## 5 Results
### Testing the Makespan Parameter with the Markov Chain Algorithm  
In this section, evaluations are described using synthetic workloads generated with the Markov chain algorithm and real application workloads. The parameter being evaluated is the makespan, based on Definition 11. Given the characteristics of the PTGs generated with the Markov chain algorithm, the largest distribution of sub-tasks was generated in the initial phase. This produced an early search for resources to make the assignments of the sub-tasks that constitute the PTG. The results obtained when evaluating the makespan with the synthetic loads are shown in Figure 4. With the initial loads, the makespan was significantly different between the two algorithms, but as the number of PTGs increased, the similarity between the two algorithms increased. Due to these results, the number of PTGs was increased substantially (from 400 to 500, 750, and 1000) to observe the behavior of the results. The greater the increase in PTGs, the greater the distance in the results. This confirms that the response speed of the UMDA is proportionally greater than that of the genetic algorithm. Therefore, the response speed of the UMDA is substantially better than the genetic algorithm with different workloads. Once the populations to be evaluated were considered, the genetic algorithm was found to be more time-consuming during the convergence process that aims for the best allocations. To determine the quality of allocations, the percentage of resources used by each algorithm (UMDA and genetic) was determined with this parameter. The values are calculated using Equation (6) in Definition 14. Figure 5 shows the percentages obtained with both algorithms. The experiment was performed when the workload was equal to 500 PTGs and the availability of resources was 100%. Then, the latter was reduced in each test, i.e., only 90% was made available in the second experiment, 80% in the third experiment, and so on, until approximately 10% of the system resources were available. The percentages obtained were variable. When there is a large amount of resources available in the target system, the genetic algorithm manages to assign more tasks that use more resources in the allocation; however, when the number of available resources decreases, it consumes more time in the convergence, but maintains a high resource allocation. The advantage observed with the UMDA is that it can provide the allocation results in less time even when the amount of resources is limited in each allocation. We only provide one graph regarding quality of allocations because this research was not meant to measure this parameter; however, for comparison purposes, it is described. Figure 4: Results of makespan parameter with the Markov chain algorithm. Testing the Makespan Parameter with the Parallel Approach for the Random Generation of Graphs on GPUs  
The parallel approach for the random generation of graphs on GPUs produces denser DAGs (high representativeness of the tasks in each level of the DAG), but shows shorter critical paths [13]. Therefore, using this method, it is possible to extract the characteristics of the PTG levels and the density of the graph more easily. The results obtained when evaluating workloads are shown in Figure 6. The performance of both algorithms was similar with loads lower than 500 PTGs, but a substantial difference was observed with workloads higher than 500 PTGs. The fundamental observation in this experiment was the convergence time consumed by the algorithms. It is clear that the UMDA provides better response times when evaluating makespan because it does not manipulate the solutions to the problem; it instead uses the joint probability distribution. Evaluating the Waiting Time Parameter Using the Markov Chain Method and the Parallel Approach for the Random Generation of Graphs on GPUs  
The evaluation parameter, waiting time, allows the observation of the time that tasks must wait before they can be served.