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.     

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.     

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.     

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。     

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.     

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.     

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.     

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.     

Barely Random Algorithms for Multiprocessor Scheduling

We consider randomized algorithms for on-line scheduling on identical machines. For two machines, a randomized algorithm achieving a competitive ratio of was found by Bartal et al. (1995). Seiden has presented a randomized algorithm which achieves competitive ratios of 1.55665, 1.65888, 1.73376, 1.78295, and 1.81681, for m=3,4,5,6,7, respectively (Seiden, 2000). A barely random algorithm is one which is a distribution over a constant number of deterministic strategies. The algorithms of Bartal et al. and Seiden are not barely random–in fact, these algorithms potentially make a random choice for each job scheduled. We present the first barely random on-line scheduling algorithms. In addition, our algorithms use less space and time than the previous algorithms, asymptotically.     

基于图匹配的多处理机调度算法

本文对具有高通讯延迟的多处理机系统（机群系统）上的任务调度算法进行了研究，与以往算法主要考虑任务图的关键路径不同，本文给出了任务图的调度与其偶图匹配的对应关系，并由此提出了一种新的启发式算法，通过模拟试验显示本算法具有较好的调度效果。     

Utilization Bounds for Multiprocessor Rate-Monotonic Scheduling

In this paper, we extend Liu and Layland's utilization bound for fixed priority scheduling on uniprocessors to homogeneous multiprocessor systems under a partitioning strategy. Assuming that tasks are pre-emptively scheduled on each processor according to fixed priorities assigned by the Rate-Monotonic policy, and allocated to processors by the First Fit algorithm, we prove that the utilization bound is (n–1)(2^1/2–1)+(m–n+1)(2^1/(m–n+1)–1), where m and n are the number of tasks and processors, respectively. This bound is valid for arbitrary utilization factors. Moreover, if all the tasks have utilization factors under a value , the previous bound is raised and the new utilization bound considering is calculated. Finally, simulation provides the average-case behavior.     

Approximation Algorithms for Multiprocessor Scheduling under Uncertainty

Motivated by applications in grid computing and project management, we study multiprocessor scheduling in scenarios where there is uncertainty in the successful execution of jobs when assigned to processors. We consider the problem of multiprocessor scheduling under uncertainty, in which we are given n unit-time jobs and m machines, a directed acyclic graph C giving the dependencies among the jobs, and for every job j and machine i, the probability p _ij of the successful completion of job j when scheduled on machine i in any given particular step. The goal of the problem is to find a schedule that minimizes the expected makespan, that is, the expected time at which all of the jobs are completed.     

非同起点加工的多机调度合成算法

针对调度h个独立任务到初始时刻并非都空闲的m台机器上加工,使得机器最长加工时间（makespan）最短的问题,改进MLPT算法以减少运行时间,改进MULTIFIT算法以减少迭代次数,提出以改进的MLPT算法结果为改进的MULTIFIT算法的初始上界的合成算法－CMM,从理论上对MLPT,MULTIFIT和CMM算法的时间复杂度和调度结果进行了分析和比较,实验结果表明：改进的MULTIFIT经MULTIFIT的平均迭代次数少;CMM在平均迭代次数方面甚至比改进的MULTIFIT还少得多且调度结果不次于MULTIFIT和MLPT的优者。     

一种粒子群优化的异构多处理器任务调度算法

为提高异构多处理器任务调度的执行效率,充分发挥多处理器并行性能,提出一种基于粒子群优化的异构多处理器任务调度算法-PSOASA算法.PSOASA算法以求得任务最短完成时间为目标,首先采用整数矩阵对粒子进行编码,并定义交换操作更新粒子状态,实现粒子搜索空间到离散空间的映射,使连续的粒子群优化算法适用于离散的异构多处理器任务调度问题,同时引入模拟退火算法,克服粒子群算法的“早熟”收敛现象,避免求得的解陷入局部最优.实验结果表明,PSOASA算法的执行效率优于目前广泛采用的遗传算法,有效地降低任务执行时间,减少了迭代次数,适用于异构多处理器环境大规模任务调度.     

Scheduling in Multiprocessor Systems with Additional Restrictions

An admissible multiprocessor preemptive scheduling problem is solved for the given execution intervals. In addition, a number of generalizations are considered—interprocessor communications are arbitrary and may vary in time; costs for processing interruptions and switches from one processor to another are taken into account; and besides the processors, additional resources are used. Algorithms based on reducing the original problem to finding paths of a specific length in a graph, a flow problem, and an integer system of linear restrictions are developed.     

多处理机系统的一种实时容错调度算法

实时容错调度算法是实时系统容错研究中的重要问题.提出了一种多处理机系统的实时任务容错调度算法.该算法结合了基版本/副版本(Primary/Backup)方法和三模冗余(TMR)方法,以保证在一个节点机失效时,任务可以在其截止时限内完成.