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

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.

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The Stochastic Optimality of SEPT in Parallel Machine Scheduling

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800003326 ◽

1994 ◽

Vol 8 (2) ◽

pp. 179-188 ◽

Cited By ~ 2

Author(s):

Cheng-Shang Chang ◽

Arie Hordijk ◽

Rhonda Righter ◽

Gideon Weiss

Keyword(s):

Processing Time ◽

Parallel Machines ◽

Concave Function ◽

Parallel Machine Scheduling ◽

Parallel Machine ◽

Preemptive Scheduling ◽

Processing Times ◽

Schur Concave ◽

Completion Times ◽

Stochastic Optimality

We consider preemptive scheduling on parallel machines where processing times of jobs are i.i.d. but jobs may already have received distinct amounts of service. We show that when processing times are increasing in likelihood ratio, SEPT (shortest expected [remaining] processing time first) stochastically minimizes any increasing and Schur-concave function of the job completion times. The same result holds when processing times are exponential with possibly different means.

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On almost optimal priority rules for preemptive scheduling of stochastic jobs on parallel machines

Advances in Applied Probability ◽

10.1017/s0001867800027166 ◽

1995 ◽

Vol 27 (03) ◽

pp. 821-839 ◽

Cited By ~ 3

Author(s):

Gideon Weiss

Keyword(s):

Single Machine ◽

Parallel Machines ◽

Preemptive Scheduling ◽

Priority Rule ◽

Priority Rules ◽

Holding Costs ◽

Processing Times ◽

Stochastic Processing ◽

Stochastic Processing Times

We consider scheduling a batch of jobs with stochastic processing times on single or parallel machines, with the objective of minimizing the expected holding costs. Preemption of jobs is allowed, and the holding costs of preempted jobs may depend on the stage of completion. We provide a new proof of the optimality of a Gittins priority rule for the single machine and use the same proof to show that the Gittins priority rule is nearly optimal for parallel machines.

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On almost optimal priority rules for preemptive scheduling of stochastic jobs on parallel machines

Advances in Applied Probability ◽

10.2307/1428135 ◽

1995 ◽

Vol 27 (3) ◽

pp. 821-839 ◽

Cited By ~ 18

Author(s):

Gideon Weiss

Keyword(s):

Single Machine ◽

Parallel Machines ◽

Preemptive Scheduling ◽

Priority Rule ◽

Priority Rules ◽

Holding Costs ◽

Processing Times ◽

Stochastic Processing ◽

Stochastic Processing Times

We consider scheduling a batch of jobs with stochastic processing times on single or parallel machines, with the objective of minimizing the expected holding costs. Preemption of jobs is allowed, and the holding costs of preempted jobs may depend on the stage of completion. We provide a new proof of the optimality of a Gittins priority rule for the single machine and use the same proof to show that the Gittins priority rule is nearly optimal for parallel machines.

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

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800002436 ◽

1992 ◽

Vol 6 (2) ◽

pp. 171-191 ◽

Cited By ~ 3

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.

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Inequalities and bounds for the scheduling of stochastic jobs on parallel machines

Journal of Applied Probability ◽

10.2307/3213879 ◽

1985 ◽

Vol 22 (3) ◽

pp. 739-744 ◽

Cited By ~ 6

Author(s):

Michael Pinedo ◽

Zvi Schechner

Keyword(s):

Processing Time ◽

Parallel Machines ◽

Random Variable ◽

Expected Time ◽

Processing Times ◽

Set Up ◽

Expected Makespan

Consider n jobs and m machines. The m machines are identical and set up in parallel. All n jobs are available at t = 0 and each job has to be processed on one of the machines; any one can do. The processing time of job j is Xj, a random variable with distribution Fj. The sequence in which the jobs start with their processing is predetermined and preemptions are not allowed. We investigate the effect of the variability of the processing times on the expected makespan and the expected time to first idleness. Bounds are presented for these quantities in case the distributions of the processing times of the jobs are new better (worse) than used.

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A bicriterion approach to preemptive scheduling of parallel machines with controllable job processing times

Discrete Applied Mathematics ◽

10.1016/0166-218x(94)00071-5 ◽

1995 ◽

Vol 63 (3) ◽

pp. 237-256 ◽

Cited By ~ 27

Author(s):

Eugeniusz Nowicki ◽

Stanisław Zdrzałka

Keyword(s):

Parallel Machines ◽

Preemptive Scheduling ◽

Processing Times

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Stochastic scheduling for a two-machine open shop

Journal of Applied Probability ◽

10.1017/s002190020010138x ◽

1997 ◽

Vol 34 (03) ◽

pp. 733-744

Author(s):

Rhonda Righter

Keyword(s):

Processing Time ◽

Stochastic Scheduling ◽

Open Shop ◽

Preemptive Scheduling ◽

Processing Times ◽

Dependent Rate ◽

One Machine

We study the problem of preemptive scheduling of jobs in a two-machine open shop. Jobs require processing on both machines, but the order does not matter. We define the D-LERPT (double longest expected remaining processing time) policy as the policy that first processes jobs that have not yet been processed by either machine (double jobs), in decreasing order of expected remaining processing times, and then processes jobs that require processing on only one machine in any order. We show that D-LERPT stochastically minimizes the makespan when preemption is not permitted and jobs (but not machines) are stochastically identical, and that D-LERPT minimizes the makespan in the increasing convex sense when preemption is permitted and the machines are stochastically identical and processing times are exponential or geometric with a job dependent rate.

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Scheduling with Discretely Compressible Processing Times to Minimize Makespan

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.575.926 ◽

2014 ◽

Vol 575 ◽

pp. 926-930

Author(s):

Shu Xia Zhang ◽

Yu Zhong Zhang

Keyword(s):

Dynamic Programming ◽

Polynomial Time ◽

Processing Time ◽

Parallel Machines ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Identical Parallel Machines ◽

Processing Times ◽

Release Times ◽

Scheduling Model

In this paper, we address the scheduling model with discretely compressible processing times, where processing any job with a compressed processing time incurs a corresponding compression cost. We consider the following problem: scheduling with discretely compressible processing times to minimize makespan with the constraint of total compression cost on identical parallel machines. Jobs may have simultaneous release times. We design a pseudo-polynomial time algorithm by approach of dynamic programming and an FPTAS.

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SCHEDULING ON TWO PARALLEL MACHINES WITH TWO DEDICATED SERVERS

The ANZIAM Journal ◽

10.1017/s1446181117000141 ◽

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.

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