Semi-online Scheduling Problems Research Articles

Semi-Online Machine Covering under a Grade of Service Provision

Two semi-online scheduling problems on two parallel identical machines under a grade of service (GoS) provision were studied. The goal is to maximize the minimum machine load. For the semi-online version where the largest processing time of all jobs is known in advance, we show that no competitive algorithm exists. For the semi-online version where the optimal offline value is known in advance, we propose an optimal algorithm with competitive ratio 2.

Optimal semi-online scheduling algorithms on two parallel identical machines under a grade of service provision

This paper investigates semi-online scheduling problems on two parallel identical machines under a grade of service (GoS) provision. We consider two different semi-online versions where the optimal offline value of the instance is known in advance or the largest processing time of all jobs is known in advance. Respectively for the two semi-online models, two algorithms with competitive ratios of 3/2 and (5+1)/2 are shown to be optimal.

Semi-online scheduling with known partial information about job sizes on two identical machines

In this paper we consider the semi-online scheduling problem with known partial information about job sizes on two identical machines, where all the jobs have processing times in the interval [ p , t p ] ( p > 0 , t ≥ 1 ) and the maximum job size is t p . The objective is to minimize the makespan. For 1 ≤ t < 4 3 and t ≥ 2 , we obtain lower bounds t + 1 2 and 4 3 on the optimal solution, respectively, which match the upper bounds given by He and Zhang (1999) in [2]. For 4 3 ≤ t < 2 , we prove that a lower bound on the optimal solution is max { 4 t + 4 3 t + 4 , 2 t t + 1 } and design an algorithm with a competitive ratio equal to this lower bound.

Optimal semi-online algorithm for scheduling with rejection on two uniform machines

In this paper we consider a semi-online scheduling problem with rejection on two uniform machines with speed 1 and s?1, respectively. A sequence of independent jobs are given and each job is characterized by its size (processing time) and its penalty, in the sense that, jobs arrive one by one and can be either rejected by paying a certain penalty or assigned to some machine. No preemption is allowed. The objective is to minimize the sum of the makespan of schedule, which is yielded by all accepted jobs and the total penalties of all rejected ones. Further, two rejection strategies are permitted thus an algorithm can propose two different schemes, from which the better solution is chosen. For the above version, we present an optimal semi-online algorithm H that achieves a competitive ratio ? H (s) as a piecewise function in terms of the speed ratio s.

An optimal semi-online algorithm for a single machine scheduling problem with bounded processing time

The single machine semi-online scheduling problem is considered with the assumption that the ratio of the longest processing time to the shortest one is not greater than γ. The goal is to minimize the total weighted completion time. We design a semi-online algorithm and prove that it has a competitive ratio of 1 + 4 γ 2 + 1 − 1 2 γ , which is also shown to be the best performance achieved by any deterministic semi-online algorithm for the problem.

A semi-online algorithm and its competitive analysis for a single machine scheduling problem with bounded processing times

The single machine semi-online scheduling problem with the objective ofminimizing total completion time is investigated with the assumption that theratio of the longest to the shortest processing time is not greater than aconstant $\gamma$. A semi-online algorithm is designed and its competitiveratio is proven to be $1+ \frac{\gamma - 1}{1 + \sqrt {1 + \gamma (\gamma -1)}}$. The competitive analysis method is as following: it starts from an arbitrary instance andmodifies the instance towards the possible structure of the worst-case instancewith respect to the given online algorithm. The modification guarantees thatthe performance ratio does not decrease. Eventually, it comes up with arelatively simple instance with a special structure, whose performance ratiocan be directly analyzed and serves as an upper bound on the competitive ratio.

OPTIMAL SEMI-ONLINE ALGORITHM FOR SCHEDULING ON A BATCH PROCESSING MACHINE

We consider two semi-online scheduling problems on a single batch (processing) machine with jobs' nondecreasing processing times and jobs' nonincreasing processing times, respectively. Our objective is to minimize the makespan. A batch processing machine can handle up to B jobs simultaneously. We study an unbounded model where B = ∞. The jobs that are processed together construct a batch, and all jobs in a batch start and complete at the same time. The processing time of a batch is given by the longest processing time of any job in the batch. Jobs arrive over time. Let pj denote the processing time of job Jj. Given job Jj and its following job Jj + 1, we assume that pj + 1 ≥ α pj, where α ≥ 1 is a constant number, for the first problem with jobs' nondecreasing processing times. For the second problem, we assume that pj + 1 ≤ α pj, where 0 &lt; α &lt; 1 is a constant number. We propose an optimal algorithm for both problems with a competitive ratio [Formula: see text] for the first problem and [Formula: see text] for the second problem.

Semi-online scheduling with known maximum job size on two uniform machines

In this paper, we investigate the semi-online scheduling problem with known maximum job size on two uniform machines with the speed ratio s?1. The objective is to minimize the makespan. Two algorithms are presented, where the first is optimal for $1\leq s\leq\sqrt{2}$ , and the second is optimal for 1.559?s?2 and $s\ge \frac{3+\sqrt{17}}{2}$ . In addition, the improvement on lower bounds is made for $2 .

Two semi-online scheduling problems on two uniform machines

This paper considers two semi-online scheduling problems, one with known optimal value and the other with known total sum, on two uniform machines with a machine speed ratio of s ≥ 1 . For the first problem, we provide an optimal algorithm for s ∈ [ 1 + 3 2 , 1 + 21 4 ] , and improved algorithms or/and lower bounds for s ∈ [ 1 + 21 4 , 3 ] , over which the optimal algorithm is unknown. As a result, the largest gap between the competitive ratio and the lower bound decreases to 0.02192. For the second problem, we also present algorithms and lower bounds for s ≥ 1 . The largest gap between the competitive ratio and the lower bound is 0.01762, and the length of the interval over which the optimal algorithm is unknown is 0.47382. Our algorithms and lower bounds for these two problems provide insights into their differences, which are unusual from the viewpoint of the known results on these two semi-online scheduling problems in the literature.

Optimal semi-online algorithms for preemptive scheduling problems with inexact partial information

In semi-online scheduling problems, we always assume that some partial additional information is exactly known in advance. This may not be true in some application. This paper considers semi-online problems on identical machines with inexact partial information. Three problems are considered, where we know in advance that the optimal value, or the largest job size are in given intervals, respectively, while their exact values are unknown. We give both lower bounds of the problems and competitive ratios of algorithms as functions of a so-called disturbance parameter r ∈[1, ∞). We establish for which r the inexact partial information is useful to improve the performance of a semi-online algorithm with respect to its pure online problem. Optimal preemptive semi-online algorithms are then obtained.

Semi-online scheduling problems on two identical machines with inexact partial information

In semi-online scheduling problems, we always assume that some partial additional information is exactly known in advance. This may not be true in some applications. This paper considers semi-online scheduling problems on two identical machines with inexact partial information. Three versions are considered, where we know in advance that the total size of all jobs, the optimal value, and the largest job size are in given intervals, respectively, while their exact values are unknown. We give both lower bounds of the problems and competitive ratios of algorithms as functions of a so-called disturbance parameter r ∈ [ 1 , ∞ ) . We establish for which r the inexact partial information is useful to improve the performance of a semi-online algorithm with respect to its pure online problem. Optimal or near optimal algorithms are then obtained.

Semi-online Machine Covering on Two Uniform Machines with Known Total Size

This paper investigates semi-online scheduling problem with known total size on two uniform machines for maximizing the minimum machine completion time. Lower bounds and optimal algorithms for every s≥1 are given, where s is the speed ratio of two machines. Both the overall competitive ratio and the competitive ratio for Open image in new window are strictly smaller than those of the optimal algorithms of the corresponding semi-online scheduling problem with known optimal value. It indicates that two related semi-online problems are different.

Extension of algorithm list scheduling for a semi-online scheduling problem

A general algorithm, called ALG, for online and semi-online scheduling problem Pm||Cmax with m ≥ 2 is introduced. For the semi-online version, it is supposed that all job have their processing times within the interval [p, rp], where p > 0,1 < r ≤ m/m − 1. ALG is a generalization of LS and is optimal in the sense that there is not an algorithm with smaller competitive ratio than that of ALG.

Semi On-Line Scheduling Problem for Maximizing the Minimum Machine Completion Time on Two Uniform Machines

In 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 \(\frac{2s+1}{s+1}\). We also claim the Cmin 2 algorithm is tight and the gap between the competitive ratio of Cmin 2 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.

Preemptive Semi–online Algorithms for Parallel Machine Scheduling with Known Total Size

This paper investigates preemptive semi-online scheduling problems on m identical parallel machines, where the total size of all jobs is known in advance. The goal is to minimize the maximum machine completion time or maximize the minimum machine completion time. For the first objective, we present an optimal semi–online algorithm with competitive ratio 1. For the second objective, we show that the competitive ratio of any semi–online algorithm is at least \( \frac{{2m - 3}} {{m - 1}} \) for any m > 2 and present optimal semi–online algorithms for m = 2, 3.

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 machinesM1,M2,M3 is known in advance. A speedsi(S1=1,s2=r,s3=s, 1≤r≤s) is associated with machineMi. Our goal is to maximizeCmin-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 algorithm cannot be improved and is the best possible for 1≤s≤2,r=1.

Optimal non-preemptive semi-online scheduling on two related machines

We consider the following non-preemptive semi-online scheduling problem. Jobs with non-increasing sizes arrive one by one to be scheduled on two uniformly related machines, with the goal of minimizing the makespan. We analyze both the optimal overall competitive ratio, and the optimal competitive ratio as a function of the speed ratio ( q ⩾ 1 ) between the two machines. We show that the greedy algorithm LPT has optimal competitive ratio 1 4 ( 1 + 17 ) ≈ 1.28 overall, but does not have optimal competitive ratio for every value of q. We determine the intervals of q where LPT is an algorithm of optimal competitive ratio, and design different algorithms of optimal competitive ratio for the intervals where it fails to be the best algorithm. As a result, we give a tight analysis of the competitive ratio for every speed ratio.

Preemptive semi-online scheduling with tightly-grouped processing times

This paper investigates a preemptive semi-online scheduling problem on m identical parallel machines where m = 2, 3. It is assumed that all jobs have their processing times in between p and rp (p > 0, r ≥ 1). The goal is to minimize the makespan. Best possible algorithms are designed for any r ≥ 1 when m = 2, 3.

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.

Optimal algorithms for semi-online preemptive scheduling problems on two uniform machines

This paper investigates two preemptive semi-online scheduling problems to minimize makespan on two uniform machines. In the first semi-online problem, we know in advance that all jobs have their processing times in between p and rp $(p > 0,r\geq 1)$. In the second semi-online problem, we know the size of the largest job in advance. We design an optimal semi-online algorithm which is optimal for every combination of machine speed ratio $s\geq 1$ and job processing time ratio $r\geq 1$ for the first problem, and an optimal semi-online algorithm for every machine speed ratio $s\geq 1$ for the second problem.