma^{2}\exp(-bt)(\exp((b+\frac{1}{2}\sigma^{2})t)+\frac{2b+ \sigma^{2}}{\sigma^{2}}-1)}{(1+\beta\exp(-bt))(2b+\sigma^{2})} \tag{5}\]  
where \(\psi(t)\) is the mean value function representing the expected cumulative number of the detected faults by time \(t\). \(b(t)\) denotes an FDR function, and \(b(t)=\frac{b}{1+\beta\exp(-bt)}\). \(b\) represents an FDR and \(\beta\) is an IF. \(b(t)\) shows an S-shaped curve change over time. Note that we propose three types of introduced faults to illustrate the phenomenon of fault introduction and the random changes in the debugging process of OSS. We assume that the number of faults detected instantaneously is related to the remaining number of faults in the software, where \"instantaneous\" refers to the number of faults that are detected in \((t,t+\Delta t)\), which is related to the number of remaining faults in the software. In this paper, we consider that there is a learning process for OSS during FD. In other words, the learning phenomenon occurs during software testing, where software testers go through a learning process. Herein, software testers refers to software developers, volunteers in the community, or members in the team. This learning process presents an S-shaped curve with the cumulative number of detected faults over the testing time. Assuming that the intensity function of the fault introduction obeys the Weibull distribution, the reason is that the shape parameter of the Weibull distribution is flexible. By changing the shape parameter of the Weibull distribution, a variety of fault data changes can be simulated. Therefore, the Weibull distribution can simulate the complex process of fault introduction for OSS. Equation (5) is an expression of the PM. It should be noted that in order to simplify the modeling calculation, a Taylor formula expansion and truncation are used in the process of model derivation. Please refer to Appendix A for a detailed model derivation process. We consider that the newly generated faults in the current version are only related to the new functional changes in the current version and have nothing to do with the remaining faults in the previous version. Furthermore, the related research indicated that faults in the current version are not related to the faults in the previous version in most cases for OSS [48]. Note that the case of fault masking [49] is considered to detect faults rather than introduce new ones in this study. In addition, the work of this paper focuses on the establishment of the RM of OSS, which is used to evaluate the reliability of OSS. In order to avoid ambiguity, we call the PM an OSS RM rather than an OSS fault model. In this paper, software reliability refers to the probability that software runs without faults for a certain amount of time in a given environment. Assuming that the FD follows an NHPP, software reliability can be expressed as follows:  
\[R(x|t)=\exp\{-\{\psi(t+x)-\psi(t)]\} \tag{6}\]  
In Equation (6), within a time interval of [\(t\), \(t+x\)], the process of software failure follows the NHPP. This is the mathematical expression formula for software reliability. Among them, \(\psi(t)\) represents the MVF; in other words, it denotes the expected cumulative number of faults detected by time \(t\). \(\psi(t)\) is also called software RM.