Approximation in Stochastic Scheduling: The Power of LP-Based Priority Policies

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Walter McKenzie
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1 Approxiation in Stochastic Scheduling: The Power of -Based Priority Policies Rolf Möhring, Andreas Schulz, Marc Uetz Setting (A P p stoch, r E( w and (B P p stoch E( w We will assue that the processing ties are independent, and that we now the realization of the processing tie, once a ob is finished. Note that we want a non anticipatory policy (decision at tie t can only be based on the inforation fro the past up to tie t, i.e. the set of obs already finished or being perfored at t, their start ties, and the conditional distribution of reaining processing ties. (Our policy will be an even further special case. We will follow an -based approach, the with the copletion ties as decision variables ost of the approach and the leas are extensions of the deterinistic case. (Note: so we are optiizing over the perforance space, not the decisions directly. We will use the solution to define the order of the obs in case (A, and ust as a bound on the optial solution in case (B. Input nuber of achines nuber of obs for each ob : release date r weight w expected processing tie p an upper bound on the coefficient of variation of all processing tie distributions,, i.e., such that Varp E(p for all. (Interpretation: standard deviation of noralized processing ties, where noralized eans scaled such that the expectation is. We will copare the algorith that aes use of this liited info only, with any algorith that has full inforation about the processing ties.

2 3 Linear Progra iniize w E Want: subect to {E } such that these are expected copletion ties with respect to soe non anticipatory policy. But we ll settle for a relaxation. Valid inequalities: We now that for deterinistic scheduling we have p ( p + p for all subsets of obs A. For stochastic scheduling with a non anticipatory policy Π, we get E(p E( Π ( E(p + (E(p Var(p for all subsets of obs A. Proof onsider any policy Π, and fix a realization of {p }. Let S denote the starting tie of ob. The deterinistic inequalities give p (S + p ( p + p for all subsets of obs A. We are going to tae expectations, and note that processing ties are independent rando variables E(p i p = E(p i E(p for all i policy is non anticipatory S and p are independent E(S p = E(S E(p Var(p = E(p E(p. So first rewrite p S Taing expectations: i,,i ( p i p + + p for all subsets of obs A. }{{} = E(p E(S = i,,i ( i E(p i E(p E(p i E(p E(p }{{} = Var(p E(p Var(p. Finally E( = E(S + E(p and we re done. When using, we can write

3 ( E(p E( ( ( + E(p + (E(p. Also, we have the trivial inequalities ( E( r + E(p. 4 Algorith with release dates We are going to consider the following algorith (policy Π: Solve the iniize w E subect to ( and ( call optial solution { }. Next, we are going to schedule the obs in the order of increasing expected copletion tie, allowing for idle tie when there is a ob that is available, but a different ob has priority and is not yet available. lai Π is a (3 / + ax{, (( / }-approxiation. Proof Renuber obs such that... n. Fix a realization of {p }. Note that Π ax r + p + p =,,..., = since we are considering ob-based priority scheduling policies (consider sae policy but start the first ob at ax r then no idle tie; bound holds. Note that r, and since... n, we have ax =,,..., r. So, that fact and taing expectations, gives E Π Also, Ep, so E Π ( + Ep + Ep. = + Ep. = So we want to bound = Ep. Note that we have inequalities (, and... n, therefore Ep = = E(p ( = ( + E(p + (E(p. = E(p + = ( + = (E(p = Ep. 3

4 We now concern two cases. (case ( + 0. Now = E(p and so E Π ( (case ( + 0. Note now that + Ep ( 4. = Ep and... n, therefore ax Ep = ax =,..., Ep = Ep =,..., = Ep = (Ep = Ep. So = E(p + = E(p ( + = (E(p = Ep ( + E(p + = ( + E Π ( + Ep = ( + ( = (3 + ( + Putting it together: we have an 3 +ax{, }-approxiation (for each separately, and so also for the weighted su, since we can solve the in O(n operations (see appendix B of paper. 5 Result P p stoch E( w (no release dates Sae approach gives + ax{, }-approxiation. But iniu ratio rule (when achine becoes idle, schedule ob with lowest Ep /w ratio gives + ( + ( /(-approxiation. Approach: Renuber obs such that Ep /w Ep /w... Ep n /w n. iniize w E subect to ( Solution (given by Edonds greedy algorith: = Ep = ( ( Ep. 4

5 (Here the fact that Ep /w Ep /w... Ep n /w n is used. Sae to start of proof of result we now so Π ax r + p + p = p + p =,,..., = E Π Ep + Ep = = = + = ( Ep ( ( + Ep ( + ( Ep Note that we don t have inequalities (, so we do not necessarily have (and thus can t conclude E ( Π + ( +(. We can say soething about the weighted su though: w E Π w ( + ( + w Ep ( + ( ( + (OPT ( + ( + ( (OPT Note: for single achine Sallest Ratio Rule (sort obs in order of non-decreasing Ep /w is optial already nown []. References [] Rolf H. Möhring, Andreas S. Schulz, and Marc Uetz. Approxiation in stochastic scheduling: the power of -based priority policies. J. AM, 46(6:94 94, 999. [] Michael H. Rothopf. Scheduling independent tass on parallel processors. Manageent Sci., : ,

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