Preemptive scheduling in overloaded systems

Theory of Computing Systems, 2010

We present a unified optimal semi-online algorithm for preemptive scheduling on uniformly related machines with the objective to minimize the makespan. This algorithm works for all types of semi-online restrictions, including the ones studied before, like sorted (decreasing) jobs, known sum of processing times, known maximal processing time, their combinations, and so on. Based on the analysis of this algorithm, we derive some global relations between various semi-online restrictions and tight bounds on the approximation ratios for a small number of machines.

Operations Research Letters, 2000

This note deals with the scheduling problem of maximizing the number of early jobs on a single machine. We investigate the on-line version of this problem in the Preemption-Restart model. This means that jobs may be preempted, but preempting results in all the work done on this job so far being lost. Thus, if the job is restarted, then it has to be done from scratch.

Lecture Notes in Computer Science, 2000

In this paper, we derive bounds on performance guarantees of online algorithms for real-time preemptive scheduling of jobs with deadlines on K machines when jobs are characterized in terms of their minimum stretch factor α (or, equivalently, their maximum execution rate r = 1/α). We consider two well known preemptive models that are of interest from practical applications: the hard real-time scheduling model in which a job must be completed if it was admitted for execution by the online scheduler, and the firm real-time scheduling model in which the scheduler is allowed not to complete a job even if it was admitted for execution by the online scheduler. In both models, the objective is to maximize the sum of execution times of the jobs that were executed to completion, preemption is allowed, and the online scheduler must immediately decide, whenever a job arrives, whether to admit it for execution or reject it. We measure the competitive ratio of any online algorithm as the ratio of the value of the objective function obtained by this algorithm to that of the best possible offline algorithm. We show that no online algorithm can have a competitive ratio greater than 1 − (1/α) + ε for hard real-time scheduling with K ≥ 1 machines and greater than 1 − (3/(4 α)) + ε for firm real-time scheduling on a single machine, where ε > 0 may be arbitrarily small, even if the algorithm is allowed to know the value of α in advance. On the other hand, we exhibit a simple online scheduler that achieves a competitive ratio of at least 1 − (1/α) in either of these models with K machines. The performance guarantee of our simple scheduler shows that it is in fact an optimal scheduler for hard real-time scheduling with K machines. We also describe an alternative scheduler for firm real-time scheduling on a single machine in which the competitive ratio does not go to zero as α approaches 1. Both of our schedulers do not know the value of α in advance.

Journal of Scheduling, 2001

In this paper, we derive bounds on performance guarantees of online algorithms for real-time preemptive scheduling of jobs with deadlines on K machines when jobs are characterized in terms of their minimum stretch factor α (or, equivalently, their maximum execution rate r = 1/α). We consider two well known preemptive models that are of interest from practical applications: the hard real-time scheduling model in which a job must be completed if it was admitted for execution by the online scheduler, and the firm real-time scheduling model in which the scheduler is allowed not to complete a job even if it was admitted for execution by the online scheduler. In both models, the objective is to maximize the sum of execution times of the jobs that were executed to completion, preemption is allowed, and the online scheduler must immediately decide, whenever a job arrives, whether to admit it for execution or reject it. We measure the competitive ratio of any online algorithm as the ratio of the value of the objective function obtained by this algorithm to that of the best possible offline algorithm. We show that no online algorithm can have a competitive ratio greater than 1 − (1/α) + ε for hard real-time scheduling with K ≥ 1 machines and greater than 1 − (3/(4 α)) + ε for firm real-time scheduling on a single machine, where ε > 0 may be arbitrarily small, even if the algorithm is allowed to know the value of α in advance. On the other hand, we exhibit a simple online scheduler that achieves a competitive ratio of at least 1 − (1/α) in either of these models with K machines. The performance guarantee of our simple scheduler shows that it is in fact an optimal scheduler for hard real-time scheduling with K machines. We also describe an alternative scheduler for firm real-time scheduling on a single machine in which the competitive ratio does not go to zero as α approaches 1. Both of our schedulers do not know the value of α in advance.

IEEE Transactions on Computers, 1969

One of the important potentials of multiprocessor systems is the ability to speed the completion of a computation by concurrently processing independent portions of the job. In this paper we consider the static scheduling of computations for a system containing two indentical processors. The object is to complete the computation in the minimum amount of time. A computation is assumed to be specified as a partially ordered set of tasks and the execution time for each task. A solution for the two-machine case with preemptive scheduling is presented.

2012

We study a maximization problem: online scheduling on m identical machines to maximize the number of early jobs. The problem is online in the sense that all jobs arrive over time. Each job's characteristics, such as processing time and due date, become known at its arrival time. We consider the preemption-restart model, in which preemption is allowed, while once a job is restarted, it loses all the progress that has been made on this job so far. If in some schedule a job is completed before or at its due date, then it is called early or on time . The objective is to maximize the number of early jobs. For m identical machines, we prove an upper bound 1 − 1/2m of competitive ratio and show that ECT earliest completion time algorithm is 1/2-competitive.

Journal of Mathematics and Computer Science

This paper considers a stochastic online scheduling problem in which a set of independent jobs are to be processed on a single machine. Each job has a processing time, which is a random variable with normal distribution. All the jobs arrive overtime, which means that the existence and the parameters of each job including its processing time specifications and weight are unknown until its release date. Moreover, the actual processing time of each job is unknown until its completion. During the processing, jobs are allowed to be preempted and restarted later. So, the processing time devoted to the job before the preemption is lost and considered as preemption penalty. The objective is to minimize the expected value of the total weighted completion time. Since the problem is strongly NP-hard, a heuristic algorithm is proposed in this paper and is validated using numerical examples. The proposed method utilizes the properties of the normal distribution but it can be used as a heuristic for other distributions, as long as their means and variances are available.

SIAM Journal on Computing, 2000

We study the problem of processor scheduling for n parallel jobs applying the method of competitive analysis. We prove that for jobs with a single phase of parallelism, a preemptive scheduling algorithm without information about job execution time can achieve a mean completion time within 2 -2 n+1 times the optimum. In other words, we prove a competitive ratio of 2 -2 n+1 . The result is extended to jobs with multiple phases of parallelism (which can be used to model jobs with sublinear speedup) and to interactive jobs (with phases during which the job has no CPU requirements) to derive solutions guaranteed to be within 4-4 n+1 times the optimum. In comparison with previous work, our assumption that job execution times are unknown prior to their completion is more realistic, our multiphased job model is more general, and our approximation ratio (for jobs with a single phase of parallelism) is tighter and cannot be improved. While this work presents theoretical results obtained using competitive analysis, we believe that the results provide insight into the performance of practical multiprocessor scheduling algorithms that operate in the absence of complete information.

2010

Tasks' scheduling has always been a central problem in the embedded real-time systems community. As in general the scheduling problem is NP-hard, researchers have been looking for efficient heuristics to solve the scheduling problem in polynomial time. One of the most important scheduling strategies is the Earliest Deadline First (EDF). It is known that EDF is optimal for uniprocessor platforms for many cases, such as: non-preemptive synchronous tasks(i.e., all tasks have the same starting time and cannot be interrupted), and preemptive asynchronous tasks (i.e., the tasks may be interrupted and may have arbitrary starting time). However, Mok showed that EDF is not optimal in multiprocessor platforms. In fact, for the multiprocessor platforms, the scheduling problem is NP-complete in most of the cases where the corresponding scheduling problem can be solved by a polynomial-time algorithm for uniprocessor platforms. Coffman and Graham identified a class of tasks for which the scheduling problem can be solved by a polynomial time algorithm, that is, two-processor platform, no resources, arbitrary partial order relations, and every task is nonpreemptive and has a unit computation time. Our paper introduces a new non-trivial and practical subclass of tasks, called urgent tasks. Briefly, a task is urgent if it is executed right after it is ready or it can only wait one unit time after it is ready. Practical examples of embedded real time systems dealing with urgent tasks are all modern building alarm systems, as these include urgent tasks such as `checking for intruders', `sending a warning signal to the security office',`informing the building's owner about a potential intrusion', and so on. By using propositional logic, we prove a new result in schedulability theory, namely that the scheduling problem for asynchronous and preemptive urgent tasks can be solved in polynomial time.

2000

We consider the problem of scheduling an unknown sequence of tasks for a single server as the tasks arrive with the goal off maximizing the total weighted value of the tasks served before their deadline is reached. This problem is faced for example by schedulers in packet communication networks when packets have deadlines and rewards associated with them. We make the simplifying assumptions that every task takes the same fixed amount of time to serve, that every task arrives with the same initial latency to its deadline. We also assume that future task arrivals are stochastically described by a Hidden Markov Model (HMM). The resulting decision problem can be formally modelled as a Partially Observable Markov Decision Process (POMDP).