Conditions for On-line Scheduling of Hard Real-Time Tasks on Multiprocessors

In this paper we consider the problem ofon-linescheduling ofhard real-timetasks onmultipleprocessors. For a given set of ready tasks, one can propose many schedules. These schedules, however, may not necessarily be suitable for on-line scheduling. A suitable on-line schedule is one which can accommodate any future task set when it arrives. The traditional approach to solve the on-line scheduling problem is to propose a heuristic, and then to prove its effectiveness by comparing it with existing heuristics using simulation. No attempt has, however, been made to obtain a condition on the current schedule which when satisfied will permit one to schedule an arbitrary future task. In this paper, we aim at developing such a condition on the current schedule for the set of ready tasks which when satisfied can guarantee an on-line schedule for any futurefeasibletask set.     

Simple and integrated heuristic algorithms for scheduling tasks with time and resource constraints

We consider the problem of scheduling a set of n tasks in a system having r resources. Each task has an arbitrary, but known, processing time and a deadline, and may request use of a number of resources. A resource can be used either in shared mode or exclusive mode. In this article, we study algorithms used for determining whether or not a set of tasks is schedulable in such a system, and if so, determining a schedule for it. This scheduling problem is known to be NP-complete and hence we methodically study a set of heuristics that can be used by such an algorithm. Due to the complexity of the problem, simple heuristics do not perform satisfactorily. However, an algorithm that uses combinations of these simple heuristics works very well compared to an optimal algorithm that takes exponential time complexity. For the combination that performs the best, we also determine the scheduling costs as a function of the size of the task set scheduled.     

Dynamic Dispatching of Cyclic Real-Time Tasks with Relative Timing Constraints

In some hard real-time systems, relative timing constraints may be imposed on task executions, in addition to the release time and deadline constraints. Relative timing constraints such as separation or relative deadline constraints may be given between start or finish times of tasks (Gerber et al., 1995; Han and Lin, 1989; Han et al., 1992; Han and Lin, 1992; Han et al., 1996).One approach in real-time scheduling is to find a total order on a set of N tasks in a scheduling window, and cyclically use this order at run time to execute tasks. However, in the presence of relative timing constraints, if the task execution times are nondeterministic with defined lower and upper bounds, it is not always possible to statically assign task start times at pre-runtime for a given task ordering (Gerber et al., 1995).We develop a technique called dynamic cyclic dispatching as an extension of a parametric dispatching mechanism in (Gerber et al., 1995). An ordered set of N tasks is assumed to be given in a scheduling window and this schedule(ordering) is cyclically repeated at runtime in consecutive scheduling windows. Relative timing constraints between tasks may be defined across scheduling window boundaries as well as within one scheduling window. A task set is defined to be dispatchable if there exists any way in which the tasks can be dispatched with all their timing constraints satisfied. An off-line algorithm is presented to check the dispatchability of a task set and to obtain parametric lower and upper bound functions for task start times if the task set is dispatchable. These parametric bound functions are evaluated at runtime to obtain a valid time interval during which a task can be started. The complexity of this off-line component is shown to be O(n ² N ³) where n is the number of tasks in a scheduling window that have relative timing constraints with tasks in the next scheduling window. An online algorithm can evaluate these bounds in O(N) time.Unlike static approaches which assign fixed start times to tasks in the scheduling window, our approach allows us to flexibly manage the slack times at runtime without sacrificing the dispatchability of tasks. Also, a wider class of relative timing constraints can be imposed to the task set compared to the traditional approaches.     

Task graph pre-scheduling, using Nash equilibrium in game theory

Prescheduling algorithms are targeted at restructuring of task graphs for optimal scheduling. Task graph scheduling is a NP-complete problem. This article offers a prescheduling algorithm for tasks to be executed on the networks of homogeneous processors. The proposed algorithm merges tasks to minimize their earliest start time while reducing the overall completion time. To this end, considering each task as a player attempting to reduce its earliest time as much as possible, we have applied the idea of Nash equilibrium in game theory to determine the most appropriate merging. Also, considering each level of a task graph as a player, seeking for distinct parallel processors to execute each of its independent tasks in parallel with the others, the idea of Nash equilibrium in game theory can be applied to determine the appropriate number of processors in a way that the overall idle time of the processors is minimized and the throughput is maximized. The communication delay will be explicitly considered in the comparisons. Our experiments with a number of known benchmarks task graphs and also two well-known problems of linear algebra, LU decomposition and Gauss–Jordan elimination, demonstrate the distinguished scheduling results provided by applying our algorithm. In our study, we consider ten scheduling algorithms: min–min, chaining, A ^?, genetic algorithms, simulated annealing, tabu search, HLFET, ISH, DSH with task duplication, and our proposed algorithm (PSGT).     

Scheduling of hard real-time multi-phase multi-thread (MPMT) periodic tasks

In this paper we study the scheduling of parallel and real-time recurrent tasks on multiprocessor platforms. Firstly, we propose a new parallel task model which allows recurrent tasks to be composed of several phases, each one composed of several threads. Each thread requires a single processor for execution and can be scheduled simultaneously. We then propose an algorithm to transpose popular Fork-Join task model to our MPMT task model. Secondly, we define several kinds of real-time schedulers that can be applied to our parallel task model. We distinguish between two scheduling classes: Hierarchical schedulers and Global Thread schedulers. We present and prove correct an exact schedulability test for each class. Lastly, we also evaluate the performance of our scheduling paradigm in comparison with Gang scheduling by means of simulations. In this work we extend the work of Lupu and Goossens in Scheduling of hard real-time multi-thread periodic tasks (Real-Time and Network Systems, 2011) which considers mono-phase multi-thread task model. We extend their previous results to a Multi-Phase Multi-Thread task model.     

Utilization Bounds for N-Processor Rate Monotone Scheduling with Static Processor Assignment

We consider the schedulability of a set of independent periodic tasks under fixed priority preemptive scheduling on homogeneous multiprocessor systems. Assuming there is no task migration between processors and each processor schedules tasks preemptively according to fixed priorities assigned by the Rate Monotonic policy, the scheduling problem reduces to assigning the set of tasks to disjoint processors in such a way that the Monotonic policy, the scheduling problem reduces to assigning the set of tasks to disjoint processors in such a way that the schedulability of the tasks on each processor can be guaranteed. In this paper we show that the worst case achievable utilization for such systems is between n(2^1/2-1) and (n+1)/(1+2^1/(n+1)), where n stands for the number of processors. The lower bound represents 41 percent of the total system capacity and the upper bound represents 50 to 66 percent depending on n. Practicality of the lower bound is demonstrated by proving it can be achieved using a First Fit scheduling algorithm.     

Approximation algorithms for general parallel task scheduling

A general parallel task scheduling problem is considered. A task can be processed in parallel on one of several alternative subsets of processors. The processing time of the task depends on the subset of processors assigned to the task. We first show the hardness of approximating the problem for both preemptive and nonpreemptive cases in the general setting. Next we focus on linear array network of m processors. We give an approximation algorithm of ratio O(logm) for nonpreemptive scheduling, and another algorithm of ratio 2 for preemptive scheduling. Finally, we give a nonpreemptive scheduling algorithm of ratio O(log²m) for m×m two-dimensional meshes.     

Fault-tolerant and real-time scheduling for mixed-criticality systems

The design and analysis of real-time scheduling algorithms for safety-critical systems is a challenging problem due to the temporal dependencies among different design constraints. This paper considers scheduling sporadic tasks with three interrelated design constraints: (i) meeting the hard deadlines of application tasks, (ii) providing fault tolerance by executing backups, and (iii) respecting the criticality of each task to facilitate system’s certification. First, a new approach to model mixed-criticality systems from the perspective of fault tolerance is proposed. Second, a uniprocessor fixed-priority scheduling algorithm, called fault-tolerant mixed-criticality (FTMC) scheduling, is designed for the proposed model. The FTMC algorithm executes backups to recover from task errors caused by hardware or software faults. Third, a sufficient schedulability test is derived, when satisfied for a (mixed-criticality) task set, guarantees that all deadlines are met even if backups are executed to recover from errors. Finally, evaluations illustrate the effectiveness of the proposed test.     

A decomposition approach to non-preemptive real-time scheduling

Consider the problem of scheduling a set ofn tasks on a uniprocessor such that a feasible schedule that satisfies each task's time constraints is generated. Traditionally, researchers have looked at all the tasks as a group and applied heuristic or enumeration search to it. We propose a new approach called thedecomposition scheduling where tasks are decomposed into a sequence of subsets. The subsets are scheduled independently, in the order of the sequence. It is proved that a feasible schedule can be generated as long as one exists for the tasks. In addition, the overall scheduling cost is reduced to the sum of the scheduling costs of the tasks in each subset.Simulation experiments were conducted to analyze the performance of decomposition scheduling approach. The results show that in many cases decomposition scheduling performs better than the traditional branch-and-bound algorithms in terms of scheduling cost, and heuristic algorithms in terms of percentage of finding feasible schedules over randomly-generated task sets.     

A note on scheduling multiprocessor tasks with identical processing times

We study the scheduling situation where n tasks with identical processing times have to be scheduled on m parallel processors. Each task is subjected to a release date and requires simultaneously a fixed number of processors. We show that, for each fixed value of m, the problem of minimizing total completion time can be solved in polynomial time. The complexity status of the corresponding problem Pm|r_i,p_i=p,size_i|∑C_i was unknown before.Scope and purposeThere has been increasing interest in multiprocessor scheduling, i.e., in scheduling models where tasks require several processors (machines) simultaneously. Many scheduling problems fit in this model and a large amount of research has been carried on theoretical multiprocessor scheduling. In this paper we study the situation where tasks, subjected to release dates, have identical processing time and we introduce a dynamic programming algorithm that can compute the minimum total completion time. Although this scheduling problem has been open in the literature for several years, our algorithm is simple and easy to understand.     

Scheduling of a meta-task with QoS requirements in heterogeneous computing systems

Due to the development of new applications and the increasing number of users with diverse needs who are exposed to heterogeneous computing (HC), providing users with quality of service (QoS) guarantees while executing applications has become a crucial problem that needs to be addressed. Motivated by this fact, this paper investigates the problem of scheduling a set of independent tasks with multiple QoS needs, which may include timeliness, reliability, security, data accuracy, and priority, in a HC system. This problem is referred to as the QoS-based scheduling problem and proved to be NP-hard. In the first part of this study, we formulate the QoS-based scheduling problem by using utility and penalty functions, where a utility function associated with a task is used to measure how much the owner of this task will benefit from a given scheduling decision, while penalty functions associated with resources are used to provide incentives to users to set their QoS requirements in accordance with their needs. In order to solve the QoS-based scheduling problem, a computationally efficient static scheduling algorithm (QSMTS_IP) which assumes time-invariant penalty functions is developed. We later extend the QSMTS_IP to the case where penalty functions are time varying. Furthermore, it is shown that the QSMTS_IP can be modified to run as a dynamic scheduling algorithm. The simulation studies carried out show that the QSMTS_IP is capable of meeting diverse QoS requirements of many users simultaneously, while minimizing the number of users whose tasks cannot be scheduled due to the scarcity of machines.     

Online scheduling of moldable parallel tasks

In this paper, we study an online scheduling problem with moldable parallel tasks on m processors. Each moldable task can be processed simultaneously on any number of processors of a parallel computer, and the processing time of a moldable task depends on the number of processors allotted to it. Tasks arrive one by one. Upon arrival of each task, the scheduler has to determine both the number of processors and the starting time for the task. Moreover, these decisions cannot be changed in the future. The objective is to attain a schedule such that the longest completion time over all tasks, i.e., the makespan, is minimized. First, we provide a general framework to show that any \(\rho \)-bounded algorithm for scheduling of rigid parallel tasks (the number of processors for a task is fixed a prior) can be extended to yield an algorithm for scheduling of moldable tasks with a competitive ratio of \(4\rho \) if the ratio \(\rho \) is known beforehand. As a consequence, we achieve the first constant competitive ratio, 26.65, for the moldable parallel tasks scheduling problem. Next, we provide an improved algorithm with a competitive ratio of at most 16.74.     

An optimal boundary fair scheduling algorithm for multiprocessor real-time systems

With the emergence of multicore processors, the research on multiprocessor real-time scheduling has caught more researchers’ attention recently. Although the topic has been studied for decades, it is still an evolving research field with many open problems. In this work, focusing on periodic real-time tasks with quantum-based computation requirements and implicit deadlines, we propose a novel optimal scheduling algorithm, namely boundary fair (Bfair), which can achieve full system utilization as the well-known Pfair scheduling algorithms. However, different from Pfair algorithms that make scheduling decisions and enforce proportional progress (i.e., fairness) for all tasks at each and every time unit, Bfair makes scheduling decisions and enforces fairness to tasks only at tasks’ period boundaries (i.e., deadlines of periodic tasks). The correctness of the Bfair algorithm to meet the deadlines of all tasks’ instances is formally proved and its performance is evaluated through extensive simulations. The results show that, compared to that of Pfair algorithms, Bfair can significantly reduce the number of scheduling points (by up to 94%) and the overhead of Bfair at each scheduling point is comparable to that of the most efficient Pfair algorithm (i.e., PD²). Moreover, by aggregating the time allocation of tasks for the time interval between consecutive period boundaries, the resulting Bfair schedule can dramatically reduce the number of required context switches and task migrations (as much as 82% and 85%, respectively) when compared to those of Pfair schedules, which in turn reduces the run-time overhead of the system.     

Multi-core real-time scheduling for generalized parallel task models

Multi-core processors offer a significant performance increase over single-core processors. They have the potential to enable computation-intensive real-time applications with stringent timing constraints that cannot be met on traditional single-core processors. However, most results in traditional multiprocessor real-time scheduling are limited to sequential programming models and ignore intra-task parallelism. In this paper, we address the problem of scheduling periodic parallel tasks with implicit deadlines on multi-core processors. We first consider a synchronous task model where each task consists of segments, each segment having an arbitrary number of parallel threads that synchronize at the end of the segment. We propose a new task decomposition method that decomposes each parallel task into a set of sequential tasks. We prove that our task decomposition achieves a resource augmentation bound of 4 and 5 when the decomposed tasks are scheduled using global EDF and partitioned deadline monotonic scheduling, respectively. Finally, we extend our analysis to a directed acyclic graph (DAG) task model where each node in the DAG has a unit execution requirement. We show how these tasks can be converted into synchronous tasks such that the same decomposition can be applied and the same augmentation bounds hold. Simulations based on synthetic workload demonstrate that the derived resource augmentation bounds are safe and sufficient.     

Earliness–tardiness minimization on a single machine to schedule preventive maintenance tasks: metaheuristic and exact methods

In this paper, we consider the problem of scheduling a set of M preventive maintenance tasks to be performed on M machines. The machines are assigned to execute production tasks. We aim to minimize the total preventive maintenance cost such that the maintenance tasks have to continuously be run during the schedule horizon. Such a constraint holds when the maintenance resources are not sufficient. We solve the problem by two exact methods and meta-heuristic algorithms. As exact procedures we used linear programming and branch and bound methods. As meta-heuristics, we propose a local search approach as well as a genetic algorithm. Computational experiments are performed on randomly generated instances to show that the proposed methods produce appropriate solutions for the problem. The computational results show that the deviation of the meta-heuristics solutions to the optimal one is very small, which confirms the effectiveness of meta-heuristics as new approaches for solving hard scheduling problems.     

An adaptive hierarchical master–worker (AHMW) framework for grids—Application to B&B algorithms

Well-suited to embarrassingly parallel applications, the master–worker (MW) paradigm has largely and successfully used in parallel distributed computing. Nevertheless, such a paradigm is very limited in scalability in large computational grids. A natural way to improve the scalability is to add a layer of masters between the master and the workers making a hierarchical MW (HMW). In most existing HMW frameworks and algorithms, only a single layer of masters is used, the hierarchy is statically built and the granularity of tasks is fixed. Such frameworks and algorithms are not adapted to grids which are volatile, heterogeneous and large scale environments. In this paper, we revisit the HMW paradigm to match such characteristics of grids. We propose a new dynamic adaptive multi-layer hierarchical MW (AHMW  ) dealing with the scalability, volatility and heterogeneity issues. The construction and deployment of the hierarchy and the task management (deployment, decomposition of work, distribution of tasks, …$\dots$) are performed in a dynamic collaborative distributed way. The framework has been applied to the parallel Branch and Bound algorithm and experimented on the Flow-Shop scheduling problem. The implementation has been performed using the ProActive grid middleware and the large experiments have been conducted using about 2000 processors from the Grid’5000 French nation-wide grid infrastructure. The results demonstrate the high scalability of the proposed approach and its efficiency in terms of deployment cost, decomposition and distribution of work and exploration time. The results show that AHMW outperforms HMW and MW in scalability and efficiency in terms of deployment and exploration time.     

Optimal rate-based scheduling on multiprocessors

The PD² Pfair/ERfair scheduling algorithm is the most efficient known algorithm for optimally scheduling periodic tasks on multiprocessors. In this paper, we prove that PD² is also optimal for scheduling “rate-based” tasks whose processing steps may be highly jittered. The rate-based task model we consider generalizes the widely-studied sporadic task model.     

采用配置完成优先策略的可重构任务调度算法

如何隐藏和减少配置时间是相依性可重构任务调度的关键问题.提出一种采用配置完成优先策略的相依性可重构任务调度算法,通过基于预配置优先级的列表调度算法,实现将后续任务的配置时间隐藏于前驱任务的运行时间中,并采用基于配置完成优先策略的配置重用机制,减少了任务调度后的配置过程,从而在总体上缩短了相依性任务集合的运行时间.仿真结果表明,该调度算法能有效避免调度死锁,并可减少相依性可重构任务的整体运行时间.     

Converging to periodic schedules for cyclic scheduling problems with resources and deadlines

Cyclic scheduling has been widely studied because of the importance of applications in manufacturing systems and in computer science. For this class of problems, a finite set of tasks with precedence relations and resource constraints must be executed repetitively while maximizing the throughput. Many applications also require that execution schedules be periodic i.e. the execution of each task is repeated with a fixed global period w.The present paper develops a new method to build periodic schedules with cumulative resource constraints, periodic release dates and deadlines. The main idea is to fix the period w, to unwind the cyclic scheduling problem for some number of iterations, and to add precedence relations so that the minimum time lag between two successive executions of any task equals w. Then, using any usual (not cyclic) scheduling algorithm to compute task starting times for the unwound problem, we prove that either the method converges to a periodic schedule of period w or it fails to compute a schedule. A non-polynomial upper bound on the number of iterations to unwind in order to guarantee that cyclic precedence relations and resource constraints are fulfilled is also provided. This method is successfully applied to a real-life problem, namely the software pipelining of inner loops on an embedded VLIW processor core by using a Graham list scheduling algorithm.     

Group-Based Pfair Scheduling

We consider the problem of supertasking in Pfair-scheduled multiprocessor systems. In this approach, a set of tasks, called component tasks, is assigned to a server task, called a supertask, which is then scheduled as an ordinary Pfair task. Whenever a supertask is scheduled, its processor time is allocated to its component tasks according to an internal scheduling algorithm. Hence, supertasking is an example of hierarchal (or group-based) scheduling. In this paper, we present a generalized framework for “reweighting” supertasks. The goal of reweighting is to assign a fraction of a processor to a given supertask so that all timing requirements of its component tasks are met. We consider the use of both fully preemptive and quantum-based scheduling within a supertask. Work supported by NSF grants CCR 9732916, CCR 9972211, CCR 9988327, ITR 0082866, CCR 0204312, and CCR 0309825. Preliminary versions of some content appeared previously in (Holman and Anderson, 2001, 2003).