## 4 Levelized Multiple Workflow Heterogeneous Earliest Finish Time
### The LMAEFT Algorithm
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Task**} & \multirow{2}{*}{**Level (\(I\))**} & \multicolumn{4}{c}{**ETC Value (\(E_{ijk}\))**} & \multicolumn{2}{c}{\(-\)} \\  & & \multicolumn{2}{c}{**VM\({}_{1}\)**} & \multicolumn{2}{c}{**VM\({}_{2}\)**} & \multicolumn{2}{c}{**VM\({}_{3}\)**} & \\ \hline T\({}_{11}\) & 1 & 47.2 & 35.4 & 44.25 & 42.28 & 207.27 \\ T\({}_{12}\) & 2 & 66.6 & 49.95 & 62.44 & 59.66 & 140.74 \\ T\({}_{13}\) & 2 & 78.8 & 59.1 & 73.875 & 70.59 & 145.01 \\ T\({}_{14}\) & 3 & 53.33 & 40 & 50 & 47.78 & 47.78 \\ T\({}_{21}\) & 1 & 25 & 18.75 & 23.44 & 22.4 & 139.05 \\ T\({}_{22}\) & 2 & 38.67 & 29 & 36.25 & 34.64 & 164.61 \\ T\({}_{23}\) & 2 & 48.67 & 36.5 & 46.625 & 43.93 & 103.34 \\ T\({}_{24}\) & 2 & 39.33 & 29.5 & 36.875 & 35.235 & 87.985 \\ T\({}_{25}\) & 3 & 44 & 33 & 41.25 & 39.42 & 39.42 \\ T\({}_{31}\) & 1 & 37.33 & 28 & 35 & 33.44 & 187.98 \\ T\({}_{32}\) & 2 & 91.336 & 68.5 & 85.626 & 81.82 & 141.22 \\ T\({}_{33}\) & 3 & 44 & 33 & 41.25 & 39.42 & 39.42 \\ T\({}_{34}\) & 3 & 40 & 30 & 37.5 & 35.83 & 35.83 \\ \hline \end{tabular}
\end{table} 
Table 1: Complete information on merged workflows with upward ranks. Equation (8), EFT\({}_{111}\)= 47.2 + 10 = 57.2, EFT\({}_{112}\) = 90 + 35.4 = 125.4, EFT\({}_{113}\) = 30 + 34.25 = 74.25 (steps 1-4, Algorithm 3). After that, Algorithm 3 returns the best VM, that is V\({}_{1}\) (Step 5, Algorithm 3). So, T\({}_{1}\) will be assigned to V\({}_{1}\) as it offers a minimum earliest finish time (steps 9-11, Algorithm 1). After the allocation of T\({}_{11}\), the available time of VMs is 57.2, 90, and 30 (Step 12, Algorithm 1). Then, T\({}_{11}\) will be removed from the \(\rho^{2}\) list (Step 13, Algorithm 1). The actual entry tasks of WF\({}_{2}\) (T\({}_{22}\) and T\({}_{21}\in\rho^{2}\)) and WF\({}_{3}\) (T\({}_{31}\in\rho^{2}\)) have no predecessor tasks, so T\({}_{31}\), T\({}_{22}\), and T\({}_{21}\) will be assigned using available time and EFT similar to T\({}_{11}\). After the assignment of all the tasks of \(\rho^{2}\), the updated available times of VMs are 95.87, 90, and 88.44, as shown in Table 2.

Afterward, the tasks of the next partition are allowed to be allocated, i.e., \(\rho^{3}\) = {T\({}_{13}\), T\({}_{32}\), T\({}_{12}\), T\({}_{23}\), T\({}_{24}\)}. Now, T\({}_{13}\) is dependent on T\({}_{11}\). T\({}_{11}\) has finished at V\({}_{1}\) with an actual finish time of 57.2. So, EST\({}_{131}\) = max {95.87, max (57.2 + 0)} = 95.87, EST\({}_{132}\) = max {90, max (57.2 + 19.98)} = 90 and EST\({}_{133}\) = max {88.44, max (57.2 + 19.98)} = 88.44. Furthermore, EFT\({}_{131}\) = 78.8 + 95.87 = 174.67, EFT\({}_{132}\) = 59.1 + 90 = 149.1, and EFT\({}_{133}\) = 73.875 + 88.44 = 162.315. V\({}_{2}\) offers the minimum EFT; thus, T\({}_{31}\) will be assigned to V\({}_{2}\). Further, T\({}_{32}\), T\({}_{12}\), T\({}_{23}\), and T\({}_{24}\)\(\in\rho^{3}\) have only a single parent. Therefore, these tasks will be assigned similarly to T\({}_{13}\). However, we can see in Figure 5 that T\({}_{23}\) has two parents (T\({}_{21}\) and T\({}_{22}\)), which have been finished at V\({}_{3}\) and V\({}_{1}\), respectively. Now, EST\({}_{231}\) = max {162.47, max ((88.44 + 13.32), (95.87 + 0))} = 162.47, EST\({}_{232}\) = max {149.1, max ((88.44 + 13.32), (95.87 + 26.64))} = 149.1, EST\({}_{233}\) = max {174.066, max ((88.44 + 0), (95.87 + 26.64))} = 174.066. Additionally, EFT\({}_{231}\) = 48.67 + 162.47 = 211.14, EFT\({}_{232}\) = 36.5 + 149.1 = 185.6, EFT\({}_{233}\) = 46.625 + 174.066 = 220.691. So, T\({}_{23}\) will be assigned to V\({}_{2}\) with the updated available times of VMs such as 201.8, 185.6, and 174.066, as shown in Table 2.

Similarly, the allocation of tasks of the fourth partition, \(\rho^{4}\) = {T\({}_{14}\), T\({}_{33}\), T\({}_{25}\), T\({}_{34}\)}, has been performed and is also presented in Table 2. The last partition has only a single task T\({}_{\rm exit}\), which is a pseudo exit task that does not affect the available time of VMs.