## 3 Results and Discussion
### Reliability Assessment of Web Service Composition  
In this paper, the authors primarily considered the errors induced by web service responses. To this end, the authors utilized a very useful Linux utility called traffic control (tc) to assign a threshold of 10 s for the response time to adequately simulate and control the network latency. If a response is so slow that its delay is noticeable by a customer (i.e., 20% or more), then the service is deemed to have effectively failed. The reliability requirement of the web service composition set by the developer is \(p=0.004\). Reliability assessment of the web services revealed the corresponding failure probabilities, as presented in Table 2--\(\hat{p}_{1}=0.081\), \(\hat{p}_{2}=0.024\), \(\hat{p}_{3}=0.096\) and \(\hat{p}_{4}=0.038\). By (6), the failure probability, \(p_{s}\), of the web service composition can, therefore, be calculated as follows. \[\hat{p}_{s}=F(\hat{p}_{1},\hat{p}_{2},\ldots,\hat{p}_{4})=0.0081 \tag{21}\]  
Substituting (21) into (19) yields  
\[\frac{a}{a+b}=0.0081 \tag{22}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Web Service** & **Failure Probability** \\ \hline \(w_{1}\) & 0.081 \\ \(w_{2}\) & 0.024 \\ \(w_{3}\) & 0.096 \\ \(w_{4}\) & 0.038 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Failure probabilities of web services. Figure 5: Web service composition with invocations. Using (20), the values of the hyperparameters were calculated to be \(a=1\) and \(b=122.5\) following MMR. Then, reliability assessment was initiated with reliability growth assessment. By (21), the reliability, \(\hat{p}_{s}\), of the web service composition was calculated to be 0.0081 and, as 0.0081 \(>0.004\), the reliability of web service composition was deemed to not satisfy the pre-defined reliability requirement, after repairing. Thus, reliability growth assessment was continued. When performing the 250 th test case, the failure occurred, then correcting it. By (8), the failure probability can be calculated as follows. \[\hat{p}_{s}^{\prime}=\frac{a_{0}+x}{a_{0}+b_{0}+n}=\frac{1+1}{1+122.5+250}=0.005 4>0.004\]  
Evidently, the current reliability of web service composition still did not satisfy the reliability requirement. Thus, testing was continued until the 1529 th test case, when the 6 th failure was observed. The corresponding failure probability was calculated as follows. \[\hat{p}_{s}^{\prime}=\frac{a_{0}+x}{a_{0}+b_{0}+n}=\frac{1+6}{1+122.5+1829}=0.0 036<0.004\]  
At this point, the reliability of the web service composition finally satisfied the reliability requirement, and therefore, the software reliability growth assessment was deemed to be complete. Meanwhile, the values of the hyperparameters of the posterior distribution of the failure probability \(p\) were calculated to be \(a_{0}=7,\text{ and }b_{0}=1760.1\). \[f(p_{s})=Beta(7,\text{ }1760.1)=\frac{p_{s}^{6}(1-p_{s})^{1759.1}}{B(7,\text{ }1760.1)}\]  
To verify whether the reliability of the web service composition satisfied the reliability requirement, the following equation was used, following (20), to calculate the maximum number of test cases required for the reliability demonstration test. \[\int_{0}^{pq}\frac{p^{q_{0}+j-1}(1-p)^{b_{0}+N_{j+1}-j-1}}{B\Big{(}a_{0}+j,b+N _{j+1}-j\Big{)}}dp\geq c\]  
Finally, the authors compared the required numbers of test cases using MMR, Bayesian Monte Carlo method (BMCM) [28], and expert scoring method (ESM) [29] to obtain a reliability demonstration requirement of \((p_{0},c)=(0.005,0.9)\). With an increasing number of failures, the maximum number of test cases corresponding to different methods was required to satisfy the requirements presented in Table 3.  
In the case of BMCM, the unavailability of prior information during the first execution of the web service composition induced a uniform distribution of the hyperparameters over the domain. In this case, \(Beta(2,2)\) was adopted to represent the lack of prior knowledge, which assigned the prior reliability centers around 0.5, with equal probabilities of failure and success.