## 3 Results and Discussion
### Reliability Assessment of Web Service Composition
Then, the Bayes formula based on the prior distribution was used to calculate the \(n\) test cases. In a beta-binomial conjugation \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Number of Failures**} & \multicolumn{3}{c}{**Number of Test Cases**} \\ \cline{2-4}  & **BMCM** & **ESM** & **MMR** \\ \hline
0 & 4602 & 3932 & 3692 \\
1 & 7635 & 5478 & 4573 \\
2 & 9402 & 7123 & 6432 \\
3 & 10,041 & 8232 & 8923 \\
4 & 11,900 & 10,345 & 10,401 \\
5 & 16,104 & 13,335 & 11,452 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Number of reliability demonstration test cases with different methods. framework, it is useful to think that the hyperparameters of the prior distribution (beta distribution) of system failure probability correspond to a certain number of pseudo observations with the properties specified by these hyperparameters [30; 31]. In the case of ESM, let us assume that there are \(n\) experts who assign judgment intervals of \([\theta_{L},\theta_{H}]\) corresponding to the failure probability, \(\theta_{k}\), associated to the \(k\)th stage of growth assessment, and that \(\overline{\theta}_{L}\) and \(\overline{\theta}_{H}\) denote the means of \(\theta_{L}\) and \(\theta_{H}\), respectively, over all experts. Then, the expectation, \(E\), and the variance, \(V\), can be calculated using the following equations. \[ \begin{array}{l}E=\frac{\overline{\theta}_{L}+\overline{\theta}_{H}}{2}\\ V=\frac{\left(\overline{\theta}_{H}-\overline{\theta}_{L}\right)^{2}}{12} \end{array} \]  
The hyperparameters, \(a\) and \(b\), can be evaluated using the following equations. \[ \begin{array}{l}a=\frac{E\left(V-nE\right)}{D}\\ b=\frac{\left(n-E\right)\left(V-nE\right)}{D}\end{array} \]  
where \(D=(n-1)E^{2}+n(E-V)\). The judgment interval, \([\theta_{L},\theta_{H}]\), of each expert reflects their judgement as elicited via interviews. Following the completion of the three steps of reliability growth assessment presented in Table 4, the posterior distribution was obtained to be Beta (61.27,324.92). The numbers of required test cases corresponding to different methods are presented in Table 3. The required number of test cases in the case of BMCM is \(n=4602\), which implies that if no failure is observed in 4602 successive test cases, the reliability of web service composition is deemed to satisfy the reliability requirement, \((p_{0},c)=(0.005,0.9)\), and the software can be accepted. However, if a failure is noticed in, say, the 3021 st test case, the assessment must be continued for 4614 (\(7635-3021=4614\)) further test cases. If no failure is observed during these iterations, the web service composition is deemed to satisfy the reliability demonstration requirement. However, if an additional failure was noticed during this stage as well at, say, the 5500 th test case, the assessment is to be continued for another additional 3902 (\(9402-5500=3902\)) test cases. The satisfaction of the reliability demonstration requirement is then adjudged once again. The process continues in an analogous fashion. The experimental data indicated that the number of test cases required by MMR was significantly lower than that required by BMCM and ESM. Corresponding to 0 failures, the numbers of test cases required by BMCM and ESM were 4602 and 3932, respectively, whereas that required by MMR was merely 3692. This represents reductions of 910 and 670, respectively, corresponding to reduction rates of 19.8% and 14.6%, respectively. Furthermore, in the case of MMR, the required number of test cases exhibited the smallest increment when the number of failures was increased from 0 to 1--for BMCM, ESM, and MMR, the required numbers of test cases increased by 65.9%, 39.3%, and 23.9%, respectively, in this case. Similarly, when the number of failures was increased from 4 to 5, the increase in the numbers of test cases required by BMCM, ESM, and MMR were observed to be 35.3%, 28.9%, and 10.1%, respectively.