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.     

Optimal preemptive semi-online scheduling on two uniform processors

We investigate a preemptive semi-online scheduling problem. Jobs with sizes within a certain range [1,r] (r?1) arrive one by one to be scheduled on two uniform parallel processors with speed 1 and s?1, respectively. The objective is to minimize makespan. We characterize the optimal competitive ratio as a function of both s and r by devising a deterministic on-line scheduling algorithm along with a matching lower bound, which also holds for randomized algorithms.     

Scheduling parallel jobs to minimize the makespan

We consider the NP-hard problem of scheduling parallel jobs with release dates on identical parallel machines to minimize the makespan. A parallel job requires simultaneously a prespecified, job-dependent number of machines when being processed. We prove that the makespan of any nonpreemptive list-schedule is within a factor of 2 of the optimal preemptive makespan. This gives the best-known approximation algorithms for both the preemptive and the nonpreemptive variant of the problem. We also show that no list-scheduling algorithm can achieve a better performance guarantee than 2 for the nonpreemptive problem, no matter which priority list is chosen. List-scheduling also works in the online setting where jobs arrive over time and the length of a job becomes known only when it completes; it therefore yields a deterministic online algorithm with competitive ratio 2 as well. In addition, we consider a different online model in which jobs arrive one by one and need to be scheduled before the next job becomes known. We show that no list-scheduling algorithm has a constant competitive ratio. Still, we present the first online algorithm for scheduling parallel jobs with a constant competitive ratio in this context. We also prove a new information-theoretic lower bound of 2.25 for the competitive ratio of any deterministic online algorithm for this model. Moreover, we show that 6/5 is a lower bound for the competitive ratio of any deterministic online algorithm of the preemptive version of the model jobs arriving over time.     

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.     

Minimizing Makespan and Preemption Costs on a System of Uniform Machines

It is well known that for preemptive scheduling on uniform machines there exist polynomial time exact algorithms, whereas for non-preemptive scheduling there are probably no such algorithms. However, it is not clear how many preemptions (in total, or per job) suffice in order to guarantee an optimal polynomial time algorithm. In this paper we investigate exactly this hardness gap, formalized as two variants of the classic preemptive scheduling problem. In generalized multiprocessor scheduling (GMS) we have a job-wise or total bound on the number of preemptions throughout a feasible schedule. We need to find a schedule that satisfies the preemption constraints, such that the maximum job completion time is minimized. In minimum preemptions scheduling (MPS) the only feasible schedules are preemptive schedules with the smallest possible makespan. The goal is to find a feasible schedule that minimizes the overall number of preemptions. Both problems are NP-hard, even for two machines and zero preemptions. For GMS, we develop polynomial time approximation schemes, distinguishing between the cases where the number of machines is fixed, or given as part of the input. Our scheme for a fixed number of machines has linear running time, and can be applied also for instances where jobs have release dates, and for instances with arbitrary preemption costs. For MPS, we derive matching lower and upper bounds on the number of preemptions required by any optimal schedule. Our results for MPS hold for any instance in which a job, J_j, can be processed simultaneously by ρ_j machines, for some ρ_j ≥ 1.     

Resource Constraints for Preemptive Job-shop Scheduling

This paper presents an experimental study of constraint propagation algorithms for preemptive scheduling. We propose generalizations of non-preemptive constraint propagation techniques (based on timetables, on disjunctive constraints, and on edge-finding) to preemptive and mixed problems, i.e., problems in which some activities can be interrupted and some cannot. Another approach relies on incremental flow-based techniques. We theoretically compare these approaches and present an experimental comparison based on a branch and bound procedure for the preemptive variant of the job-shop scheduling problem. We show that both edge-finding and flow-based techniques allow the resolution of hard problem instances, including the preemptive variant of the famous FT10.     

Preemptive and Non-preemptive On-line Algorithms for Scheduling with Rejection on Two Uniform Machines

In this paper, we consider the problem of on-line scheduling a job sequence on two uniform machines. A job can be either rejected, in which case we pay its penalty, or scheduled on machines, in which case it contributes its processing time to the makspan of the constructed schedule. The objective is to minimize the sum of the makespan of the schedule for all accepted jobs and the penalties of all rejected jobs. Both preemptive and non-preemptive versions are considered. For the preemptive version, we present an optimal on-line algorithm with a competitive ratio for any s≥1, where s is the machine speed ratio. For the non-preemptive version, we present an improved lower bound. Moreover, as an optimal algorithm for s≥1.6180 is known, we present a modified version of the known algorithm, and show that it becomes optimal for any 1.3852≤s<1.6180 and has a smaller competitive ratio than that of original version for any 1≤s<1.3852. The maximum gap between its competitive ratio and the lower bound is 0.0534.     

A new heuristic for scheduling the two-stage flowshop with additional resources

This paper deals with the problem of preemptive scheduling in a two-stage flowshop with parallel unrelated machines at the first stage and a single machine at the second stage. At the first stage, jobs use some additional resources which are available in limited quantities at any time. The resource requirements are of 0–1 type. The objective is the minimization of makespan. The problem is NP-hard. Heuristic algorithms are proposed which solve to optimality the resource constrained scheduling problem at the first stage of the flowshop, and at the same time, minimize the makespan in the flowshop by selecting appropriate jobs for simultaneous processing. Several rules of job selection are considered. The performance of the proposed heuristic algorithms is analyzed by comparing solutions with the lower bound on the optimal makespan. The extensive computational experiment shows that the proposed heuristic algorithms are able to produce near-optimal solutions in short computational time.     

Timing Analysis of Compound Scheduling Policies: Application to Posix1003.1B

The analysis of fixed priority preemptive scheduling has been extended in various ways to improve its usefulness for the design of real-time systems. In this paper, we define the layered preemptive priority scheduling policy which generalizes fixed preemptive priorities by combination with other policies in a layered structure. In particular, the combination with the Round Robin scheduling policy is studied. Its compliance with Posix 1003.1b requirements is shown and its timing analysis is provided. For this purpose and as a basis for the analysis of other policies, the concept of majorizing work arrival function, is introduced to synthesize essential ideas used in existing analysis of the fixed preemptive priority policy.If critical resources are protected by semaphores, the Priority Ceiling Protocol (PCP) can be used under fixed preemptive priorities to control resulting priority inversions. An extension of the PCP is proposed for Round Robin, to allow a global control of priority inversions under the layered priority policy and to prevent deadlocks. The initial timing analysis is extended to account for the effects of the protocol. The results are illustrated by a small test case.     

一种基于阈值的嵌入式实时系统调度算法

基于阈值的双优先级调度算法结合了抢占式与非抢占式调度算法的优点,可以提高任务集的调度成功率,并减少由于任务切换引起的系统开销。对阈值的分配是调度算法的核心。在基本优先级已知的条件下,基于回溯技术的阈值分配算法利用低端任务阈值单向影响高端任务最大响应时间的特性,可以在有限的时间内为任务集找出一组具有极大值特征的阈值。该组阈值可以将任务切换次数降至最低。     

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.     

Normal-form preemption sequences for an open problem in scheduling theory

Structural properties of optimal preemptive schedules have been studied in a number of recent papers with a primary focus on two structural parameters: the minimum number of preemptions necessary, and a tight lower bound on shifts, i.e., the sizes of intervals bounded by the times created by preemptions, job starts, or completions. These two parameters have been investigated for a large class of preemptive scheduling problems, but so far only rough bounds for these parameters have been derived for specific problems. This paper sharpens the bounds on these structural parameters for a well-known open problem in the theory of preemptive scheduling: Instances consist of in-trees of n unit-execution-time jobs with release dates, and the objective is to minimize the total completion time on two processors. This is among the current, tantalizing “threshold” problems of scheduling theory: Our literature survey reveals that any significant generalization leads to an NP-hard problem, but that any significant, but slight simplification leads to tractable problem with a polynomial-time solution. For the above problem, we show that the number of preemptions necessary for optimality need not exceed \(2n-1\); that the number must be of order \({\varOmega }(\log n)\) for some instances; and that the minimum shift need not be less than \(2^{-2n+1}.\) These bounds are obtained by combinatorial analysis of optimal preemptive schedules rather than by the analysis of polytope corners for linear-program formulations of the problem, an approach to be found in earlier papers. The bounds immediately follow from a fundamental structural property called normality, by which minimal shifts of a job are exponentially decreasing functions. In particular, the first interval between a preempted job’s start and its preemption must be a multiple of 1 / 2, the second such interval must be a multiple of 1 / 4, and in general, the i-th preemption must occur at a multiple of \(2^{-i}\). We expect the new structural properties to play a prominent role in finally settling a vexing, still-open question of complexity.     

Worst-case performance of critical path type algorithms

The critical path method remains one of the most popular approaches in practical scheduling. Being developed for the makespan problem this method can also be generalized to the maximum lateness problem. For the unit execution time task system and parallel processors this generalization is known as the Brucker–Garey–Johnson algorithm. We characterize this algorithm by introducing an upper bound on the deviation of the criterion from its optimal value. The bound is stated in terms of parameters characterizing the problem, namely number of processors, the length of the longest path, and the total required processing time. We also derive a similar bound for the preemptive version of the Brucker– Garey–Johnson algorithm.     

信息物理系统中基于保护阈值的实时调度算法研究

信息物理融合系统CPS是一种融合计算、通信与控制的新型复杂实时分布式系统,系统中计算过程和物理过程在开放环境下持续交互、深度融合。为了对物理世界的信息作出实时反馈,系统一般会采用抢占式调度的方法,保障关键任务能够在截止期前完成。但是,分布式环境中抢占式调度方式容易导致频繁的任务切换,影响系统的实时性。提出了基于保护阈值的调度算法,通过建立保护阈值模型,最大化低优先级任务的执行时间,减少任务切换次数。通过实验验证,算法有效地减少了任务切换次数,提高了CPS系统的实时性能。     

Preemptive scheduling on two parallel machines with a single server

This paper addresses a preemptive scheduling problem on two parallel machines with a single server. Each job has to be loaded (setup) by the server before being processed on the machines. The preemption is allowed in this paper. The goal is to minimize the makespan. We first show that it is no of use to preempt the job during its setup time. Namely, every optimal preemptive schedule can be converted to another optimal schedule where all the setup times are non-preemptively performed on one machine. We then present an algorithm with a tight bound of 4/3 for the general case. Furthermore, we show that the algorithm can produce optimal schedules for two special cases: equal processing times and equal setup times, which are NP-hard in the non-preemptive version.     

Single machine scheduling with simple linear deterioration to minimize total absolute deviation of completion times

We consider a single machine scheduling problem with simple linear deterioration. Job processing times are assumed to be a simple linear function of a job-dependent growth rate and the job's starting time. We seek an optimal schedule, so as to minimize the total absolute deviation of completion times (TADC). We prove several interesting properties of an optimal schedule, and introduce two efficient heuristics which are tested against a lower bound.     

信息物理融合系统的动态多优先级调度

信息物理融合系统(Cyber-physical Systems,CPS)的复杂和异构性给设计者带来了不少挑战,其中任务的多样性使得传统的调度策略不能满足CPS的性能需求.提出了专门针对基于大规模传感器网络的CPS的动态多优先级调度策略.根据任务类型分配4级缓存队列:第1级是来自控制器待处理的实时任务,拥有最高的可抢占式优先级;第2级是来自控制器待转发的实时任务,拥有次高的可抢占式优先级;第3级是来自其他节点待转发的非实时任务,拥有第三高的非抢占式优先级;第4级是来自本地待发送的非实时任务,拥有最低的非抢占式优先级.设计了抢占与非抢占混合的动态调度策略来减少任务的平均等待时间,加入了等待时间阈值机制来保证第4级任务的公平性.通过理论分析和仿真实验对调度策略的性能做了评价.仿真结果显示,动态多优先级调度策略在提高系统性能和稳定性上要优于传统优先级调度.     

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     

Heuristic algorithms for preemptive scheduling in a two-stage hybrid flowshop with additional renewable resources at each stage

This paper deals with the problem of preemptive scheduling in a two-stage flowshop with parallel unrelated machines and renewable resources at both the stages. The resource requirements are of a 0–1 type. The objective is the minimization of makespan. The problem is NP-hard. Four heuristic algorithms using linear programming are proposed for solving this problem. Performance of the algorithms is analyzed experimentally by comparing heuristic solutions with the lower bound on the optimal makespan. Statistical comparative analysis of the developed algorithms is carried out. The results of a computational experiment show that the proposed algorithms are able to produce good quality solutions in a small amount of computation time.