Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T14:26:29.686Z Has data issue: false hasContentIssue false
  • English
  • Français

Scheduling Independent Tasks to Minimize the Makespan on Identical Machines

Published online by Cambridge University Press:  27 July 2009

John Bruno ,
Edward G. Coffman Jr  and
Peter Downey
John Bruno
Affiliation:
Department of Computer Science, University of California, Santa Barbara, California 93106
Edward G. Coffman Jr
Affiliation:
AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, New Jersey 07974
Peter Downey
Affiliation:
Department of Computer Science, University of Arizona, Tucson, Arizona 85721
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider scheduling n tasks on m parallel machines where the task processing times are i.i.d. random variables with a common distribution function F. Scheduling is done by an a priori assignment of tasks to machines. We show that if the distribution function F is a Pólya frequency function of order 2 (decreasing reverse hazard rate) then the assignment that attempts to place an equal number of tasks on each machine achieves the stochastically smallest makespan among all assignments. The condition embraces many important distributions, such as the gamma and truncated normal distributions. Assuming that the task processing times have a common density that is a Pólya frequency function of order 2 (increasing likelihood ratio), then we find that flatter schedules have stochastically smaller makespans in the sense of the “joint” likelihood ratio.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 9 , Issue 3 , July 1995 , pp. 447 - 456
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing: Probability models. New York: Holt, Rinehart and Winston.Google Scholar
2.Chang, C.S. (1992). A new ordering for stochastic majorization: Theory and applications. Advances in Applied Probability 24: 604–634.CrossRefGoogle Scholar
3.Karlin, S. (1968). Totalpositivity.Vol. 1. Stanford, CA: Stanford University Press.Google Scholar
4.Makowski, A.M. & Nelson, R. (1993). Characterizing majorization on – NK, with applications to multiprocessor scheduling. Technical Report ISR 93-227, Institute for Systems Research, University of Maryland, College Park, MD.Google Scholar
5.Marshall, A.W. & Olkin, I. (1979). Inequalities: Theory of majorization and its applications. San Diego: Academic Press.Google Scholar
6.Ross, S.M. (1983). Stochastic processes. New York: John Wiley & Sons.Google Scholar
7.Schoenberg, I.J. (1951). On Pólya frequency functions, I. The totally positive functions and their Laplace transforms. Journal d'Analyse Mathimatique 1: 331–374.CrossRefGoogle Scholar
8.Shaked, M. & Shanthikumar, J.G. (1994). Stochastic orders and their applications. San Diego: Academic Press.Google Scholar
9.Shanthikumar, J.G. & Yao, D.D. (1991). Bivariate characterization of some stochastic order relations. Advances in Applied Probability 23: 642–659.CrossRefGoogle Scholar