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 $\mathcal{P}=\mathcal{NP}$ .     

Some algorithms of solving minimax multiprocessor scheduling problem

A problem of constructing schedules of minimal length without interrupts and switches is considered for a multiprocessor system, in which the job execution time depends on the processor assigned to the job. To solve this problem, the branch and bound method is developed. The method is based on efficient algorithms for calculating lower and upper bounds of minimal length of the schedule. For the particular case when processors are identical, their number is fixed and the directive deadline is given, a pseudo-polynomial algorithm is proposed for constructing the admissible schedule. The number of processors required for efficient parallelizing of the algorithm is found.     

On the worst-case ratio of a compound multiprocessor scheduling algorithm

The basic problem of nonpreemptive scheduling of independent tasks on identical processors is studied. The well-known heuristics LPT and Multifit are combined to an algorithm Mix which has a better worst-case ratio than each of its components. Exact ratios for the case of two and three processors are given.     

Optimal and heuristic solution methods for a multiprocessor machine scheduling problem

This paper deals with the scheduling of operations on a multiprocessor machine in the context of shoe manufacturing. Multiprocessor machines are composed of several parallel processors. Unlike parallel machines, the entire machine needs to be stopped whenever a single processor needs a setup. Our industrial partner, one of the major winter-shoe manufacturers in Canada, offers a relatively large variety of models. For each of these models, all the sizes proposed by this manufacturer must be produced in various quantities. Our objective is to schedule the production of the required sizes on the machine's different processors in order to minimize the global makespan, which includes both the production time and the set up time. We first present a mathematical formulation of the problem. Then, we introduce a decomposition procedure based on the mathematical model, and we demonstrate that this procedure is very efficient for small- and medium-size instances. We also propose two construction heuristics and two improvement heuristics for larger problems, and we report the results of our extensive computational experiments to demonstrate the quality of the proposed heuristics in terms of reducing production time and computational effort.     

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.     

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.

On project scheduling problem

Consideration was given to the resource-constrained project scheduling problem and its special cases. The existing lower estimates of the objective function—minimization of the project time—were compared. It was hypothesized that the optimal value of the objective function of the nonpreemptive resource-constrained project scheduling problem is at most twice as great as that of the objective function with preemption. The hypothesis was proved for the cases of parallel machines and no precedence relation.     

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     

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     

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     

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.     

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。     

A new hybrid GA solution to combinatorial optimization problems— an application to the multiprocessor scheduling problem

The multiprocessor scheduling problem is one of the classic examples of NP-hard combinatorial optimization problems. Several polynomial time optimization algorithms have been proposed for approximating the multiprocessor scheduling problem. In this paper, we suggest a geneticizedknowledge genetic algorithm (gkGA) as an efficient heuristic approach for solving the multiprocessor scheduling and other combinatorial optimization problems. The basic idea behind the gkGA approach is that knowledge of the heuristics to be used in the GA is also geneticized alongiside the genetic chromosomes. We start by providing four conversion schemes based on heuristics for converting chromosomes into priority lists. Through experimental evaluation, we observe that the performance of our GA based on each of these schemes is instance-dependent. However, if we simultaneously incorporate these schemes into our GA through the gkGA approach, simulation results show that the approach is not problem-dependent, and that the approach outperforms that of the previous GA. We also show the effectiveness of the gkGA approach compared with other conventional schemes through experimental evaluation. This work was presented, in part, at the Second International Symposium on Artifiical Life and Robotics, Oita, Japan, February 18–20, 1997     

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.     

基于粒子群优化算法的异构多处理器任务调度

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,有效地降低任务的执行时间,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.