Optimal Time-Critical Scheduling via Resource Augmentation

We consider two fundamental problems in dynamic scheduling: scheduling to meet deadlines in a preemptive multiprocessor setting, and scheduling to provide good response time in a number of scheduling environments. When viewed from the perspective of traditional worst-case analysis, no good on-line algorithms exist for these problems, and for some variants no good off-line algorithms exist unless P = NP . We study these problems using a relaxed notion of competitive analysis, introduced by Kalyanasundaram and Pruhs, in which the on-line algorithm is allowed more resources than the optimal off-line algorithm to which it is compared. Using this approach, we establish that several well-known on-line algorithms, that have poor performance from an absolute worst-case perspective, are optimal for the problems in question when allowed moderately more resources. For optimization of average flow time, these are the first results of any sort, for any NP -hard version of the problem, that indicate that it might be possible to design good approximation algorithms.     

Optimal Time-Critical Scheduling via Resource Augmentation

Abstract. We consider two fundamental problems in dynamic scheduling: scheduling to meet deadlines in a preemptive multiprocessor setting, and scheduling to provide good response time in a number of scheduling environments. When viewed from the perspective of traditional worst-case analysis, no good on-line algorithms exist for these problems, and for some variants no good off-line algorithms exist unless P = NP . We study these problems using a relaxed notion of competitive analysis, introduced by Kalyanasundaram and Pruhs, in which the on-line algorithm is allowed more resources than the optimal off-line algorithm to which it is compared. Using this approach, we establish that several well-known on-line algorithms, that have poor performance from an absolute worst-case perspective, are optimal for the problems in question when allowed moderately more resources. For optimization of average flow time, these are the first results of any sort, for any NP -hard version of the problem, that indicate that it might be possible to design good approximation algorithms.     

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.     

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.     

Optimal and online preemptive scheduling on uniformly related machines

We consider the problem of preemptive scheduling on uniformly related machines. We present a semi-online algorithm which, if the optimal makespan is given in advance, produces an optimal schedule. Using the standard doubling technique, this yields a 4-competitive deterministic and an e≈2.71-competitive randomized online algorithm. In addition, it matches the performance of the previously known algorithms for the offline case, with a considerably simpler proof. Finally, we study the performance of greedy heuristics for the same problem.     

有中断时间代价的一致并行机抢先调度问题

提出了一种具有中断时间代价的抢先调度问题(P|ptmn(δ)|C_max):在抢先调度中,一个任务发生一次中断,其总的执行时间会增加一个δ.该问题在工程任务分配、分布式计算和网络通信等实际问题中有着广泛的应用背景.证明了这是一个NP-hard问题,给出了一个时间复杂度为O(nlogn+m)的脱线近似算法LPT-Wrap,其近似比小于等于1.40825,并分析了P|ptmn(δ)|C_max的在线特性,给出一个线性时间复杂度的在线近似算法,其竞争比为2.     

Balanced Scheduling toward Loss-Free Packet Queuing and Delay Fairness

In current networks, packet losses can occur if routers do not provide sufficiently large buffers. This paper studies how many buffers should be provided in a router to eliminate packet losses. We assume a network router has m incoming queues, each corresponding to a single traffic stream, and must schedule at any time on-line from which queue to take the next packet to send out. To exclude packet losses with a small amount of buffers, the maximum queue length must be kept low over the entire scheduling period. We call this new on-line problem the balanced scheduling problem (BSP). By competitive analysis, we measure the power of on-line scheduling algorithms to prevent packet losses. We show that a simple greedy algorithm is Θ(log m)-competitive which is asymptotically optimal, while Round-Robin scheduling is not better than m-competitive, as actually is any deterministic on-line algorithm for BSP. We also give a polynomial time algorithm for solving off-line BSP optimally. We also study another on-line balancing problem that tries to balance the delay among the m traffic streams.     

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.     

异构系统中一种基于可用性的抢占式任务调度算法*

针对大多数现有的异构系统调度算法没有考虑由多类任务特别是抢占式任务所引起的可用性需求的不足,在现有基于可用性的非抢占式任务调度算法的基础上,通过计算任务的平均等待时间来确定优先级等级,对异构系统中多类抢占式任务的可用性约束的调度问题进行了探索,提出了一种基于可用性的抢占式优先调度算法P-SSAC。该算法在不增加硬件代价的前提条件下通过调度增加了系统的可用性,缩短了任务的平均等待时间,同时该算法可对抢占式的任务进行有效调度。仿真实验结果表明,该算法有效实现了异构系统可用性和任务等待时间之间的折中。     

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.     

Approximation Algorithms for Average Stretch Scheduling

We study the basic problem of preemptive scheduling of a stream of jobs on a single processor. Consider an on-line stream of jobs, and let the ith job arrive at time r(i) and have processing time p(i). If C(i) is the completion time of job i, then the flow time of i is C(i) − r(i) and the stretch of i is the ratio of its flow time to its processing time; that is, . Flow time measures the time that a job is in the system regardless of the service it requests; the stretch measure relies on the intuition that a job that requires a long service time must be prepared to wait longer than jobs that require small service times. We present the improved algorithmic results for the average stretch metric in preemptive uniprocessor scheduling. Our first result is an off-line polynomial-time approximation scheme (PTAS) for average stretch scheduling. This improves upon the 2-approximation achieved by the on-line algorithm srpt that always schedules a job with the shortest remaining processing time. In a recent work, Chekuri and Khanna (Proc. 34th Ann. Symp. Theory Comput., 297–305, 2002) have presented approximation algorithms for weighted flow time, which is a more general metric than average stretch; their result also yields a PTAS for average stretch. Our second set of results considers the impact of incomplete knowledge of job sizes on the performance of on-line scheduling algorithms. We show that a constant-factor competitive ratio for average stretch is achievable even if the processing times (or remaining processing times) of jobs are known only to within a constant factor of accuracy.     

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).     

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.     

The EDL Server for Scheduling Periodic and Soft Aperiodic Tasks with Resource Constraints

In this paper, we are concerned with the problem of serving soft aperiodic tasks on a uniprocessor system where periodic tasks are scheduled on a dynamic-priority, preemptive basis and exclusively access to critical sections. Scheduling of tasks is handled by the Dynamic Priority Ceiling Protocol working with an Earliest Deadline scheduler. Our analysis determines the maximum processing time which may be stolen from periodic tasks without jeopardizing both their timing constraints and resource consistency. It provides the basis for an on-line scheduling algorithm, the EDL Server, to deal with the minimization of response times for soft aperiodic tasks.     

Dynamic scheduling of real-time tasks under precedence constraints

While scheduling theory has been developed over a long period of time, it is important to note that most results concern problems with static characteristics. However, a real-time system is dynamic and requires on-line and adaptive scheduling strategies. So, an important aspect of real-time systems research is to devise methods flexible enough to react to a dynamic change of processor load and to attempt to schedule all the tasks judiciously.In this paper, we are particularly concerned with the problem of scheduling tasks which are of two kinds: periodic and sporadic, on a monoprocessor machine. Periodic tasks are independent, run cyclically and their characteristics are known in advance. In addition, we allow for the unpredictable occurrence of aperiodic task groups, with timing and precedence constraints. Clearly, the main problem is to devise a schedulability test that makes it possible to decide whether a new occurring group can be accepted, without upsetting the tight timing behavior requirements.We present an optimal acceptance test which returns a decision on the basis of the current state of processor load and by considering tasks to be scheduled according to the well known preemptive algorithm Earliest Deadline. The acceptance algorithm and a complexity analysis are presented.     

An O(n 2 ) algorithm for scheduling equal-length preemptive jobs on a single machine to minimize total tardiness

In this paper, we study the problem of scheduling n equal-length preemptive jobs on a single machine to minimize total tardiness, subject to release dates. The complexity status of this problem has remained open to date. We provide an O(n²) time algorithm to solve the problem.     

New Algorithms for Disk Scheduling

Abstract. Processor speed and memory capacity are increasing several times faster than disk speed. This disparity suggests that disk I/ O performance could become an important bottleneck. Methods are needed for using disks more efficiently. Past analysis of disk scheduling algorithms has largely been experimental and little attempt has been made to develop algorithms with provable performance guarantees. We consider the following disk scheduling problem. Given a set of requests on a computer disk and a convex reachability function that determines how fast the disk head travels between tracks, our goal is to schedule the disk head so that it services all the requests in the shortest time possible. We present a 3/2 -approximation algorithm (with a constant additive term). For the special case in which the reachability function is linear we present an optimal polynomial-time solution. The disk scheduling problem is related to the special case of the Asymmetric Traveling Salesman Problem with the triangle inequality (ATSP-Δ ) in which all distances are either 0 or some constant α . We show how to find the optimal tour in polynomial time and describe how this gives another approximation algorithm for the disk scheduling problem. Finally we consider the on-line version of the problem in which uniformly distributed requests arrive over time. We present an algorithm related to the above ATSP-Δ .     

Hadoop集群环境下集成抢占式调度策略的本地性调度算法设计

在Hadoop集群环境下本地性调度算法是提高数据本地性的算法。本地性调度算法的调度策略的本质是提高数据本地性,减少网络传输开销,避免阻塞。但是由于Map任务的完成时间不同,Reduce任务存在的等待现象影响了作业的平均完成时间,使得作业的完成时间增加,进而引起系统的性能参数不佳。因此提出在保留原算法数据本地性要求的基础上集成可抢占式的调度方法。在Reduce任务等待时,挂起该任务并释放资源给其他Map任务,当Map任务完成到一定程度后,重新调度Reduce任务。基于上述调度策略设计了集成抢占式策略的本地性调度。为了对改进的算法进行验证,通过实验对本地性调度算法和集成抢占式本地性调度算法进行比较。实验结果表明,在相同数据上,集成抢占式本地性调度算法的平均完成时间有明显的降低。     

Dimensioning and on-line scheduling in Lambda Grids using divisible load concepts

Due to the large amounts of data required to be processed by the typical Grid job, it is conceivable that the use of optical transport networks in Grid deployment (hence the term “Lambda Grid”) will increase. The exact topology of the interconnecting network is obtained by solving a dimensioning problem, and the outcome of this strongly depends on both the expected workload characteristics and Grid scheduling policy. Solving this combined scheduling and dimensioning problem using straightforward ILP modelling is cumbersome; however, for steady-state Grid operation, Divisible Load Theory (DLT) can yield scalable formulations of this problem. In this paper, the on-line hierarchical scheduling on a lambda Grid of workload approaching the Grid’s capacity in a two-tier Grid mode of operation is studied. A number of these algorithms are goal-driven, in the sense that target per-resource goals are obtained from the off-line solution to the Divisible Load model. We compare these on-line multiresource scheduling policies for different workloads, Grid interconnection topologies and Grid parameters. We show that these algorithms perform well in the studied scenarios when compared to a fully centralized scheduling algorithm.