## 4 Numerical Examples  
### Fault Data Sets for Oss  
The fault data sets used in this paper are collected from four projects of Apache OSS products ([[https://issues.apache.org/jira/issues](https://issues.apache.org/jira/issues)]([https://issues.apache.org/jira/issues](https://issues.apache.org/jira/issues)), accessed on 5 January 2023), KNOX, NIFI, TEZ, and BIGTOP. Each project of OSS has three successive versions. The first fault data set (DS1), collected from the KNOX project of Apache OSS products, has three subsets, KNOX 0.3.0 (DS1-1), KNOX 0.4.0 (DS1-2), and KNOX 0.5.0 (DS1-3). The second fault data set, collected from the NIFI project of Apache OSS products, has three subsets, NIFI 1.2.0 (DS2-1), NIFI 1.3.0 (DS2-2), and NIFI 1.4.0 (DS2-3). The third fault data set, collected from the TEZ project of Apache OSS products, has three subsets, TEZ 0.2.0 (DS3-1), TEZ 0.3.0 (DS3-2), and TEZ 0.4.0 (DS3-3). The fourth fault data set (DS) was obtained from the BIGTOP project of Apache OSS products. These fault data sets include three successive software versions: BIGTOP 0.3.0 (DS4-1) from October 2011 to April 2012, BIGTOP 0.4.0 (DS4-2) from September 2011 to October 2012, and BIGTOP 0.5.0 (DS4-3) from September 2011 to December 2012. Note that the attributes of the faults in bug tracking systems include Type, Status, Resolution, etc. The type of fault data collected by us include all standard issue types and all substandard issue types. The fault data status includes OPEN, INPROGRESS, REOPENED, RESOLVED, AND CLOSED. The fault data resolution excludes Duplicate, Invalid, Not A Problem, Cannot Reproduce, and Not A Bug. Table 1 lists the detailed information on the fault data sets that are used in this paper. Table 2 shows all software RMs for comparison. In Table 2, there are six CSS RMs, the G-O model, the Delayed S-shaped model, the Inflection S-shaped model, the Yamada Imperfect-2 model, the P-N-Z model, and the Weibull distribution model, and three OSS RMs, the Wang model, the Li model, and the PM. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Fault Data Sets**} &
\begin{tabular}{c} **Total Number of** \\ **Detected Faults** \\ \end{tabular} & **Total Time** & **Time Period of Collected Fault Data Sets** \\ \hline \multirow{3}{*}{DS1(KNOX)} & KNOX 0.3.0 (DS1-1) & 85 & 33 weeks & From March 2013 to November 2013 \\  & KNOX 0.4.0 (DS1-2) & 117 & 73 weeks & From March 2013 to August 2014 \\  & KNOX 0.5.0 (DS1-3) & 80 & 83 weeks & From April 2013 to October 2014 \\ \hline \multirow{3}{*}{DS2(NIFI)} & NIFI 1.2.0 (DS2-1) & 396 & 39 months & From December 2014 to January 2018 \\  & NIFI 1.3.0 (DS2-2) & 111 & 179 weeks & From March 2015 to July 2018 \\  & NIFI 1.4.0 (DS2-3) & 201 & 168 weeks & From December 2014 to January 2018 \\ \hline \multirow{3}{*}{DS3(TEZ)} & TEZ 0.2.0 (DS3-1) & 406 & 237 days & From 19 April 2013 to 1 December 2013 \\  & TEZ 0.3.0 (DS3-2) & 130 & 328 days & From 19 April 2013 to 1 March 2014 \\ \cline{1-1}  & TEZ 0.4.0 (DS3-3) & 72 & 164 days & From 8 October 2013 to 30 March 2014 \\ \hline \multirow{3}{*}{DS4(BIGTOP)} & BIGTOP 0.3.0 (DS5-1) & 92 & 164 days & From October 2011 to April 2012 \\ \cline{1-1}  & BIGTOP 0.4.0 (DS5-2) & 237 & 385 days & From September 2011 to October 2012 \\ \cline{1-1}  & BIGTOP 0.5.0 (DS5-3) & 96 & 66 weeks & From September 2011 to December 2012 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The information on the fault data sets. In this paper, we collected fault data sets using time (days or weeks) in the bug tracking system (please see Table 1). We used the collected fault data sets to fit and estimate the model parameters' values. We can predict the number of faults in a software using the model. Thus, graphs in the paper depict the evolution across time (days or weeks). Although the fault data set used in this experiment is relatively old, using it to evaluate and validate the performance of the model still has practical reference value and guiding significance for modern software testing.