Abstract
In this article the on-line problem of scheduling parallel tasks on related computational clusters with processors of different capacities was studied in average case. In the problem the objective is to make a schedule on k clusters with w processors each of N tasks, where the task i,i≤N requires time hi on cluster with nominal capacity of processors and wi≤w processors. We presume for all 1≤i≤N that wi has uniform distribution on (0,w] and that hi has uniform distribution on (0,1]. The processors on different clusters have different capacities v1,…,vk. The task with nominal time hi will require wi processors be computed in time hi/vj on cluster number j. Let sum volume of computations W be the sum of volumes of computations for each task. Let L be the minimal time at which all clusters will compute all the tasks, assigned to them, where each task is assigned to one cluster. The expected value of free volume of computations E(Vsp) is used to analyze the quality of an algorithm, where Vsp=∑1≤i≤kviL-W. It was shown that for every algorithm for scheduling parallel tasks on related clusters E(Vsp)=Ω(w√N). An online scheduling algorithm Limited Hash Scheduling was proposed that distribute tasks to limited areas. This algorithm works in a closed-end mode and has a mathematical expectation of a free calculation volume equal to O(w√(N ln N)). The idea of the algorithm is to schedule tasks of close required number of required processors into different limited in time and the number of allowed to use processors areas on clusters.
Highlights
In this article the on-line problem of scheduling parallel tasks on related computational clusters with processors of different capacities was studied in average case
In the problem the objective is to make a schedule on clusters with processors each of tasks, where the task, requires time on cluster with nominal capacity of processors and processors
On-line algorithm for scheduling parallel tasks on related computational clusters with processors of different capacities and its average-case analysis
Summary
Задача построения расписаний выполнения параллельных задач на группах кластеров с процессорами различной производительности (далее – задача построения расписаний), объединённых, например, сетями Grid[15] с одинаковым числом процессоров на кластерах ранее изучалась лишь в худшем случае. Если представить каждую задачу в виде прямоугольника, ширине которого соответствует число процессоров, необходимых для решения задачи, а высоте прямоугольника без учёта сжатия- нормализованное время исполнения задачи(на кластере единичной производительности), то задача построения расписаний выполнения параллельных задач может рассматриваться, аналогично работе [10], как задача упаковки прямоугольников в несколько полубесконечных полос, в которой, попадая в полосу с номером , прямоугольник сжимается по высоте в раз, где -скорость -ой полосы. Жуку удалось получить оценку R 2e для задачи построения расписаний на группах кластеров с процессорами различной производительности с различным числом процессоров на кластерах, обобщив алгоритм из работы [4] для задачи упаковки прямоугольников в полосы разной ширины на случай задачи построения расписания на кластере. Ω w √ , где N- число задач, V - свободный объём вычислений
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