scholarly journals Online Non-preemptive Scheduling on Unrelated Machines with Rejections

2021 ◽  
Vol 8 (2) ◽  
pp. 1-22
Author(s):  
Giorgio Lucarelli ◽  
Benjamin Moseley ◽  
Nguyen Kim Thang ◽  
Abhinav Srivastav ◽  
Denis Trystram
Keyword(s):  
Lower Bounds ◽  
Competitive Ratio ◽  
Poor Performance ◽  
Flow Time ◽  
Theory And Practice ◽  
Worst Case Analysis ◽  
Preemptive Scheduling ◽  
Scheduling Problems ◽  
Worst Case ◽  
Small Constant

When a computer system schedules jobs there is typically a significant cost associated with preempting a job during execution. This cost can be incurred from the expensive task of saving the memory’s state or from loading data into and out of memory. Thus, it is desirable to schedule jobs non-preemptively to avoid the costs of preemption. There is a need for non-preemptive system schedulers for desktops, servers, and data centers. Despite this need, there is a gap between theory and practice. Indeed, few non-preemptive online schedulers are known to have strong theoretical guarantees. This gap is likely due to strong lower bounds on any online algorithm for popular objectives. Indeed, typical worst-case analysis approaches, and even resource-augmented approaches such as speed augmentation, result in all algorithms having poor performance guarantees. This article considers online non-preemptive scheduling problems in the worst-case rejection model where the algorithm is allowed to reject a small fraction of jobs. By rejecting only a few jobs, this article shows that the strong lower bounds can be circumvented. This approach can be used to discover algorithmic scheduling policies with desirable worst-case guarantees. Specifically, the article presents algorithms for the following three objectives: minimizing the total flow-time, minimizing the total weighted flow-time plus energy where energy is a convex function, and minimizing the total energy under the deadline constraints. The algorithms for the first two problems have a small constant competitive ratio while rejecting only a constant fraction of jobs. For the last problem, we present a constant competitive ratio without rejection. Beyond specific results, the article asserts that alternative models beyond speed augmentation should be explored to aid in the discovery of good schedulers in the face of the requirement of being online and non-preemptive.

scholarly journals Packing into the smallest square: Worst-case analysis of lower bounds

2006 ◽  
Vol 3 (4) ◽  
pp. 317-326 ◽  
Author(s):  
Alberto Caprara ◽  
Andrea Lodi ◽  
Silvano Martello ◽  
Michele Monaci
Keyword(s):  
Lower Bounds ◽  
Case Analysis ◽  
Worst Case Analysis ◽  
Worst Case

Scheduling two classes of exponential jobs on parallel processors: structural results and worst-case analysis

1991 ◽  
Vol 23 (4) ◽  
pp. 925-944 ◽  
Author(s):  
Cheng-Shang Chang ◽  
Randolph Nelson ◽  
Michael Pinedo
Keyword(s):  
Case Analysis ◽  
Parallel Processors ◽  
Worst Case Analysis ◽  
Weighted Sum ◽  
Scheduling Problems ◽  
Worst Case ◽  
Total Cost ◽  
Class 1 ◽  
Completion Times ◽  
Sum Of Completion Times

In this paper, we consider scheduling problems with m machines in parallel and two classes of job. We assume that all jobs are present at time 0 and there are no further arrivals. The service times of class 1 (2) jobs are independent and exponentially distributed with mean . Each class 1 (2) job incurs a cost c1 (c2) per unit of time until it leaves the system. The objective is to minimize the expected total cost, that is the expected weighted sum of completion times. We show that the optimal policy among all preemptive policies is of threshold type. Based on these structural results, we also show that the ratio of the expected weighted sum of completion times under the cµ-rule to that under the optimal rule is less than 1·71.

Scheduling two classes of exponential jobs on parallel processors: structural results and worst-case analysis

1991 ◽  
Vol 23 (04) ◽  
pp. 925-944 ◽  
Author(s):  
Cheng-Shang Chang ◽  
Randolph Nelson ◽  
Michael Pinedo
Keyword(s):  
Case Analysis ◽  
Parallel Processors ◽  
Worst Case Analysis ◽  
Weighted Sum ◽  
Scheduling Problems ◽  
Worst Case ◽  
Total Cost ◽  
Class 1 ◽  
Completion Times ◽  
Sum Of Completion Times

In this paper, we consider scheduling problems with m machines in parallel and two classes of job. We assume that all jobs are present at time 0 and there are no further arrivals. The service times of class 1 (2) jobs are independent and exponentially distributed with mean . Each class 1 (2) job incurs a cost c 1 (c 2) per unit of time until it leaves the system. The objective is to minimize the expected total cost, that is the expected weighted sum of completion times. We show that the optimal policy among all preemptive policies is of threshold type. Based on these structural results, we also show that the ratio of the expected weighted sum of completion times under the cµ-rule to that under the optimal rule is less than 1·71.

Online Job Dispatching and Scheduling to Minimize Job Completion Time and to Meet Deadlines

2018 ◽  
Vol 18 (04) ◽  
pp. 1850012
Author(s):  
YUPENG LI
Keyword(s):  
Competitive Ratio ◽  
Completion Time ◽  
Online Algorithm ◽  
Performance Metrics ◽  
Case Analysis ◽  
Deterministic Algorithm ◽  
Worst Case Analysis ◽  
Worst Case ◽  
Job Dispatching ◽  
Shed Light

In this paper, we study the problem of job dispatching and scheduling, where each job consists of a set of tasks. Each task is processed by a set of machines simultaneously. We consider two important performance metrics, the average job completion time (JCT), and the number of deadline-aware jobs that meet their deadlines. The goal is to minimize the former and maximize the latter. We first propose OneJ to minimize the job completion time (JCT) when there is exactly one single job in the system. Then, we propose an online algorithm called MultiJ, taking OneJ as a subroutine, to minimize the average JCT, and prove it has a good competitive ratio. We then derive another online algorithm QuickJ to maximize the number of jobs that can meet their deadlines. We show that QuickJ is competitive via a worst case analysis. We also conjecture that the competitive ratio of QuickJ is likely to be the best one that any deterministic algorithm can achieve. We also shed light on several important merits of MultiJ and QuickJ, such as no severe coordination overhead, scalability, work conservation, and no job starvation.

scholarly journals Scheduling in the Random-Order Model

Algorithmica ◽  
2021 ◽  
Author(s):  
Susanne Albers ◽  
Maximilian Janke
Keyword(s):  
Competitive Ratio ◽  
High Probability ◽  
Online Algorithm ◽  
Fundamental Problem ◽  
Random Order ◽  
Random Permutation ◽  
Worst Case Analysis ◽  
Makespan Minimization ◽  
Order Model ◽  
Worst Case

AbstractMakespan minimization on identical machines is a fundamental problem in online scheduling. The goal is to assign a sequence of jobs to m identical parallel machines so as to minimize the maximum completion time of any job. Already in the 1960s, Graham showed that Greedy is $$(2-1/m)$$ ( 2 - 1 / m ) -competitive. The best deterministic online algorithm currently known achieves a competitive ratio of 1.9201. No deterministic online strategy can obtain a competitiveness smaller than 1.88. In this paper, we study online makespan minimization in the popular random-order model, where the jobs of a given input arrive as a random permutation. It is known that Greedy does not attain a competitive factor asymptotically smaller than 2 in this setting. We present the first improved performance guarantees. Specifically, we develop a deterministic online algorithm that achieves a competitive ratio of 1.8478. The result relies on a new analysis approach. We identify a set of properties that a random permutation of the input jobs satisfies with high probability. Then we conduct a worst-case analysis of our algorithm, for the respective class of permutations. The analysis implies that the stated competitiveness holds not only in expectation but with high probability. Moreover, it provides mathematical evidence that job sequences leading to higher performance ratios are extremely rare, pathological inputs. We complement the results by lower bounds, for the random-order model. We show that no deterministic online algorithm can achieve a competitive ratio smaller than 4/3. Moreover, no deterministic online algorithm can attain a competitiveness smaller than 3/2 with high probability.

scholarly journals Efficient Mechanism Design for Online Scheduling (Extended Abstract)

2017 ◽  
Author(s):  
Xujin Chen ◽  
Xiaodong Hu ◽  
Tie-Yan Liu ◽  
Weidong Ma ◽  
Tao Qin ◽  
...  
Keyword(s):  
Mechanism Design ◽  
Lower Bounds ◽  
Competitive Ratio ◽  
Online Scheduling ◽  
Equal Length ◽  
The Other ◽  
Release Time ◽  
Incentive Compatible ◽  
Small Constant ◽  
Unequal Length

This work concerns the mechanism design for online scheduling in a strategic setting. In this setting, each job is owned by a self-interested agent who may misreport the release time, deadline, length, and value of her job, while we need to determine not only the schedule of the jobs, but also the payment of each agent. We focus on the design of incentive compatible (IC) mechanisms, and study the maximization of social welfare (i.e., the aggregated value of completed jobs) by competitive analysis. We first derive two lower bounds on the competitive ratio of any deterministic IC mechanism to characterize the landscape of our research: one bound is 5, which holds for equal-length jobs; the other bound is $\frac{\kappa}{\ln\kappa}+1-o(1)$, which holds for unequal-length jobs, where $\kappa$ is the maximum ratio between lengths of any two jobs. We then propose a deterministic IC mechanism and show that such a simple mechanism works very well for two models: (1) In the preemption-restart model, the mechanism can achieve the optimal competitive ratio of 5 for equal-length jobs and a near optimal ratio of $(\frac{1}{(1-\epsilon)^2}+o(1)) \frac{\kappa}{\ln\kappa}$ for unequal-length jobs, where $0<\epsilon<1$ is a small constant; (2) In the preemption-resume model, the mechanism can achieve the optimal competitive ratio of 5 for equal-length jobs and a near optimal competitive ratio (within factor 2) for unequal-length jobs.

scholarly journals A SORTING IMPROVEMENT IN THE HEURISTIC BASED ON REMAINING AVAILABLE AND PROCESSING PERIODS TO MINIMIZE TOTAL TARDINESS IN PROGRESSIVE IDLING-FREE 1-MACHINE PREEMPTIVE SCHEDULING

2021 ◽  
Author(s):  
Vadim V. Romanuke
Keyword(s):  
Negative Impact ◽  
Computational Study ◽  
Total Tardiness ◽  
Release Dates ◽  
Preemptive Scheduling ◽  
Scheduling Problem ◽  
Scheduling Problems ◽  
Due Dates ◽  
Worst Case ◽  
Priority Weights

Background. In preemptive job scheduling, which is a part of the flow-shop sequencing tasks, one of the most crucial goals is to obtain a schedule whose total tardiness would be minimal. Total tardiness minimization is commonly reduced to solving a combinatorial problem which becomes practically intractable as the number of jobs and the numbers of their processing periods increase. To cope with this challenge, heuristics are used. A heuristic, in which the decisive ratio is the reciprocal of the maximum of a pair of the remaining processing period and remaining available period, is closely the best one. However, the heuristic may produce schedules of a few jobs whose total tardiness is 25 % greater than the minimum or even worse. Therefore, this heuristic needs a corrective branch which would further try to minimize total tardiness under certain conditions. Objective. The goal is to ascertain what is to be corrected in the heuristic so that the total tardiness value could be obtained lesser. The heuristic will be applied to tight-tardy progressive idling-free 1-machine preemptive scheduling, where the release dates are given in ascending order starting from 1 to the number of jobs, and the due dates are tightly set after the release dates. In this scheduling problem, the inaccuracy of finding the minimal total tardiness has the strongest negative impact, so this is almost the worst case, which defines the accuracy limit of the heuristic and positively serves just as the principle of minimax guaranteeing decreasing losses in the worst conditions. Methods. The heuristic sorts maximal decisive ratios by release dates, where the scheduling preference is given to the earliest job. To achieve the said goal, three other sorting approaches are presented and a computational study is carried out with applying each of the four heuristic approaches to minimize total tardiness. For this, two series of 266000 and 1064000 scheduling problems are generated. Results. The earliest-job sorting ensures a heuristically minimal total tardiness value in more than 97.6 % of scheduling problems, but it fails to minimize total tardiness in no less than 2.2 % of the cases. Nevertheless, a sorting approach with minimizing remaining processing periods produces a heuristically minimal total tardiness for almost any scheduling problem. If an exception occurs, this sorting approach “loses” to the other sorting approaches very little. Moreover, the exceptions are quite rare as it has been registered just a one scheduling problem (out of 31914 cases followed by a sole “win” of a heuristic version) whose minimal total tardiness is achieved by the earliest-job sorting. Conclusions. The best heuristic version is that one which uses the sorting approach with minimizing remaining processing periods. This, however, is confirmed only for the case where jobs do not have any priorities. The case when jobs have their priority weights is to be yet analyzed.

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