## 4 Levelized Multiple Workflow Heterogeneous Earliest Finish Time
### The LMAEFT Algorithm
### Statistical Testing

The statistical analyses used in this section to determine the significance level of the hypothesis on sample datasets of Sections 5.1-5.3 of the simulation investigation, displayed in Figures 8-10, were performed using SPSS Statistics 20. The hypotheses are as follows:

* Hypothesis 1 (For Comparing samples)
* This states that there is no significant difference between the various models when considering the variable X. * This states that there is a significant difference between the various models when considering the variable X.

Figure 10: Varying the number of virtual machines. Where X consists of the parameters, i.e., makespan, flowtime (FT), and average utilization (Us). Let us take the simulation samples for LMHEFT, HEFT-1, HEFT-2, and DLS on makespan by varying the number of workflows as presented in Figure 8a. In SPSS, some tests, such as the normality tests and homogeneity (equality of variance) tests, are performed to check the applicability of parametric or non-parametric tests on the data samples LMHEFT, HEFT-2, HEFT-1, and DLS. The confidence interval is 95%. The data samples for models on the makespan are given in Table 5.

The Kolmogorov-Smirnov and Shapiro-Wilk tests are used to test the normality of the samples. The Shapiro-Wilk test is preferred for small samples, while the other is recommended for larger samples [80; 81]. The results of the Kolmogorov-Smirnov and Shapiro-Wilk tests for the data with a sample size of six are provided in Table 6. The majority of the significant values in Table 6 are greater than 0.05. Additionally, the normal Q-Q plots are also presented in Figures 11-14 to further verify the normality of each model's samples, i.e., LMHEFT, HEFT-2, HEFT-1, and DLS. From the numerical results and normal Q-Q plots, it is concluded that the samples are normally distributed. The Levene test is used to measure the equality of variances of samples used in the statistical analysis. Further, the results for homogeneity of variances are presented in Table 7 by using the Levene test. The Sig. value 0.002 is less than 0.05, which confirms that the sample variances are not homogeneous. The samples of the LMHEFT, HEFT-1, DLS, and HEFT-2 models are all independent of the makespan, flowtime, and utilization. The samples are also greater than two in number. Based on the above verification, the data samples are not homogeneous. It is observed from the above table that the equality of variance assumption is violated, which is a pre-condition for the parametric test. The significance level between the samples for hypothesis testing can thus be tested using the non-parametric test, namely, the Kruskal-Wallis test [80; 82; 83]. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{**Kolmogorov–Smirnov**} & \multicolumn{3}{c}{**Shapiro–Wilk**} \\ \cline{3-8}  & & **Statistic** & **df** & **Sig.** & **Statistic** & **df** & **Sig.** \\ \hline \multirow{4}{*}{Algorithm} & LMHEFT & 0.218 & 6 & 0.200 * & 0.871 & 6 & 0.229 \\ \cline{2-8}  & HEFT-1 & 0.255 & 6 & 0.200 * & 0.847 & 6 & 0.149 \\ \cline{1-1} \cline{2-8}  & HEFT-2 & 0.207 & 6 & 0.200 * & 0.901 & 6 & 0.379 \\ \cline{1-1} \cline{2-8}  & DLS & 0.251 & 6 & 0.200 * & 0.819 & 6 & 0.087 \\ \hline \hline \end{tabular}
* This is a lower bound of the true significance.

\end{table} 
Table 6: Normality tests for models for makespan on a varying number of workflows. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**S. No. ** & **Batch Size** & **LMHEFT** & **HEFT-1** & **HEFT-2** & **DLS** \\ \hline
1 & 4 & 31,856,647.9 & 54,132,829.43 & 31,784,467.61 & 193,039,779.7 \\ \hline
2 & 8 & 35,042,724.8 & 96,602,445.47 & 35,458,823.07 & 208,631,874.1 \\ \hline
3 & 16 & 42,678,420.3



Figure 11: Normal Q-Q plot for makespan of LMHEFT. Figure 12: Normal Q-Q plot for makespan of HEFT-1. Figure 13: Normal Q-Q plot for makespan of HEFT-2. The statistics of the Kruskal-Wallis test are shown in Table 8. On the makespan of the samples under consideration, the significant (Sig.) value, in this case, is 0.003.