## 1 Introduction
The results obtained from the evaluations show that the UMDA exceeds the genetic algorithm in the scheduling speed and resource allocation but significantly lacks a process for task grouping in the clusters, causing a disaggregation of sub-tasks that belong to a PTG. The genetic algorithm has slower scheduling and allocation times but it achieves closer sub-task groupings of PTGs, which optimizes communication times between tasks. This research, which extends a previously published method [8], includes six new contributions, as described below. First, in [8], only one algorithm for random task generation was used, and thus, we used the following. * The Markov chain algorithm [14] for the random generation of PTGs, whose main feature is the generation of PTGs with a greater distribution of sub-tasks in the initial phase, allowing a rapid consumption of resources and observing the behavior of the heuristic algorithms. * The parallel approach for the random generation of graphs on Graphics Processing Units (GPUs) [15], shows a thinning of the width of the PTG, and each level produces a substantial number of edges, indicating the parallel applications with high rates of communication between them, allowing evaluation of the communication times of the target system [13]. Second, two performance parameters--makespan and waiting time--are evaluated separately with each of the above algorithms to evaluate the proposed scheduler in terms of its response times with different workloads as follows,  
* the makespan parameter, which is the most widely used parameter in the literature to evaluate a schedulers' performance, which provides a perspective on the total time each of the PTGs remains in the HDCS from their arrival in the queue to the end of their execution, and
* the waiting time, which is the time that a PTG waits for scheduling and allocation in the queue of the HDCS. This parameter provides an evaluation of the scheduler's response times. Third, the number of clusters in the target system is increased to evaluate the growth of the resource matrix, and the response times of the algorithm when many processors remain available. Fourth, an example is provided of the array method to show how it works and to achieve an understanding of its concepts and algorithms. Fifth, the quality parameter of the allocations is evaluated to verify the percentage of allocated (used) resources of the HDCS once each of the algorithms (UMDA and genetic) performs the scheduling and allocation. Evaluating the quality of allocations is important because it measures the performance of the algorithm when increasing and decreasing the number of resources available in the HDCS; it also allows observing the behavior of each of the algorithms with a minimum or maximum amount of resources, which represents the optimization of resources. Sixth, the section on basic definitions is extended for a broader understanding of the proposed scheduler. The remainder of this paper is structured as follows. Section 2 presents a set of terms and the UMDA (the base algorithm used in this paper). Section 3 summarizes related studies that deal with scheduling problems and critical paths, and studies that proposed solutions to the layering problem in Directed Acyclic Graph (DAG) tasks. Section 4 explains the array method with an example using the basic concepts in Section 2; then, Section 5 describes the experiments and the results obtained on the hardware and software platforms. Section 6 provides a discussion of this work. Section 7 outlines the subsequent studies that are currently being carried out, and Section 8 describes the materials and methods used to carry out this research. Finally, Section 7 presents the conclusions of this work.