Scheduling Independent Multiprocessor Tasks

Abstract. We study the problem of scheduling a set of n independent multiprocessor tasks with prespecified processor allocations on a fixed number of processors. We propose a linear time algorithm that finds a schedule of minimum makespan in the preemptive model, and a linear time approximation algorithm that finds a schedule of makespan within a factor of (1+\eps) of optimal in the non-preemptive model. We extend our results by obtaining a polynomial time approximation scheme for the parallel processors variant of the multiprocessor task model.     

独立多处理机任务静态调度问题的近似算法

研究独立多处理机任务静态调度问题P_m|fix|C_max,即在m个处理机系统中调度n个多处理机任务,每个任务指派到所需一组处理机上不可剥夺地执行.该问题应用广泛但早已证明为NP难问题,而且也不存在常数近似算法.分析了问题P_m|fix|C_max和其中所有任务都是单位处理机时间的特殊情形P_m|fix,p=1|C_max的调度,并利用实例划分(split scheduling,简称SS)、首次满足优先(first fit,简称FF)和最大宽度优先(large wide first,简称LWF)等方法,构造了问题P_m|fix,p=1|C_max的√2m +1近似算法和问题P_m|fix|C_max的2√m 近似算法,优于目前已有文献的最好结果.     

A Genetic Algorithm for Hybrid Flow-shop Scheduling with Multiprocessor Tasks

The hybrid flow-shop scheduling problem with multiprocessor tasks finds its applications in real-time machine-vision systems among others. Motivated by this application and the computational complexity of the problem, we propose a genetic algorithm in this paper. We first describe the implementation details, which include a new crossover operator. We then perform a preliminary test to set the best values of the control parameters, namely the population size, crossover rate and mutation rate. Next, given these values, we carry out an extensive computational experiment to evaluate the performance of four versions of the proposed genetic algorithm in terms of the percentage deviation of the solution from the lower bound value. The results of the experiments demonstrate that the genetic algorithm performs the best when the new crossover operator is used along with the insertion mutation. This genetic algorithm also outperforms the tabu search algorithm proposed in the literature for the same problem.     

Guaranteeing Response Times for Aperiodic Tasks in Global Multiprocessor Scheduling

We provide a constant time schedulability test for an on-line multiprocessor server handling aperiodic tasks. Dhall's effect is avoided by dividing the tasks in two priority classes based on task utilization: heavy and light. We prove that if the load on the multiprocessor server stays below U _threshold = 3 ? √7 ≈ 35.425%, the server can accept an incoming aperiodic task and guarantee that the deadlines of all accepted tasks will be met. The same number 35.425% is also a threshold for a task to be characterized as heavy.The bound U _threshold = 3 ? √7≈ 35.425% is easy-to-use, but not sharp if we know the number of processors in the multiprocessor system. Assuming the server to be equipped with m processors, we calculate a formula for the sharp bound U _threshold (m), which converges to U _threshold from above as m → ∞.The results are based on a utilization function u(x) = 2(1 ? x)/(2 + √2+2x). By using this function, the performance of the multiprocessor server can in some cases be improved beyond U _threshold(m) by paying the extra overhead of monitoring the individual utilization of the current tasks.     

同等并行处理机上独立任务的调度

n个独立任务在m个同等并行处理机上处理,使总完成时间最小的非抢先调度是确定性调度理论的一个基本问题.文中提出一种算法--Bound Fit算法,它的最坏情况性能至少和MULTIFIT算法一样甚至更好,而所需的时间却比MULTIFIT算法少.     

减链约束多处理器任务在三处理器中的调度

研究三个并行处理器环境中,具有递减链约束的多处理器任务的调度问题,调度目标 是最小化总处理时间,假设单项任务需单位处理时间.首先给出了减链调度问题的最优化性质 与条件,并说明了减链调度问题仍然是NP难的.随后基于两段flow-shop问题的Johnson's算 法的修正和减链调度问题最优化性质,提出了一个启发式算法,并从分析和仿真计算两方面说 明该算法是有效的和高效的.     

The On-Line Multiprocessor Scheduling Problem with Known Sum of the Tasks

In this paper we investigate a semi on-line multiprocessor scheduling problem. The problem is the classical on-line multiprocessor problem where the total sum of the tasks is known in advance. We show an asymptotic lower bound on the performance ratio of any algorithm (as the number of processors gets large), and present an algorithm which has performance ratio at most for any number of processors. When compared with known general lower bounds, this result indicates that the information on the sum of tasks substantially improves the performance ratio of on-line algorithms.     

多处理机容错系统中实时任务的轮转式调度算法

基于多处理机实时系统的“主从备份技术”,文章提出一种采用轮转式调度策略实现容错调度的算法。模拟结果表明,该算法可达到较均衡的任务分布,提高了CPU利用率。     

单层树型网格下独立任务的周期性调度

提出单层树型网格下单位独立任务的周期性调度方法,单位独立任务是大小相等的独立任务.首先,为单层树型网格下的单位独立任务调度建立线性规划模型,通过分析整数线性规划求解过程,发现一个单层树型网格平台在节点构成不同时,分别具有非饱和态、临界态或冗余态特征;并且,随着网格节点上任务数的增多,线性规划最优解呈线性增长,任务调度具有周期性特性.据此给出非饱和态、临界态或冗余态网格的定义、性质和判定方法,推导出单位独立任务调度的周期长度.最后,分析了周期性调度的时间复杂性,提出一种周期性调度算法Periodic-Sched.实验结果表明,周期性调度是有效的.单位独立任务的周期性调度将大规模的任务调度问题简化为一个周期内的任务调度,降低了调度问题的复杂度.该调度方法适用于对Hadoop平台的Map任务进行调度.     

树型网格计算环境下的独立任务调度

任务调度是实现高性能网格计算的一个基本问题,然而,设计和实现高效的调度算法是非常具有挑战性的.讨论了在网格资源计算能力和网络通信速度异构的树型计算网格环境下,独立任务的调度问题.与实现最小化任务总的执行时间不同(该问题已被证明是NP难题),为该任务调度问题建立了整数线性规划模型,并从该线性规划模型中得到最优任务分配方案??各计算节点最优任务分配数.然后,基于最优任务分配方案,构造了两种动态的需求驱动的任务分配启发式算法:OPCHATA(optimization-based priority-computation heuristic algorithm for task allocation)和OPBHATA(optimization-basedpriority-bandwidth heuristic algorithm for task allocation).实验结果表明:在异构的树型计算网格环境下实现大量独立任务调度时,该算法的性能明显优于其他算法.     

容错多处理机中一种高效的实时调度算法

针对基于主副版本容错的多处理机中独立的、抢占性的硬实时任务,提出了一种高效的调度算法——TPFTRM(task partition based fault tolerant rate-monotonic)算法.该算法将单机实时RM 算法扩展到容错多处理机上,并且调度过程中从不使用主动执行的任务副版本,而仅使用被动执行和主副重叠方式执行的任务副版本,从而最大限度地利用副版本重叠和分离技术提高了算法调度性能.此外,TPFTRM 根据任务负载不同将任务集合划分成两个不相交的子集进行分配;还根据处理机调度的任务版本不同,将处理机集合划分成3 个不相交的子集进行调度,从而使TPFTRM 调度算法便于理解、实现以及减少了调度所需要的运行时间.模拟实验对各种具有不同周期和任务负载的任务集合进行了调度测试.实验结果表明,TPFTRM与目前所知同类算法相比,在调度相同参数的任务集合时不仅明显减少了调度所需要的处理机数目,还减少了调度所需要的运行时间,从而证实了TPFTRM 算法的高效性.     

面向多处理器的实时周期任务容错调度算法研究

主副版本策略是多处理器系统实时任务调度中处理容错问题的一种重要方式.根据分布式控制系统的特点,本文提出一种改进的FTRMBF算法-PR-FTRMBF,以提高系统周期任务的可调度性.在FTRMBF等已有的调度算法中,当没有处理器分配给当前副版本时,将为副版本分配新的处理器;本文提出的改进算法则以回溯的方式重新分配主版本.在保证系统实时性能和容错能力的前提下,节省了处理器数目.仿真实验表明,与FTRMBF算法相比,改进算法显著提高了系统任务的可调度性.     

A New Approach for Scheduling of Parallelizable Tasks in Real-Time Multiprocessor Systems

In a parallelizable task model, a task can be parallelized and the component tasks can be executed concurrently on multiple processors. We use this parallelism in tasks to meet their deadlines and also obtain better processor utilisation compared to non-parallelized tasks. Non-preemptive parallelizable task scheduling combines the advantages of higher schedulability and lower scheduling overhead offered by the preemptive and non-preemptive task scheduling models, respectively. We propose a new approach to maximize the benefits from task parallelization. It involves checking the schedulability of periodic tasks (if necessary, by parallelizing them) off-line and run-time scheduling of the schedulable periodic tasks together with dynamically arriving aperiodic tasks. To avoid the run-time anomaly that may occur when the actual computation time of a task is less than its worst case computation time, we propose efficient run-time mechanisms.We have carried out extensive simulation to study the effectiveness of the proposed approach by comparing the schedulability offered by it with that of dynamic scheduling using Earliest Deadline First (EDF), and by comparing its storage efficiency with that of the static table-driven approach. We found that the schedulability offered by parallelizable task scheduling is always higher than that of the EDF algorithm for a wide variety of task parameters and the storage overhead incurred by it is less than 3.6% of the static table-driven approach even under heavy task loads.     

异构环境下独立任务调度算法的研究

本文基于Min-min算法和Sufferage算法提出了基于任务调度损失的最小最早完成时间算法(Sufferage Min-min,SMM).该算法将任务调度损失引入Min-min算法,选取最早完成时间较小的k个任务,再优先对其中任务调度损失最大的一个进行调度.SMM算法克服了Min-min算法单纯追求局部最优而缺少全局意识的缺点.测试表明,SMM算法可以做到调度跨度低与平均等待时间小的统一,在综合性能上较Min-min算法有所提高.     

Online Randomized Multiprocessor Scheduling

The use of randomization in online multiprocessor scheduling is studied. The problem of scheduling independent jobs on m machines online originates with Graham [16]. While the deterministic case of this problem has been studied extensively, little work has been done on the randomized case. For m= 2 a randomized 4/3-competitive algorithm was found by Bartal et al. A randomized algorithm for m ≥ 3 is presented. It achieves competitive ratios of 1.55665, 1.65888, 1.73376, 1.78295, and 1.81681, for m = 3, 4, 5, 6,7 , respectively. These competitive ratios are less than the best deterministic lower bound for m=3,4,5 and less than the best known deterministic competitive ratio for m = 3,4,5,6,7 . Two models of multiprocessor scheduling with rejection are studied. The first is the model of Bartal et al. Two randomized algorithms for this model are presented. The first algorithm performs well for small m , achieving competitive ratios of 3/2 , , for m=2,3,4 , respectively. The second algorithm outperforms the first for m ≥ 5 . It beats the deterministic algorithm of Bartal et al. for all m = 5 ,. . ., 1000 . It is conjectured that this is true for all m . The second model differs in that preemption is allowed. For this model, randomized algorithms are presented which outperform the best deterministic algorithm for small m . Received August 11, 1997; revised February 25, 1998.     

Efficiency of the Methods of Scheduling Complex Sets of Tasks in Nonuniform Multiprocessor Computer Systems

In order to minimize in a nonuniform multiprocessor computer system the time of execution of each particular set of user-defined tasks having random execution times, the methodology of mathematical forecasting of the execution times of complex sets of interrelated tasks was used for the first time to examine the problem of formal estimation and choice of a criterion (discipline) from their predefined set for dynamic scheduling of tasks to the processors of nonuniform multiprocessor computer systems.     

PLUFS: 一种开销敏感的周期任务在线多处理器节能实时调度算法

现有周期任务多处理器节能调度算法虽然在考虑处理器实际开销情况下可以实现较好的节能效果,但仍不能保证最优可调度性.针对嵌入式实时系统中不可忽视的状态切换开销,提出一种开销敏感的周期任务在线多处理器节能实时调度算法PLUFS.该算法通过TL面流调度模型与处理器实际切换开销模型相结合,在每个TL面的初始时刻、任务结束执行时刻实现节能调度,在不违反周期任务集最优可调度性的前提下,达到实时约束与能耗节余的合理折中.经过理论证明和模拟实验,结果表明：PLUFS算法不仅保证了周期任务集的最优可调度性,而且节能效果整体优于现有算法,能耗节余比现有算法提高约10%～20%.     

Linear Programming-Based Affinity Scheduling of Independent Tasks on Heterogeneous Computing Systems

Resource management systems (RMS) are an important component in heterogeneous computing (HC) systems. One of the jobs of an RMS is the mapping of arriving tasks onto the machines of the HC system. Many different mapping heuristics have been proposed in recent years. However, most of these heuristics suffer from several limitations. One of these limitations is the performance degradation that results from using outdated global information about the status of all machines in the HC system. This paper proposes several heuristics which address this limitation by only requiring partial information in making the mapping decisions. These heuristics utilize the solution to a linear programming (LP) problem which maximizes the system capacity. Simulation results show that our heuristics perform very competitively while requiring dramatically less information.     

一种独立任务的同型机调度快速算法

如何将n个独立任务调度到m台同型机上加工,使总完成时间最短,是一个复杂问题.通过分析Bound Fit预备算法的性质,结合MULTIFIT和Bound Fit提出QUICKFIT算法;对相同机器数和任务数,QUICKFIT能用比MULTIFIT和Bound Fit都少的迭代次数得到相同的总完成时间.实验结果表明,任务机器比越大,QUICKFIT算法的性能就越优于MULTIFIT和Bound Fit.绝大多数情况下,总完成时间等于MULTIFIT和Bound Fit中的最小者.该算法适用于大规模同型机调度.     

A Note on Scheduling Tall/Small Multiprocessor Tasks with Unit Processing Time to Minimize Maximum Tardiness

We study the scheduling situation where n tasks, subjected to release dates and due dates, have to be scheduled on m parallel processors. We show that, when tasks have unit processing times and either require 1 or m processors simultaneously, the minimum maximal tardiness can be computed in polynomial time. Two algorithms are described. The first one is based on a linear programming formulation of the problem while the second one is a combinatorial algorithm. The complexity status of this tall/small task scheduling problem P|r _i ,p _i =1, size _i {1,m}|T _max was unknown before, even for two processors.