Preemptive Scheduling of Stochastic Jobs with a Two-Stage Processing Time Distribution on M + 1 Parallel Machines

1992 ◽  
Vol 6 (2) ◽  
pp. 171-191 ◽  
Author(s):  
Chueng-Chiu Huang ◽  
Gideon Weiss
Keyword(s):  
Processing Time ◽  
Parallel Machines ◽  
Optimal Solution ◽  
General Distribution ◽  
Constant Independent ◽  
Expected Number ◽  
Total Flowtime ◽  
Optimal Decisions ◽  
Identical Machines ◽  
Parallel Identical Machines

We analyze the optimal preemptive sequencing of n jobs on M + 1 parallel identical machines to minimize expected total flowtime. The running times of the jobs are independent samples from the distribution Pr(X = H) = p, Pr(X = H + T) = 1 − p, where H, T are random variables of general distribution. Preemption of a job is allowed when H is completed. This problem does not have a simple optimal solution. We show that the scheme of shortest expected remaining processing time first (SERPT) is close to optimal in two senses. The expected flowtime under SERPT and under the optimal policy differ by no more than a constant, independent of the number of jobs, and the expected number of optimal decisions that are not according to SERPT is bounded by a constant, independent of the number of jobs.

SCHEDULING ON TWO PARALLEL MACHINES WITH TWO DEDICATED SERVERS

2017 ◽  
Vol 58 (3-4) ◽  
pp. 314-323
Author(s):  
YIWEI JIANG ◽  
PING ZHOU ◽  
HUIJUAN WANG ◽  
JUELIANG HU
Keyword(s):  
Processing Time ◽  
Parallel Machines ◽  
List Scheduling ◽  
Np Hard ◽  
Worst Case ◽  
Identical Machines ◽  
Loading And Unloading ◽  
Parallel Identical Machines ◽  
Dedicated Servers ◽  
Nonpreemptive Scheduling

We study a nonpreemptive scheduling on two parallel identical machines with a dedicated loading server and a dedicated unloading server. Each job has to be loaded by the loading server before being processed on one of the machines and unloaded immediately by the unloading server after its processing. The loading and unloading times are both equal to one unit of time. The goal is to minimize the makespan. Since the problem is NP-hard, we apply the classical list scheduling and largest processing time heuristics, and show that they have worst-case ratios, $8/5$ and $6/5$, respectively.

scholarly journals Scheduling jobs with stochastically ordered processing times on parallel machines to minimize expected flowtime

1986 ◽  
Vol 23 (03) ◽  
pp. 841-847 ◽  
Author(s):  
R. R. Weber ◽  
P. Varaiya ◽  
J. Walrand
Keyword(s):  
Processing Time ◽  
Parallel Machines ◽  
Preemptive Scheduling ◽  
Total Reward ◽  
Processing Times ◽  
Identical Machines ◽  
Scheduling Strategies

A number of jobs are to be processed using a number of identical machines which operate in parallel. The processing times of the jobs are stochastic, but have known distributions which are stochastically ordered. A reward r(t) is acquired when a job is completed at time t. The function r(t) is assumed to be convex and decreasing in t. It is shown that within the class of non-preemptive scheduling strategies the strategy SEPT maximizes the expected total reward. This strategy is one which whenever a machine becomes available starts processing the remaining job with the shortest expected processing time. In particular, for r(t) = – t, this strategy minimizes the expected flowtime.

scholarly journals Closing the Gap for Makespan Scheduling via Sparsification Techniques

2020 ◽  
Vol 45 (4) ◽  
pp. 1371-1392 ◽  
Author(s):  
Klaus Jansen ◽  
Kim-Manuel Klein ◽  
José Verschae
Keyword(s):  
Parallel Machines ◽  
Optimal Solution ◽  
Structural Result ◽  
Packing Problems ◽  
Running Time ◽  
Linear Dependency ◽  
Identical Machines ◽  
Programming Techniques ◽  
Closing The Gap ◽  
Related Machines

Makespan scheduling on identical machines is one of the most basic and fundamental packing problems studied in the discrete optimization literature. It asks for an assignment of n jobs to a set of m identical machines that minimizes the makespan. The problem is strongly NP-hard, and thus we do not expect a ([Formula: see text])-approximation algorithm with a running time that depends polynomially on [Formula: see text]. It has been recently shown that a subexponential running time on [Formula: see text] would imply that the Exponential Time Hypothesis (ETH) fails. A long sequence of algorithms have been developed that try to obtain low dependencies on [Formula: see text], the better of which achieves a quadratic running time on the exponent. In this paper we obtain an algorithm with an almost-linear dependency on [Formula: see text] in the exponent, which is tight under ETH up to logarithmic factors. Our main technical contribution is a new structural result on the configuration-IP integer linear program. More precisely, we show the existence of a highly symmetric and sparse optimal solution, in which all but a constant number of machines are assigned a configuration with small support. This structure can then be exploited by integer programming techniques and enumeration. We believe that our structural result is of independent interest and should find applications to other settings. We exemplify this by applying our structural results to the minimum makespan problem on related machines and to a larger class of objective functions on parallel machines. For all these cases, we obtain an efficient PTAS with running time with an almost-linear dependency on [Formula: see text] and polynomial in n.

Scheduling Stochastic Jobs with a Two-Point Distribution on Two Parallel Machines

1989 ◽  
Vol 3 (1) ◽  
pp. 89-116 ◽  
Author(s):  
E.G. Coffman ◽  
M. Hofri ◽  
G. Weiss
Keyword(s):  
Optimal Policy ◽  
Hazard Rate ◽  
Processing Time ◽  
Parallel Machines ◽  
Flow Time ◽  
Constant Independent ◽  
Total Flow ◽  
Point Distribution ◽  
Two Machines ◽  
Expected Increase

We analyze the optimal preemptive sequencing of n jobs on two machines to minimize expected total flow time. The running times of the jobs are independent samples from the distribution Pr(X = 1) = p, Pr(X = κ + 1) = 1 − p. We verify that the shortest-expected-remaining-processing-time (SERPT) policy, which is optimal for independent and identically distributed (i.i.d.) running times with a monotone hazard-rate distribution, is not optimal for this distribution. However, we prove that if p ≥ 1/κ, then the number of decisions where SERPT and an optimal policy disagree is bounded by a constant independent of n. For p < 1/k, we prove that the expected number of such decisions has a similar bound. In addition, bounds on the expected increase in flow times under SERPT are derived; these bounds are also independent of n.

scholarly journals Scheduling preemptable jobs on identical processors under varying availability of an additional continuous resource

2016 ◽  
Vol 26 (3) ◽  
pp. 693-706 ◽  
Author(s):  
Rafał Różycki ◽  
Grzegorz Waligóra ◽  
Jan Węglarz
Keyword(s):  
Processing Speed ◽  
Optimal Solution ◽  
Renewable Resource ◽  
Mathematical Programming Problem ◽  
Scheduling Problems ◽  
Two Phase ◽  
Schedule Length ◽  
Identical Machines ◽  
Parallel Identical Machines ◽  
Continuous Scheduling

Abstract In this work we consider a problem of scheduling preemptable, independent jobs, characterized by the fact that their processing speeds depend on the amounts of a continuous, renewable resource allocated to jobs at a time. Jobs are scheduled on parallel, identical machines, with the criterion of minimization of the schedule length. Since two categories of resources occur in the problem: discrete (set of machines) and continuous, it is generally called a discrete-continuous scheduling problem. The model studied in this paper allows the total available amount of the continuous resource to vary over time, which is a practically important generalization that has not been considered yet for discrete-continuous scheduling problems. For this model we give some properties of optimal schedules on a basis of which we propose a general methodology for solving the considered class of problems. The methodology uses a two-phase approach in which, firstly, an assignment of machines to jobs is defined and, secondly, for this assignment an optimal continuous resource allocation is found by solving an appropriate mathematical programming problem. In the approach various cases are considered, following from assumptions made on the form of the processing speed functions of jobs. For each case an iterative algorithm is designed, leading to an optimal solution in a finite number of steps.

Semi-Online Machine Covering under a Grade of Service Provision

2011 ◽  
Vol 101-102 ◽  
pp. 484-487
Author(s):  
Yong Wu ◽  
Min Ji ◽  
Qi Fan Yang
Keyword(s):  
Service Provision ◽  
Competitive Ratio ◽  
Processing Time ◽  
Optimal Algorithm ◽  
Online Version ◽  
Scheduling Problems ◽  
Grade Of Service ◽  
Identical Machines ◽  
Machine Covering ◽  
Parallel Identical Machines

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.

scholarly journals Scheduling jobs with stochastically ordered processing times on parallel machines to minimize expected flowtime

1986 ◽  
Vol 23 (3) ◽  
pp. 841-847 ◽  
Author(s):  
R. R. Weber ◽  
P. Varaiya ◽  
J. Walrand
Keyword(s):  
Processing Time ◽  
Parallel Machines ◽  
Preemptive Scheduling ◽  
Total Reward ◽  
Processing Times ◽  
Identical Machines ◽  
Scheduling Strategies

A number of jobs are to be processed using a number of identical machines which operate in parallel. The processing times of the jobs are stochastic, but have known distributions which are stochastically ordered. A reward r(t) is acquired when a job is completed at time t. The function r(t) is assumed to be convex and decreasing in t. It is shown that within the class of non-preemptive scheduling strategies the strategy SEPT maximizes the expected total reward. This strategy is one which whenever a machine becomes available starts processing the remaining job with the shortest expected processing time. In particular, for r(t) = – t, this strategy minimizes the expected flowtime.

scholarly journals Parallel machines scheduling with time-dependent deterioration, using meta-heuristic algorithms

2021 ◽  
Vol 3 (3) ◽  
Author(s):  
Jaber Kalaki Juybari ◽  
Somayyeh Kalaki Juybari ◽  
Reza Hasanzadeh
Keyword(s):  
Large Scale ◽  
Processing Time ◽  
Parallel Machines ◽  
Heuristic Algorithms ◽  
Optimal Solution ◽  
Time Dependent ◽  
Mixed Integer ◽  
Small Scale ◽  
Simulating Annealing ◽  
Parallel Machines Scheduling

AbstractIn this paper, we consider the identical parallel machines scheduling problem with exponential time-dependent deterioration. The meaning of time-dependent deterioration is that the processing time of a job is not a constant and depends on the scheduled activities. In other words, when a job is processed later, it needs more processing time compared to the jobs processed earlier. The main purpose is to minimize the makespan. To reach this aim, we developed a mixed integer programming formulation for the problem. We solved problem in small scale using GAMS software, while due to the fact that in larger scales the aforesaid case is a complex and intricate optimized problem which is NP-hard, it is not possible to solve it by standard calculating techniques (in logical calculating times); we applied the meta-heuristic genetic algorithm, simulating annealing and artificial immune system, and their performance has been evaluated. In the end, we showed that solving the problem in small scale, with the meta-heuristic algorithms (GA, SA, and AIS) equals the optimal solution (GAMS), And on a large scale, at a time of approximately equal solution, meta-heuristic algorithm simulating annealing, provides a more optimal solution.

scholarly journals Semi-Online Scheduling on Two Machines withGoSLevels and Partial Information of Processing Time

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Taibo Luo ◽  
Yinfeng Xu
Keyword(s):  
Processing Time ◽  
Parallel Machines ◽  
Partial Information ◽  
Online Scheduling ◽  
Scheduling Problems ◽  
Total Processing Time ◽  
Grade Of Service ◽  
Two Machines

This paper investigates semi-online scheduling problems on two parallel machines under a grade of service (GoS) provision subject to minimize the makespan. We consider three different semi-online versions with knowing the total processing time of the jobs with higherGoSlevel, knowing the total processing time of the jobs with lowerGoSlevel, or knowing both in advance. Respectively, for the three semi-online versions, we develop algorithms with competitive ratios of3/2,20/13, and4/3which are shown to be optimal.

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