Abstract
We consider a stochastic scheduling problem in which n jobs are to be scheduled on m identical processors which operate in parallel. The processing times of the jobs are not known in advance but they have known distributions with hazard rates ρ 1, (t), …, ρ n (t). It is desired to minimize the expected value of к(C), where C i is the time at which job i is completed C = (C 1, …, C n ), and к(C) is increasing and concave in C. Suppose processor i first becomes available at time τ i . We prove that if there is a single static list priority policy which is optimal for every τ = (τ 1, …, τ m ), then the minimal expected cost must be increasing and concave in τ. Moreover, if к(C) is supermodular in C then this cost is also supermodular in τ. This result is used to prove that processing jobs according to the static list priority order (1,2,…,n) minimizes the expected value of ∑w i h(C i ), when h(·) is a nondecreasing, concave function, w 1 ≥ … ≥ w n , and ρ 1 (t 1)w 1 ≥ … ≥ ρ n (t n )w n for all t 1, …, t n .
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Bruno and P. Downey And G. N. Frederickson, Sequencing tasks with exponential service times to minimize the expected flowtime or makespan, J. Ass. Comp. Mach., 28 (1981), pp. 100–113.
K. D. Glazebrook, Scheduling tasks with exponential service times on parallel processors, J. Appl. Prob., 16 (1979), pp. 658–689.
T. Kampke, Optimalitatsaussagen fur Spezeille Stochastische Schedulingprobleme, Diploma-Mathematiker, Rheinisch-Westfalischen Technischen Hochschule Aachen, 1985.
T. Kampke, On the optimality of static priority policies in stochastic scheduling on parallel machines,J. Appl. Prob., 24 (1987) (to appear).
L. Lovasz, Submodular functions and convexity, Mathematical Programming, the State of the Art, eds A. Bachem et al, Springer, Berlin, 1983, pp. 235–257.
R. R. Weber and P. Nash, An optimal strategy in multi-server stochastic scheduling, J. R. Statist. Soc., B 40 (1979), pp. 323–328.
R. R. Weber, Scheduling jobs with stochastic processing requirements on parallel machines to minimize makespan or flowtime, J. Appl. Prob., 19 (1982), pp. 167–182.
R. R. Weber and P. Varaiya and J. Walrand, Scheduling jobs with stochastically ordered processing times on parallel machines to minimize expected flowtime, J. Appl. Prob., 23 (1986), pp. 841–847.
G. Weiss and M. Pinedo, Scheduling tasks with exponential service times on non-identical processors to minimize various cost functions, J. Appl. Prob., 17 (1980), pp. 187–202.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Springer-Verlag New York Inc.
About this paper
Cite this paper
Weber, R.R. (1988). Stochastic Scheduling on Parallel Processors and Minimization of Concave Functions of Completion Times. In: Fleming, W., Lions, PL. (eds) Stochastic Differential Systems, Stochastic Control Theory and Applications. The IMA Volumes in Mathematics and Its Applications, vol 10. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8762-6_34
Download citation
DOI: https://doi.org/10.1007/978-1-4613-8762-6_34
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-8764-0
Online ISBN: 978-1-4613-8762-6
eBook Packages: Springer Book Archive