On parallelizing the multiprocessor scheduling problem

Existing heuristics for scheduling a node and edge weighted directed task graph to multiple processors can produce satisfactory solutions but incur high time complexities, which tend to exacerbate in more realistic environments with relaxed assumptions. Consequently, these heuristics do not scale well and cannot handle problems of moderate sizes. A natural approach to reducing complexity, while aiming for a similar or potentially better solution, is to parallelize the scheduling algorithm. This can be done by partitioning the task graphs and concurrently generating partial schedules for the partitioned parts, which are then concatenated to obtain the final schedule. The problem, however, is nontrivial as there exists dependencies among the nodes of a task graph which must be preserved for generating a valid schedule. Moreover, the time clock for scheduling is global for all the processors (that are executing the parallel scheduling algorithm), making the inherent parallelism invisible. In this paper, we introduce a parallel algorithm that is guided by a systematic partitioning of the task graph to perform scheduling using multiple processors. The algorithm schedules both the tasks and messages, and is suitable for graphs with arbitrary computation and communication costs, and is applicable to systems with arbitrary network topologies using homogeneous or heterogeneous processors. We have implemented the algorithm on the Intel Paragon and compared it with three closely related algorithms. The experimental results indicate that our algorithm yields higher quality solutions while using an order of magnitude smaller scheduling times. The algorithm also exhibits an interesting trade-off between the solution quality and speedup while scaling well with the problem size     

Declustering: a new multiprocessor scheduling technique

The authors present a new compile-time scheduling heuristic called declustering, which schedules acyclic precedence graphs that fit the synchronous data flow (SDF) model onto multiprocessor architectures. This technique accounts for interprocessor communication (IPC) overheads and considers interconnection constraints in the architecture so that shared resource contention can be avoided. The algorithm initially invokes a new clustering method that uses graph-analysis techniques to isolate parallelism instances. When constructing an initial set of clusters, this procedure explicitly addresses the tradeoff between exploiting parallelism and incurring communication cost. By hierarchically combining these clusters and then systematically decomposing this hierarchy, the declustering method exposes parallelism instances in order of importance and attains a cluster granularity that fits the characteristics of the architecture. It is shown that declustering retains the clustering advantage of avoiding IPC, yet overcomes the inflexibility associated with traditional clustering approaches     

Multi-objective fuzzy multiprocessor flowshop scheduling

This paper considers a bi-objective hybrid flowshop scheduling problems with fuzzy tasks’ operation times, due dates and sequence-dependent setup times. To solve this problem, we propose a bi-level algorithm to minimize two criteria, namely makespan, and sum of the earliness and tardiness, simultaneously. In the first level, the population will be decomposed into several sub-populations in parallel and each sub-population is designed for a scalar bi-objective. In the second level, non-dominant solutions obtained from sub-population bi-objective random key genetic algorithm (SBG) in the first level will be unified as one big population. In the second level, for improving the Pareto-front obtained by SBG, based on the search in Pareto space concept, a particle swarm optimization (PSO) is proposed. We use a defuzzification function to rank the Bell-shaped fuzzy numbers. The non-dominated sets obtained from each of levels and an algorithm presented previously in literature are compared. The computational results showed that PSO performs better than others and obtained superior results.     

On multiprocessor temperature-aware scheduling problems

We study temperature-aware scheduling problems under the model introduced in [Chrobak et al. AAIM 2008], where unit-length jobs of given heat contributions and common release dates are to be scheduled on a set of parallel identical processors. We consider three optimization criteria: makespan, maximum temperature and (weighted) average temperature. On the positive side, we present polynomial time approximation algorithms for the minimization of the makespan and the maximum temperature, as well as, optimal polynomial time algorithms for minimizing the average temperature and the weighted average temperature. On the negative side, we prove that there is no approximation algorithm of absolute ratio $\frac{4}{3}-\epsilon $ for the problem of minimizing the makespan for any $\epsilon > 0$ , unless $\mathcal{P}=\mathcal{NP}$ .     

Optimal virtual cluster-based multiprocessor scheduling

Scheduling of constrained deadline sporadic task systems on multiprocessor platforms is an area which has received much attention in the recent past. It is widely believed that finding an optimal scheduler is hard, and therefore most studies have focused on developing algorithms with good processor utilization bounds. These algorithms can be broadly classified into two categories: partitioned scheduling in which tasks are statically assigned to individual processors, and global scheduling in which each task is allowed to execute on any processor in the platform. In this paper we consider a third, more general, approach called cluster-based scheduling. In this approach each task is statically assigned to a processor cluster, tasks in each cluster are globally scheduled among themselves, and clusters in turn are scheduled on the multiprocessor platform. We develop techniques to support such cluster-based scheduling algorithms, and also consider properties that minimize total processor utilization of individual clusters. In the last part of this paper, we develop new virtual cluster-based scheduling algorithms. For implicit deadline sporadic task systems, we develop an optimal scheduling algorithm that is neither Pfair nor ERfair. We also show that the processor utilization bound of us-edf{m/(2m−1)} can be improved by using virtual clustering. Since neither partitioned nor global strategies dominate over the other, cluster-based scheduling is a natural direction for research towards achieving improved processor utilization bounds.

A genetic algorithm for multiprocessor scheduling

The problem of multiprocessor scheduling can be stated as finding a schedule for a general task graph to be executed on a multiprocessor system so that the schedule length can be minimized. This scheduling problem is known to be NP-hard, and methods based on heuristic search have been proposed to obtain optimal and suboptimal solutions. Genetic algorithms have recently received much attention as a class of robust stochastic search algorithms for various optimization problems. In this paper, an efficient method based on genetic algorithms is developed to solve the multiprocessor scheduling problem. The representation of the search node is based on the order of the tasks being executed in each individual processor. The genetic operator proposed is based on the precedence relations between the tasks in the task graph. Simulation results comparing the proposed genetic algorithm, the list scheduling algorithm, and the optimal schedule using random task graphs, and a robot inverse dynamics computational task graph are presented     

Slack-based multiprocessor scheduling of aperiodic real-time tasks

We provide a constant time schedulability test and priority assignment algorithm for an on-line multiprocessor server handling aperiodic tasks. The so called Dhall’s effect is avoided by dividing tasks in two priority classes based on their utilization: heavy and light. The improvement in this paper is due to assigning priority of light tasks based on slack—not on deadlines. We prove that if the load on the multiprocessor stays below \((3 - \sqrt{5} )/2 \approx 38.197\%\), the server can accept an incoming aperiodic task and guarantee that the deadlines of all accepted tasks will be met. This is better than the current state-of-the-art algorithm where the priorities of light tasks are based on deadlines (the corresponding bound is in that case 35.425%).The bound \((3 - \sqrt{5} )/2\) can be improved if the number of processors m is known. There is a formula for the sharp bound \(U_{\mathit{threshold}}(m) = \frac{3m - 2 - \sqrt{5m^{2} - 8m + 4}}{2(m - 1)}\), which converges to \((3 - \sqrt{5} )/2\) from above as m→∞. For m≥3, the bound is higher (i.e., better) than the corresponding sharp bound for the state-of-the-art algorithm where the priorities of light tasks are based on deadlines.A simulation study also indicates that when m>3 the best effort behavior of the priority assignment scheme suggested here is better than that of the traditional scheme where priorities are based on deadlines.     

Generalized multiprocessor scheduling and applications to matrixcomputations

The paper considerably extends the multiprocessor scheduling techniques of G.N.S. Prasanna and B.R. Musicus (1995; 1991) and applies it to matrix arithmetic compilation. Using optimal control theory in the special case where the speedup function of each task is p^α (where p is the amount of processing power applied to the task), closed form solution for task graphs formed from parallel and series connections was derived by G.N.S. Prasanna and B.R. Musicus (1995; 1991). The paper extends these results for arbitrary DAGS. The optimality conditions impose nonlinear constraints on the flow of processing power from predecessors to successors, and on the finishing times of siblings. The paper presents a fast algorithm for determining and solving these nonlinear equations. The algorithm utilizes the structure of the finishing time equations to efficiently run a conjugate gradient minimization, leading to the optimal solution. The algorithm has been tested on a variety of DAGs commonly encountered in matrix arithmetic. The results show that if the p^α speedup assumption holds, the schedules produced are superior to heuristic approaches. The algorithm has been applied to compiling matrix arithmetic (K.P. Belkhale and P. Banerjee, 1993), for the MIT Alewife machine, a distributed shared memory multiprocessor. While matrix arithmetic tasks do not exactly satisfy the p^α speedup assumptions, the algorithm can be applied as a good heuristic. The results show that the schedules produced by our algorithm are faster than alternative heuristic techniques     

多处理器固定优先级算法的可调度性分析

针对多处理器实时调度中的固定优先级(FP)调度算法,提出了一种改进的可调度性判定方法。引入Baruah的最早截止期优先(EDF)窗口分析框架,将高优先级任务带入作业的最大数量限定为m-1(m为处理器个数),进而对任务的干涉上界进行重新界定,并由此得到一个更加紧密的可调度性判定充分条件。仿真实验结果表明,该方法增加了通过判定任务集的数量,体现出更优的可调度判定性能。     

粒子群优化多处理机任务调度算法

提出基于粒子群优化的多处理机调度算法,采用列表调度,同时把粒子群的矢量表达方式转换为基于调度优先级的模型。调度结果显示能提高全局搜索能力,加快进化速度,优于模拟退火等启发式算法结果。     

Generalized tardiness bounds for global multiprocessor scheduling

We consider the issue of deadline tardiness under global multiprocessor scheduling algorithms. We present a general tardiness-bound derivation that is applicable to a wide variety of such algorithms (including some whose tardiness behavior has not been analyzed before). Our derivation is very general: job priorities may change rather arbitrarily at runtime, capacity restrictions may exist on certain processors, and, under certain conditions, non-preemptive regions are allowed. Our results show that, with the exception of static-priority algorithms, most global algorithms considered previously have bounded tardiness. In addition, our results provide a simple means for checking whether tardiness is bounded under newly-developed algorithms.     

多处理器系统的在线节能调度算法

随着多处理器系统计算性能的提高,能耗管理已变得越来越重要,如何满足实时约束并有效降低能耗成为实时调度中的一个重要问题。基于多处理器计算系统,针对随机到达的任务,提出一种在线节能调度算法(OLEAS)。该算法在满足任务截止期限的前提下,尽量将任务调度到产生能耗最少的处理器,当某个任务在所有处理器上都不能满足截止期限要求时,则调整处理器之间的部分任务,使之尽量满足截止期限要求。同时,OLEAS尽量使单个处理器上的任务按平均电压/频率执行,以降低能耗,只有当新到任务不满足截止期限要求时,才逐个调高前面任务的电压/频率。模拟实验比较了OLEAS、最早完成时间优先(EFF)、最高电压节能(HVEA)、最低电压节能(LVEA)、贪心最小能耗(MEG)和最小能耗最小完成时间(ME-MC)的性能,结果表明OLEAS在满足任务截止期限和节省能耗方面具有明显的综合优势     

基于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——FPSOTTS算法。该算法以求得任务最短完成时间为目标,首先通过建立新的编码方式和粒子更新公式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;同时通过引入禁忌算法进行局部搜索,克服粒子群算法的早熟收敛现象,避免陷入局部最优。实验结果表明,FPSOTTS算法的执行效率优于Min-min算法和遗传算法,有效地降低任务的执行时间。FP-SOTTS算法很好地解决了异构多处理器任务调度问题,并且适合于大规模并行任务调度。     

An incremental genetic algorithm approach to multiprocessor scheduling

We have developed a genetic algorithm (GA) approach to the problem of task scheduling for multiprocessor systems. Our approach requires minimal problem specific information and no problem specific operators or repair mechanisms. Key features of our system include a flexible, adaptive problem representation and an incremental fitness function. Comparison with traditional scheduling methods indicates that the GA is competitive in terms of solution quality if it has sufficient resources to perform its search. Studies in a nonstationary environment show the GA is able to automatically adapt to changing targets.     

求解多处理机调度问题的蚁群算法

蚁群算法是受自然界中的蚂蚁觅食行为启发而设计的智能优化算法,特别适合处理离散型的组合优化问题。提出一种求解多处理机调度的蚁群算法,利用一个蚂蚁代表一个处理机来选择任务,并通过分析关键路径及每个任务的最早、最迟开始时间来确定每个任务的紧迫程度,让蚂蚁以此来选择任务。实验证明,该算法可比传统算法取得有更好运行效率的调度策略。     

Multi-objective energy aware multiprocessor scheduling using bat intelligence

In this paper, a new heuristic called bat intelligence (BI) is introduced for solving energy aware multiprocessor scheduling problems. Bat intelligence is a novel optimization method that models prey hunting behaviors of bats. Bat intelligence and genetic algorithm (GA) are used to solve single-objective multiprocessor scheduling problem using, makespan, tardiness, and energy consumption as objective functions. Bat intelligence shows considerable improvement in terms of solution quality when compared with GA. Different combinations of these objectives are used to solve bi-objective multiprocessor scheduling problems, (makespan vs. energy, and also tardiness vs. energy). Tri-objective multiprocessor scheduling problem is also presented at the end. To generate desirable efficient alternatives, a Normalized Weighted Additive Utility Function is used. Simulation shows that BI identifies a set of efficient solutions that correspond to the assigned weights. The computational simulation also shows conflicting relationships between makespan and energy, and also between tardiness and energy.