基于双曲线边界的多处理器实时任务可调度性判定

Lopez等学者求解出基于单调速率算法和首次适应分派策略的多处理器实时任务可调度性判定边界.该边界在所有O(m)复杂度的判定边界中是最优的.基于Bini等学者针对单处理器提出的双曲线可调度性判定方法,给出了一种多处理器实时任务可调度性判定边界.新边界在相当数量的利用率分布下明显优于已有边界.新边界与已有边界具有相容性,所以虽然新边界无法在所有情况下超越已有边界,但在实际应用中可联合两种边界进行判定,在不增加计算复杂度的同时全面提高可调度任务集的数量.     

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.     

Reduction-based schedulability analysis of distributed systems with cycles in the task graph

A significant problem with no simple solutions in current real-time literature is analyzing the end-to-end schedulability of tasks in distributed systems with cycles in the task graph. Prior approaches including network calculus and holistic schedulability analysis work best for acyclic task flows. They involve iterative solutions or offer no solutions at all when flows are non-acyclic. This paper demonstrates the construction of the first generalized closed-form expression for schedulability analysis in distributed task systems with non-acyclic flows. The approach is a significant extension to our previous work on schedulability in Directed Acyclic Graphs. Our main result is a bound on end-to-end delay for a task in a distributed system with non-acyclic task flows. The delay bound allows one of several schedulability tests to be performed. Using the end-to-end delay bound, we extend the delay composition algebra developed for acyclic distributed systems in prior work, to handle loops in the task graph as well. Evaluation shows that the schedulability tests thus constructed are less pessimistic than prior approaches for large distributed systems.     

Implementation and evaluation of global and partitioned scheduling in a real-time OS

In this work, we provide an experimental comparison between Global-EDF and Partitioned-EDF, considering the run-time overhead of a real-time operating system (RTOS). Recent works have confirmed that OS implementation aspects, such as the choice of scheduling data structures and interrupt handling mechanisms, impact real-time schedulability as much as scheduling theoretic aspects. However, these studies used real-time patches applied into a general-purpose OS. By measuring the run-time overhead of an RTOS designed from scratch, we show how close the schedulability ratio of task sets is to the theoretical hard real-time schedulability tests. Moreover, we show how a well-designed object-oriented RTOS allows code reuse of scheduling components (e.g., thread, scheduling criteria, and schedulers) and easy real-time scheduling extensions. We compare our RTOS to a real-time patch for Linux in terms of the task set schedulability ratio of several generated task sets. In some cases, Global-EDF considering the overhead of the RTOS is superior to Partitioned-EDF considering the overhead of the patched Linux, which clearly shows how different OSs impact hard real-time schedulers.     

Exact speedup factors and sub-optimality for non-preemptive scheduling

Fixed priority scheduling is used in many real-time systems; however, both preemptive and non-preemptive variants (FP-P and FP-NP) are known to be sub-optimal when compared to an optimal uniprocessor scheduling algorithm such as preemptive earliest deadline first (EDF-P). In this paper, we investigate the sub-optimality of fixed priority non-preemptive scheduling. Specifically, we derive the exact processor speed-up factor required to guarantee the feasibility under FP-NP (i.e. schedulability assuming an optimal priority assignment) of any task set that is feasible under EDF-P. As a consequence of this work, we also derive a lower bound on the sub-optimality of non-preemptive EDF (EDF-NP). As this lower bound matches a recently published upper bound for the same quantity, it closes the exact sub-optimality for EDF-NP. It is known that neither preemptive, nor non-preemptive fixed priority scheduling dominates the other, in other words, there are task sets that are feasible on a processor of unit speed under FP-P that are not feasible under FP-NP and vice-versa. Hence comparing these two algorithms, there are non-trivial speedup factors in both directions. We derive the exact speed-up factor required to guarantee the FP-NP feasibility of any FP-P feasible task set. Further, we derive the exact speed-up factor required to guarantee FP-P feasibility of any constrained-deadline FP-NP feasible task set.     

基于负载计算的多处理器全局EDF判定方法

在多处理器实时调度过程中,干涉上界的取值对于可调度性判定的性能具有较大影响。为此,针对实时系统的最早截止期优先调度算法,引入任务松弛的有关概念,提出一种基于负载计算的可调度性判定方法。通过减小问题区间内带入作业的工作负载取值,增加任务集通过可调度性判定的可能。实验结果表明,随着处理器数量的增加,该判定方法较传统方法有5%~10%的性能提升。     

多处理器EPDFPfair算法的可调度性判定

针对多处理器实时调度中的最早伪时限优先(EPDF)Pfair算法,分析了EPDF算法在M个处理器平台上的可调度利用率约束,根据基于利用率的充分可调度性判定,提出了一种改进的可调度性判定方法。这种方法可以得到更多的可调度任务集,从而使得满足判定的强实时系统和使用tie-breaking规则困难的动态任务系统的调度有较小的开销。实验结果表明,改进的可调度性判定方法增加了判为可调度的任务集数量,具有较好的性能。     

A multiframe model for real-time tasks

The well known periodic task model of C.L. Liu and J.W. Layland (1973) assumes a worst case execution time bound for every task and may be too pessimistic if the worst case execution time of a task is much longer than the average. We give a multiframe real time task model which allows the execution time of a task to vary from one instance to another by specifying the execution time of a task in terms of a sequence of numbers. We investigate the schedulability problem for this model for the preemptive fixed priority scheduling policy. We show that a significant improvement in the utilization bound can be established in our model     

Using an adversary simulator to evaluate global EDF scheduling of sporadic task sets on multiprocessors

Schedulability analysis of real-time multiprocessor systems is usually based on sufficient but not necessary tests that produce pessimistic results. One difficulty in evaluating the effectiveness of sufficient schedulability tests has been distinguishing the cause of a task set failing the test, i.e., finding out whether the task set is in fact not schedulable or it is actually schedulable but the test itself is too pessimistic. Necessary schedulability tests help to distinguish between these two situations, since if a task set fails in the test then it is guaranteed to be unschedulable. An adversary simulator is a scheduling simulator that uses the non-determinism of the task model to generate scenarios that will stress a specific scheduling algorithm, improving the odds of a deadline miss. In this paper we describe a new adversary simulator algorithm for sporadic task sets executed on multiprocessors scheduled by Global Earliest Deadline First (G-EDF). It is shown that this new adversary simulator is more effective as a necessary test than existing approaches. We also estimate the uncertainty regarding G-EDF by applying to the same task sets a well-known sufficient schedulability test from the literature and the necessary schedulability test based on the adversary simulator.     

Laxity dynamics and LLF schedulability analysis on multiprocessor platforms

LLF (Least Laxity First) scheduling, which assigns a higher priority to a task with a smaller laxity, has been known as an optimal preemptive scheduling algorithm on a single processor platform. However, little work has been made to illuminate its characteristics upon multiprocessor platforms. In this paper, we identify the dynamics of laxity from the system??s viewpoint and translate the dynamics into LLF multiprocessor schedulability analysis. More specifically, we first characterize laxity properties under LLF scheduling, focusing on laxity dynamics associated with a deadline miss. These laxity dynamics describe a lower bound, which leads to the deadline miss, on the number of tasks of certain laxity values at certain time instants. This lower bound is significant because it represents invariants for highly dynamic system parameters (laxity values). Since the laxity of a task is dependent of the amount of interference of higher-priority tasks, we can then derive a set of conditions to check whether a given task system can go into the laxity dynamics towards a deadline miss. This way, to the author??s best knowledge, we propose the first LLF multiprocessor schedulability test based on its own laxity properties. We also develop an improved schedulability test that exploits slack values. We mathematically prove that the proposed LLF tests dominate the state-of-the-art EDZL tests. We also present simulation results to evaluate schedulability performance of both the original and improved LLF tests in a quantitative manner.     

Schedulability Analysis of Global Scheduling Algorithms on Multiprocessor Platforms

This paper addresses the schedulability problem of periodic and sporadic real-time task sets with constrained deadlines preemptively scheduled on a multiprocessor platform composed by identical processors. We assume that a global work-conserving scheduler is used and migration from one processor to another is allowed during a task lifetime. First, a general method to derive schedulability conditions for multiprocessor real-time systems will be presented. The analysis will be applied to two typical scheduling algorithms: earliest deadline first (EDF) and fixed priority (FP). Then, the derived schedulability conditions will be tightened, refining the analysis with a simple and effective technique that significantly improves the percentage of accepted task sets. The effectiveness of the proposed test is shown through an extensive set of synthetic experiments.     

应用于实时系统的DMS算法可调度性分析

固定优先级任务的可调度性判定是实时系统调度理论研究的核心问题之一。本文提出了一种可行的DMS可调度性判定方法——确切性判定方法（precised schedulability test algorithln,简称PSTA）,利用DMS调度的充要条件,保证任何任务集均可被判定,并且判定结果是确切的。首先给出了DMS调度模型,介绍了可调度性判定的基本思想,然后进一步通过实验提出并证明了PSTA相关的定理。     

一种改进的RM可调度性判定算法

固定优先级任务可调度性判定是实时系统调度理论研究的核心问题之一.目前已有的各种判定方法可归结为两大类:多项式时间调度判定和确切性判定.多项式时间调度判定通常采用调度充分条件来进行,为此,许多理想条件下基于RM(rate monotonic)调度算法的CPU利用率最小上界被提了出来.确切性判定利用RM调度的充要条件,保证任何任务集均可被判定,并且判定结果是确切的.但是由于时间复杂度较差,确切性判定方法难以实现在线分析.提出了一种改进的RM可调度性判定方法(improved schedulability test algorithm,简称ISTA).首先介绍了任务调度空间这一概念,并提出了二叉树表示,然后进一步提出了相关的剪枝理论.在此基础上,研究了任务之间可调度性的相关性及其对判定任务集可调度性的影响,提出并证明了相关的定理.最后基于提出的定理,给出了一种改进的伪多项式时间可调度性判定算法,并与已有的判定方法进行了比较.仿真结果表明,该算法平均性能作为任务集内任务个数的函数具有显著提高.     

Minimum and maximum utilization bounds for multiprocessor rate monotonic scheduling

The utilization bound for real-time rate monotonic (RM) scheduling on uniprocessors is extended to multiprocessors with partitioning-based scheduling. This allows fast schedulability tests to be performed on multiprocessors and quantifies the influence of key parameters, such as the number of processors and task sizes on the schedulability of the system. The multiprocessor utilization bound is a function of the allocation algorithm, so among all the allocation algorithms there exists at least one allocation algorithm providing the minimum multiprocessor utilization bound, and one allocation algorithm providing the maximum multiprocessor utilization bound. We prove that the multiprocessor utilization bound associated with the allocation heuristic worst fit (WF) coincides with that minimum if we use Liu and Layland's bound (LLB) as the uniprocessor schedulability condition. In addition, we present a class of allocation algorithms sharing the same multiprocessor utilization bound which coincides with the aforementioned maximum using LLB. The heuristics first fit decreasing (FFD) and best fit decreasing (BFD) belong to this class. Thus, not even an optimal allocation algorithm can guarantee a higher multiprocessor utilization bound than that of FFD and BFD using LLB. Finally, the pessimism of the multiprocessor utilization bounds is estimated through extensive simulations.     

单调速率任务分配算法利用率的界限分析

通过对单调速率任务分配算法调度策略和可调度条件的分析，在多处理器周期任务抢占调度模型基础上，细致刻画了任务分配算法如何分配任务的行为。依据Liu和Layland定理，给出多处理器下任务分配算法的最小RM利用率界的定理。仿真结果表明，分配算法的利用率界是不同特征任务集选择不同分配算法进行任务划分的关键，通过对任务集总利用率与算法利用率界的比较,判断使用该算法对任务集是否可以产生可行分配。     

Generalized rate monotonic schedulability bounds using relative period ratios

One of the most well-studied scheduling algorithms for real-time systems is the Rate Monotonic (RM) scheduling for periodic tasks. In this paper we derive a generalized RM schedulability bound by considering relative period ratios among tasks in a system. We show that schedulability bounds published earlier are special cases of our generalized bound. Our new bound may provide a higher value than earlier results.     

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。     

Demand-based schedulability analysis for real-time multi-core scheduling

In real-time systems, schedulability analysis has been widely studied to provide offline guarantees on temporal correctness, producing many analysis methods. The demand-based schedulability analysis method has a great potential for high schedulability performance and broad applicability. However, such a potential is not yet fully realized for real-time multi-core scheduling mainly due to (i) the difficulty of calculating the resource demand under dynamic priority scheduling algorithms that are favorable to multi-cores, and (ii) the lack of understanding how to combine the analysis framework with deadline-miss conditions specialized for those scheduling algorithms. Addressing those two issues, to the best of our knowledge, this paper presents the first demand-based schedulability analysis for dynamic job-priority scheduling algorithms: EDZL (Earliest Deadline first until Zero-Laxity) and LLF (Least Laxity First), which are known to be effective for real-time multi-core scheduling. To this end, we first derive demand bound functions that compute the maximum possible amount of resource demand of jobs of each task while the priority of each job can change dynamically under EDZL and LLF. Then, we develop demand-based schedulability analyses for EDZL and LLF, by incorporating those new demand bound functions into the existing demand-based analysis framework. Finally, we combine the framework with additional deadline-miss conditions specialized for those two laxity-based dynamic job-priority scheduling algorithms, yielding tighter schedulability analyses. Via simulations, we demonstrate that the proposed schedulability analyses outperform the existing schedulability analyses for EDZL and LLF.     

基于单调速率的可调度性判定改进算法

在单调速率调度策略的基础上,提出一种改进的任务集可调度性判定算法。该算法通过设定时钟变量模拟调度过程中的系统时钟,在时钟变量值增长过程中,根据任务优先级从高到低的顺序,分析各个任务的截止时间限的满足情况,判定任务的可调度性,从而确定任务集的可调度性。通过实例分析及与现有判定方法的比较,验证了该算法的正确性和高效性。     

Optimal task execution times for periodic tasks using nonlinear constrained optimization

Designing real-time systems is a challenging task and many conflicting issues arise in the process. Among them, the most fundamental one is the adjustment of appropriate values for task parameters such as task periods, deadlines, and computation times that directly influence the system feasibility. Task periods and deadlines are generally known at design stage and remains fixed throughout, however, task computation times fluctuates significantly. For a better quality of service or higher system utilization, higher task computation values are required, while this flexibility comes at the price of system infeasibility. To the best of our knowledge, no optimal solution exists for extracting the optimal task computation times in a given range so that the overall system remains feasible under a specific scheduling algorithm. In this paper, we present a generalized bound on the task schedulability defined as a nonlinear inequality h _i ≤0 in the space of the execution times c _i . Based on this bound, the adjustment problem of tasks execution times, which determines the optimum c _i for a better system performance while still meeting all temporal requirements, is addressed by solving the standard nonlinear constrained optimization problem. Simulations on synthetic task sets are presented to compare the performance of our work with the most celebrated result, i.e., LL-bound by Liu and Layland in (J. ACM 20(1):40–61, 1973).