Deterministic and randomized scheduling problems under the lp norm on two identical machines

Parallel machine scheduling problems, which are important discrete optimization problems, may occur in many applications. For example, load balancing in network communication channel assignment, parallel processing in large-size computing, task arrangement in flexible manufacturing systems, etc., are multiprocessor scheduling problem. In the traditional parallel machine scheduling problems, it is assumed that the problems are considered in offline or online environment. But in practice, problems are often not really offline or online but somehow in-between. This means that, with respect to the online problem, some further information about the tasks is available, which allows the improvement of the performance of the best possible algorithms. Problems of this class are called semi-online ones. In this paper, the semi-online problem P2|decr|lp (p>1) is considered where jobs come in non-increasing order of their processing times and the objective is to minimize the sum of the lp norm of every machine’s load. It is shown that LS algorithm is optimal for any lp norm, which extends the results known in the literature. Furthermore, randomized lower bounds for the problems P2|online|lp and P2|decr|lp are presented.

Semi on-line scheduling for maximizing the minimum machine completion time on three uniform machines

The paper investigates a semi on-line scheduling problem wherein the largest processing time of jobs done by three uniform machines M1, M2, M3 is known in advance. A speed si (s1=1, s2=r, s3=s, 1≤r≤s) is associated with machine Mi. Our goal is to maximize Cmin?the minimum workload of the three machines. We present a min3 algorithm and prove its competitive ratio is max{r+1,(3s+r+1)/(1+r+s)}, with the lower bound being at least max{2,r}. We also claim the competitive ratio of min3 algo- rithm cannot be improved and is the best possible for 1≤s≤2, r=1.

ON-LINE PROBLEMS OF MINIMIZING MAKESPAN ON A SINGLE BATCH PROCESSING MACHINE WITH NONIDENTICAL JOB SIZES

The on-line problem of scheduling on a batch processing machine with nonidentical job sizes to minimize makespan is considered. The batch processing machine can process a number of jobs simultaneously as long as the total size of these jobs being processed does not exceed the machine capacity. The processing time of a batch is given by the longest processing time of any job in the batch. Each job becomes available at its arrival time, which is unknown in advance, and its processing time becomes known upon its arrival. The paper deals with two variants： the case only with two distinct arrival times and the general case. For the first case, an on-line algorithm with competitive ratio 119/44 is given. For the latter one, a simple algorithm with competitive ratio 3 is given. For both variants the better ratios can be obtained if the problem satisfies proportional assumption.

On-Line Scheduling for Jobs with Arbitrary Release Times on Parallel Related Uniform Machines

A parallel related uniform machine system consists of m machines with different processing speeds. The speed of any machine is independent on jobs. In this paper, we consider online scheduling for jobs with arbitrary release times on the parallel uniform machine system. The jobs appear over list in terms of order. An order includes the processing size and releasing time of a job. For this model, an algorithm with competitive ratio of 12 is addressed in this paper.

Online Parallel-Batch Machines Scheduling with a Single Server

We consider parallel-batch machines scheduling problem with a single server to minimize the maximum completion time.Jobs arrive over time.Every batch has to be loaded by the sever before being processed on machines.The loading(setup)operation of a batch occurs only when some machine is idle,and the server can perform only one setup operation every time.For some special case,we provide a best possible online algorithm with competitive ratio■.For general case,we give another online algorithm with competitive ratio 3.

Competitive Analysis of Two Special Online Device Replacement Problems

When a new investment opportunity of purchasing a new device occurs, the investors must decide whether or not and when to buy this device in an online fashion. That is, the online player must make an investment decision while neither future demand for orders nor future investment opportunities are known. This problem which generalizes the basic leasing problem has been introduced by Azar et al., and then two special cases have been studied by Damaschke. In the so-called equal prices model a 2-competitive algorithm is devised and a 1.618 lower bound is given. Here we make use of an averaging technique and obtain a better tight lower bound of 2, in other words, this lower bound cannot be improved. Furthermore, another special case which only considers two-stage device replacement is studied in this paper. Accounting for the interest rate is an essential feature of any reasonable financial model. Therefore, we explore the two-stage model with and without the interest rate respectively. In addition, we introduce the risk-reward model to analyze this problem and improve the competitive ratio performance.

Semi-Online Algorithms for Scheduling with Machine Cost

In this paper, we consider the following semi-online List Model problem with known total size. We are given a sequence of independent jobs with positive sizes, which must be assigned to be processed on machines. No machines are initially provided, and when a job is revealed the algorithm has the option to purchase new machines. By normalizing all job sizes and machine cost, we assume that the cost of purchasing one machine is 1. We further know the total size of all jobs in advance. The objective is to minimize the sum of the makespan and the number of machines to be purchased. Both non-preemptive and preemptive versions are considered. For the non-preemptive version, we present a new lower bound 6/5 which improves the known lower bound 1.161. For the preemptive version, we present an optimal semi-online algorithm with a competitive ratio of 1 in the case that the total size is not greater than 4, and an algorithm with a competitive ratio of 5/4 otherwise, while a lower bound 1.0957 is also presented for general case.

Preemptive Semi-Online Scheduling with Tightly-Grouped Processing Times

This paper investigates a preemptive semi-online scheduling problem onm identical parallel machines wherem=2,3. It is assumed that all jobs have their processing times in betweenp andrp (p > 0,r ≥1). The goal is to minimize the makespan. Best possible algorithms are designed for anyr≥1 whenm=2,3. Keywords semi-online - scheduling - preemption - competitive ratio Regular PaperThis research is supported by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE. China, and the National Natural Science Foundation of China (Grant Nos. 10271110 and 60021201).Yong He received his B.S., M.S., and Ph.D. degrees all from Zhejiang University in 1989, 1992, 1996, respectively. He is currently a professor and Ph.D. supervisor at Department of Mathematics, Zhejiang University. His current research interests include combinatorial and network optimization, scheduling theory, computational biology, mathematical modeling, etc.Yi-Wei Jiang received his B.S. degree from Zhejiang University in 2002. He is currently a Ph.D. candidate of Zhejiang University. His current interests include scheduling theory and online algorithms.

ON-LINE SCHEDULING WITH REJECTION ON IDENTICAL PARALLEL MACHINES

In this paper, we consider the on-line scheduling of unit time jobs with rejection on rn identical parallel machines. The objective is to minimize the total completion time of the accepted jobs plus the total penalty of the rejected jobs. We give an on-line algorithm for the problem with competitive ratio 1/2 （2 ＋√3） ≈ 1.86602.

Online Scheduling on Bounded Batch Machines to Minimize the Maximum Weighted Completion Time

We investigate the online scheduling problem on identical parallel-batch machines to minimize the maximum weighted completion time.In this problem,jobs arrive over time and the processing times(of the jobs)are identical,and the batch capacity is bounded.For this problem,we provide a best possible online algorithm with a competitive ratio of(√5+1)/2.Moreover,when restricted to dense-algorithms,we present a best possible dense-algorithm with a competitive ratio of 2.

SEMI ON-LINE SCHEDULING PROBLEM FOR MAXIMIZING THE MINIMUM MACHINE COMPLETION TIME ON TWO UNIFORM MACHINES

Ia this paper, we consider a semi on-line version on two uniform machines Mi, i = 1, 2, where the processing time of the largest job is known in advance. A speed si（s1 = 1, 1 ≤s2 = s） is associated with machine Mi. Our goal is to maximize the Cmin. We give a Cmin 2 algorithm and prove its competitive ratio is at most 2s＋1/s＋1 We also claim the Cmin 2 algorithm is tight and the gap between the competitive ratio of Cmin2 algorithm and the optimal value is not greater than 0.555. It is obvious that our result coincides with that given by He for s =1.

ON-LINE SCHEDULING OF UNIT TIME JOBS WITH REJECTION ON UNIFORM MACHINES

The authors consider the problem of on-line scheduling of unit execution time jobs on uniform machines with rejection penalty. The jobs arrive one by one and can be either accepted and scheduled, or be rejected. The objective is to minimize the total completion time of the accepted jobs and the total penalty of the rejection jobs. The authors propose an on-line algorithm and prove that the competitive ratio is 1/2 （2 W √3） ≈ 1.86602.

Online Over Time Scheduling on Parallel-Batch Machines:A Survey

Online scheduling is a rapidly developed branch in scheduling theory.In this paper,we present an extensive survey for online over time scheduling on parallel-batch machines.Some open problems are proposed for further research.

Online Scheduling on Two Uniform Unbounded Parallel-Batch Machines to Minimize Makespan

We study an online(over time)scheduling problem on two uniform unbounded parallel-batch machines with processing speed 1 and v(0<≤1),respectively,to minimize the makespan.We first show that no online algorithm can have a competitive ratio less than 1+αL,whereαL=(√5−1)/2≈0.618 if 0≤1/2,andαL=α2(v)is the positive root ofα^(3)+3α^(2)+(3−1/v)α−1/v=0 if1/2≤1.This lower bound is still valid when all jobs have the same processing times.Then,we provide an online algorithm with a competitive ratio at most min{(√5+1)/2,√2/v}.When the jobs have the same processing times,we present the best possible online algorithm with a competitive ratio 1+αL.

Online Scheduling with Unit Processing Times and Processing Set Restrictions

This paper investigates online scheduling problems with processing set restrictions.There is a sequence of jobs that are revealed one by one over list,where each job has unit processing time.A job can only be processed by a subset of the machines,namely the processing set of the job.The objective is to minimize the makespan.For m machines with two different processing sets,we propose an optimal algorithm with a competitive ratio of 3/2.For three machines with inclusive processing sets,an optimal algorithm with competitive ratio 5/3 is provided.

Optimal Preemptive Online Algorithms for Scheduling with Known Largest Size on Two Uniform Machines

In this paper, we consider the seml-online preemptive scheduling problem with known largest job sizes on two uniform machines. Our goal is to maximize the continuous period of time （starting from time zero） when both machines are busy, which is equivalent to maximizing the minimum machine completion time if idle time is not introduced. We design optimal deterministic semi-online algorithms for every machine speed ratio s ∈ [1, ∞）, and show that idle time is required to achieve the optimality during the assignment procedure of the algorithm for any s 〉 （s^2 ＋ 3s ＋ 1）/（s^2 ＋ 2s ＋ 1）. The competitive ratio of the algorithms is （s^2 ＋ 3s ＋ 1）/（s^2 ＋ 2s ＋ 1）, which matches the randomized lower bound for every s ≥ 1. Hence randomization does not help for the discussed preemptive scheduling problem.

2-Space Bounded Online Cube and Hypercube Packing

We consider the problem of packing d-dimensional cubes into the minimum number of 2-space bounded unit cubes. Given a sequence of items, each of which is a d-dimensional （d ≥ 3） hypercube with side length not greater than 1 and an infinite number of d-dimensional （d ≥ 3） hypercube bins with unit length on each side, we want to pack all of the items in the sequence into the minimum number of bins. The constraint is that only two bins are active at anytime during the packing process. Each item should be orthogonally packed without overlapping other items. Items are given in an online manner without the knowledge of or information about the subsequent items. We extend the technique of brick partitioning for square packing and obtain two results： a three-dimensional box and d-dimensional hyperbox partitioning schemes for cube and hypercube packing, respectively. We design 32 5.43-competitive and 32/21·2d-competitive algorithms for cube and hypercube packing, respectively. To the best of our knowledge these are the first known results on 2-space bounded cube and hypercube packing.

A Better Semi-online Algorithm for Q3/s_1=s_2≤s_3/C_(min) with the Known Largest Size

This paper investigates the semi-online machine covering problem on three special uniform machines with the known largest size. Denote by sj the speed of each machine, j = 1, 2, 3. Assume 0 〈 s1 = s2 = r 〈 t = s3, and let s = t/r be the speed ratio. An algorithm with competitive ratio max（2, 3s＋6/s＋6 is presented. We also show the lower bound is at least max（2, 38 3s/s＋6）. For s ≤ 6, the algorithm is an optimal algorithm with the competitive ratio 2. Besides, its overall competitive ratio is 3 which matches the overall lower bound. The algorithm and the lower bound in this paper improve the results of Luo and Sun.

An Intelligent Energy Management System for Large-Scale Charging of Electric Vehicles

The problem of large-scale charging of electric vehicles(EVs)with consumer-imposed charging deadlines is considered.An architecture for the intelligent energy management system(iEMS)is introduced.The iEMS consists of an admission control and pricing module,a scheduling module that determines the charging sequence,and a power dispatch module that draws power from a mix of storage,local renewable energy sources,and purchased power from the grid.A threshold admission and greedy scheduling(TAGS)policy is proposed to maximize operation profit.The performance of TAGS is analyzed and evaluated based on average and worst-case performance measures and the optimality of TAGS is established for some instances.Numerical simulations demonstrate that TAGS achieves noticeable performance gains over benchmark techniques.