Scheduling Jobs with Stochastically Ordered Processing Times on Parallel Machines to Minimize Expected Flowtime Author(s): R. R. Weber, P. Varaiya, J. Walrand Source: Journal of Applied Probability, Vol. 23, No. 3, (Sep., 1986), pp. 841-847 Published by: Applied Probability Trust Stable URL: http://www.jstor.org/stable/3214023 Accessed: 29/07/2008 19:31 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=apt. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
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J. Appl.Prob.23, 841-847(1986) Printedin Israel ? AppliedProbabilityTrust1986
SCHEDULING JOBS WITH STOCHASTICALLY ORDERED PROCESSING TIMES ON PARALLEL MACHINES TO MINIMIZE EXPECTED FLOWTIME
R. R. WEBER,* Universityof Cambridge P. VARAIYA,** Universityof California,Berkeley J. WALRAND,** Universityof California,Berkeley
Abstract A numberof jobs are to be processedusinga numberof identicalmachines which operate in parallel.The processingtimes of the jobs are stochastic,but have known distributionswhich are stochasticallyordered. A reward r(t) is acquiredwhen a job is completedat time t. The functionr(t) is assumedto be convex and decreasingin t. It is shown that withinthe class of non-preemptive schedulingstrategiesthe strategySEPTmaximizesthe expected total reward. This strategy is one which whenever a machine becomes available starts processingthe remainingjob with the shortest expected processingtime. In particular,for r(t)= - t, this strategyminimizesthe expected flowtime. STOCHASTIC SCHEDULING
1. Introduction The processing times of n jobs are random variables which are stochastically ordered as Xi, T2- T){r(T2+
c)+
, + c, T2, ', R,(T2 + c
Tm; L*)}
+, X
,
m
;L*)}]
and dR(T;Lk';
c)/dc
= i(T,+
R+ (T + c,2, C,
c
- E[i(Tl
+c)+
*?T?
;m;(1)+
R1,(T + c, T2+ XI,,
L*)
*,
; L*)],
where the inequality follows from Lemma 3. Thus dA(c)/dc
_ E[1(XI < r2 + 1(X, - {r(T
l){r(Ti
+ XI
-2T - TI){r(T2+
+ C)+ R,(rT + X, +
C, T2,
C)+ Ri(T2 + C T , X *,
;L*)}
*,
Tm
T
; L*)}
+ C)+ RI(TI + C, 2+ XI, * T, Tm ; L*)}].
Using part (a) of Lemma 2, Tr,_ T2 and i(t) non-decreasing, it is easy to check that the expression over which the above expectation is taken is non-negative. This completes the proof of the theorem. 3. Discussion We have shown that when the jobs have processing times which are stochastically ordered, then the non-preemptive SEPT strategy maximizes the
Schedulingjobswithstochasticallyorderedprocessingtimes
847
expected rewardwithin the class of non-preemptivestrategies. Examiningthe proof of Theorem 1, particularlyLemma 3, it can be seen that the result is still true for some models in which the reward obtained on completing each job differs from job to job. The reward obtained upon completing job i may be generalizedto any convex, decreasingfunction ri(t), providedthat when job i is stochastically shorter than job j the inequality ri(t)
