Stochastic Online Scheduling on Parallel Machines

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1 Stochastic Online Scheduling on Parallel Machines Nicole Megow 1, Marc Uetz 2, and Tark Vredeveld 3 1 Technische Universit at Berlin, Institut f ur Matheatik, Strasse des 17. Juni 136, Berlin, Gerany negow@ath.tu-berlin.de 2 Maastricht University, Departent of Quantitative Econoics, P.O.Box 616, 6200 MD Maastricht, The Netherlands.uetz@ke.uniaas.nl 3 Konrad-Zuse-Zentru f ur Inforationstechnik Berlin, Departent Optiization, Takustr. 7, Berlin, Gerany vredeveld@zib.de Abstract. We consider a non-preeptive, stochastic parallel achine scheduling odel with the goal to iniize the weighted copletion ties of obs. In contrast to the classical stochastic odel where obs with their processing tie distributions are known beforehand, we assue that obs appear one by one, and every ob ust be assigned to a achine online. We propose a siple online scheduling policy for that odel, and prove a perforance guarantee that atches the currently best known perforance guarantee for stochastic parallel achine scheduling. For the ore general odel with ob release dates we derive an analogous result, and for NBUE distributed processing ties we even iprove upon the previously best known perforance guarantee for stochastic parallel achine scheduling. Moreover, we derive soe lower bounds on approxiation. 1 Introduction Non-preeptive parallel achine scheduling to iniize the weighted copletion ties of obs, P w C in the three- eld notation of Graha et al. [6], is one of the classical probles in scheduling theory. This proble plays a role whenever any obs ust be processed on a liited nuber of achines, with typical applications, e.g., in parallel coputing [2] or copiler optiization [3]. The ain characteristic of the odel of stochastic scheduling is the fact that the processing ties of obs are subect to uctuations, and becoe known only upon copletion of the obs. Their respective distributions are assued to be given beforehand. This usually requires the notion of scheduling policies instead of siple schedules. Research partially supported by the DFG Research Center Matheatics for key technologies (FZT 86) in Berlin. G. Persiano and R. Solis-Oba (Eds.): WAOA 2004, LNCS 3351, pp , c Springer-Verlag Berlin Heidelberg 2005

2 168 N. Megow, M. Uetz, and T. Vredeveld Stochastic scheduling. Stochastic achine scheduling odels have been addressed ainly since the 1980s [4]. Let us brie y recall the concept of a scheduling policy as introduced by M ohring et al. [10]. Roughly spoken, at any tie t, such a policy speci es which action to perfor, in particular which obs to start at t.in order to decide, it ay utilize the coplete inforation contained in the partial schedule up to tie t. However, it ust not utilize any inforation about the future. An optial scheduling policy is one that iniizes the obective function value in expectation. Notice that, in general, a scheduling policy need not yield a xed assignent of obs to achines. With the exception of the papers by Weiss [18, 19], the rst approxiation algoriths for stochastic achine scheduling have been derived only recently [11, 13, 14, 16]. In the papers [11, 14], the expected perforance of a linear prograing based list scheduling policy is copared against the expected perforance of an optial scheduling policy. The results are constant-factor approxiations for probles with or without release dates [11], and also with precedence constraints [14]. The approach is based upon the solution of linear prograing relaxations, and for the odels with release dates or precedence constraints, their solutions are used in order to de ne corresponding list scheduling policies. Recently, another type of analysis has been pursued by Steger et al. in the papers [13, 16], where the expected ratio of the perforance of the (W)SEPT rule 1 over the optiu solution is analyzed. This approach ay indeed have advantages over the previous approach, naely in ters of averaging over different realizations of processing ties, and we refer to [13, 16] for a discussion. One of the ain differences, however, is the fact that it uses a coparison against the off-line optiu, whereas the approach in [11, 14] copares against the on-line optiu. Nevertheless, restricted to odels without release dates or precedence constraints, constant-factor approxiation results for the expected ratio have been obtained for the (W)SEPT rule on parallel achines [13, 16]. Stochastic online scheduling. In this paper, we follow the approach taken by M ohring et al. [11]. In other words, we copare the expected outcoe of a certain scheduling policy against the expected outcoe of an optial scheduling policy. In contrast to the previously entioned work on stochastic scheduling, however, we consider a odel where obs have to be assigned to achines online. More precisely, obs are presented to the scheduler one by one, with their weights w and expected processing ties E [P ], and without knowledge about obs that ight appear in the future, or their nuber, they ust be assigned to a achine. This assignent cannot be revised later. Once all obs have been assigned this way, there is freedo concerning the scheduling of obs on every single achine; of course, still within the restrictions that obs ust not be preepted, and 1 In the WSEPT rule, obs are scheduled greedily in the order of non-increasing ratios w /E [P ], where w is the weight of ob, P its processing tie distribution, and E [P ] is its expectation. For unit weights, this equals SEPT; shortest expected processing tie rst.

3 Stochastic Online Scheduling on Parallel Machines 169 that their actual processing ties becoe known only upon copletion. For convenience, let us denote this odel Sos, for stochastic online scheduling. Discussion of the odel. As a atter of fact, the solution of LP relaxations is crucial for the work of M ohring et al. [11] or Skutella and Uetz [14]. For odels with release dates or precedence constraints, optial LP solutions are not only required for the purpose of analysis, but also to de ne the corresponding list scheduling policies. In order to set up these LP relaxations, it is required to know beforehand the set of obs, their expected processing ties E [P ], as well as a unifor upper bound on the squared coefficient of variation of all processing ties distributions CV [P ] 2 = Var [P ]/E [P ] 2 for all obs. One critique of this approach is the fact that in practical applications, parts of this data ight not be available. Even worse, in an online setting there is no knowledge about obs that ight appear in the future. In that case, algoriths that rst require the solution of sophisticated LP relaxations ight be useless. The Sos odel as proposed in this paper can be seen as a rst step in the direction of sipler, cobinatorial algoriths for odels with stochastic processing ties. It is a two-phase odel, where the rst phase consists of an online assignent of obs to achines. In this phase, whenever a ob is presented to the scheduler, the only inforation that is disclosed is its weight w and its expected processing tie E [P ]. In the odel with release dates, it is also the release tie r. The second phase consists of the actual process of scheduling the obs over tie, processing ties being realized according to the respective distributions. Yet, we copare the expected outcoe of the online stochastic scheduling policy to the expected outcoe of an optial scheduling policy, according to the de nition of general scheduling policies by M ohring et al. [10]. In coparison to classical online odels, we ake two rearks. First, like in classical online optiization, the adversary in the Sos odel ay choose an arbitrary sequence of obs in the rst phase. These obs are assued to be stochastic, with corresponding processing tie distributions (deterinistic obs being a special case). However, in the second phase, the actual processing ties are realized according to the exogenous probability distributions, thus they are not under control of the adversary. Moreover, given the exogenously controlled processing ties, the best the adversary can do is in fact to use an optial stochastic scheduling policy. In this view, our odel indeed incorporates soe of the ideas by Koutsoupias and Papadiitriou in [8]. Results and ethodology. We derive worst case perforance guarantees for the expected perforance of very siple, cobinatorial online scheduling policies for odels with and without release dates. For the odel without release dates, P E [ w C ], this is a perforance guarantee of 1+ ( + 1)( 1),

4 170 N. Megow, M. Uetz, and T. Vredeveld atching the previously best known perforance guarantee of [11] for the perforance of the WSEPT rule. Note, however, that this bound holds even though we use a restricted scheduling policy that rst has to assign obs to achines online, without knowledge of the obs to coe. For the odel with release dates, P r E [ w C ] we prove a ore coplicated perforance guarantee for a class of processing tie distributions that we call δ-nbue. They generalize NBUE distributions 2, which are contained as a special case. For NBUE distributions, we obtain a perforance bound strictly less than , where (5 + 5)/ Thereby, we iprove upon the previously best known perforance guarantee of 4 1/ for NBUE distributions, which was derived for an LP based list scheduling policy [11]. Again, notice that this iproved bound holds even though we use a restricted policy that rst has to assign obs to achines online, without knowledge of the obs to coe. Our results are achieved by the following, quite siple Sos policy. Once the obs have been assigned to the achines, we assue that on every achine the obs are processed in the WSEPT order 3. To ake the online decisions on achine assignents in the rst phase, at any tie when a ob is presented, we assign it to that achine where it causes the inial increase in total expected obective value; given the obs that have been assigned so far, and given that the obs on each achine will be scheduled in WSEPT order. Intuitively, the reason why we can recover (or iprove, respectively) the previous best known results in stochastic achine scheduling is the following: On the one hand, we restrict the full power of scheduling policies by xing achine assignents beforehand. On the other hand, it is precisely this xed achine assignent, together with an averaging arguent over the nuber of achines, that allows an iproveent in the analysis in coparison to general scheduling policies. We ention that, to obtain our results, we in fact utilize one of the LP based lower bounds of [11]. 2 Model Definition, Notation, and Preliinaries Let n be the nuber of obs, index 1,...,n denote a ob, with associated weight w and processing tie distribution P.ByE [P ] we denote its expected processing tie, and p denotes a particular realization of P. In the odel with release dates, r denotes the earliest point in tie when ob can be started. Given a schedule of start ties S 1,...,S n for a particular realization p =(p 1,...,p n ) of processing ties, C = S + p is the copletion tie of 2 A distribution X is called NBUE, new better than used in expectation, if E [X t X>t] E[X] for all t>0. 3 In the case with release dates, this is in fact a odi ed version of the WSEPT order, that will be explained later.

5 Stochastic Online Scheduling on Parallel Machines 171 ob, =1,...,n. Each ob ust be processed non-preeptively, on any of the achines, and each achine can process at ost one ob at a tie. The goal is to nd a scheduling policy that iniizes the expected value of the weighted copletion ties of obs, E [ w C ]. A scheduling policy eventually yields a feasible -achine schedule for each realization p of the processing ties. For a given policy, denoted by Π, let S Π(p) and C Π (p) denote the start and copletion ties of ob for a given realization p, and let S Π(P) and CΠ (P) denote the associated rando variables. Unless there is danger of abiguity, we also write S and C, for short. We let E [ Z Π] [ ] = E w C Π (P) denote the expected perforance of a scheduling policy Π. Then, if OPT is an optial scheduling policy according to the ost general de nition of stochastic scheduling policies in [10], we say that a policy Π is a ρ approxiation if, for soe ρ 1, E [ Z Π] ρ E [ Z OPT]. We assue that the obs are presented to the scheduler one by one, in the order 1,...,n. However, the nuber of obs n is not known before. When a ob is presented, the scheduler is infored about its weight w and its expected processing tie E [p ] (in the case with release dates, also its release date r ). When ob appears, it ust be assigned to a achine i 1,..., iediately, and this decision ust not be revised later. For a given ob W, and a given subset of obs W, let us de ne by H() the obs in W that have a higher priority in the WSEPT ordering, that is H()= { k W w k E [P k ] w }. E [P ] Notice that, by convention, H() contains ob, too. Accordingly, de ne { w k L()= k W E [P k ] < w } E [P ] as those obs that have lower priority in the WSEPT order. It is clear that the online scheduling policies for the Sos odel can in fact be interpreted as a subclass of stochastic scheduling policies in general. This because, assuing a classical input for a stochastic scheduling proble where all (stochastic) input data is disclosed at the outset, the only thing we require in the Sos odel is a xed assignent of obs to achines beforehand. Therefore, the expected perforance of an optial Sos policy is no less than the expected perforance of an optial policy for a corresponding classical stochastic proble. (The latter being de ned by the input after the online phase.) Hence, lower bounds on the expected value of an optial policy known fro stochastic scheduling carry over to the online setting. We crucially exploit that fact, and will utilize the following lower bound on the expected perforance E [ Z OPT] of

6 172 N. Megow, M. Uetz, and T. Vredeveld an optial stochastic scheduling policy. It is a generalization of a lower bound by Eastan et al. [5] for the deterinistic setting. Lea 1 (M ohring et al. [11]). For any instance of P E [ w C ], we have that E [Z opt ] w k H() E [P k ] ( 1)( 1) w E [P ], where bounds the squared coefficient of variation of the processing ties, that is, Var[P ]/E[P ] 2 for all obs =1,...,n and soe 0. This lea indeed plays a crucial role in achieving perforance guarantees for the Sos policies presented in the following sections. Clearly, the clai of Lea 1 also applies to the setting with release dates P r E [ w C ]. 3 Lower Bounds on Approxiation The requireent of a xed assignent of obs to achines beforehand ay be interpreted as ignoring the additional inforation on the outcoe of the stochastic process (that is, the actual realization of processing ties), at least with respect to assignents of obs to achines. In the following, we therefore give a lower bound on the expected perforance E [ Z FIX] of an optial stochastic scheduling policy FIX that assigns obs to achines beforehand. A fortiori, this lower bound holds for the best possible Sos policy, too. Theore 1. For stochastic parallel achine scheduling with unit weights and i.i.d. exponential processing ties, P p exp(1) E [ C ], there exist instances such that E [ Z FIX] 3( 2 1) E [ Z OPT] ε, for any ε>0. Here, 3( 2 1) Hence, no policy that uses xed assignents of obs to achines can perfor better in general. Notice that the Theore is forulated for the special case of exponentially distributed processing ties. Stronger bounds could probably be obtained for arbitrary distributions. However, since our perforance guarantees, as in [11], will depend on the coefficient of variation of the processing ties, we are particularly interested in lower bounds for classes of distributions where this coefficient of variation is sall. The coefficient of variation of exponentially distributed rando variables equals 1. For exaple, for the case of = 2 achines, we get a lower bound of 8/7 1.14, and for that case our perforance bound equals 2 1/ =1.5. Proof (of Theore 1). For siplicity, we will prove a slightly worse lower bound. Let us consider an instance with achines and n = + /2 exponentially distributed obs, p exp(1). The optial stochastic scheduling policy is SEPT,

7 Stochastic Online Scheduling on Parallel Machines 173 shortest expected processing tie rst [1, 20], and the expected perforance is (see, e.g., [17 Cor ]) E [ Z OPT] = E [ Z SEPT] = E [ C SEPT ] n = +. =+1 The best xed assignent policy assigns 2 obs each to /2 of the achines, and 1ob each to /2 of the achines. Hence, there are obs with E [C ]=1, and /2 obs with E [C ] = 2. The expected perforance for the best xed assignent policy FIX is E [ Z FIX] = E [ C FIX ] = +2 /2. For sall values =2,3,4..., we calculate E [ Z FIX] /E [ Z OPT] =8/7, 7/6,32/27,... It is easy to see that E [ Z FIX] E [ Z OPT] = o( 2 ), and for we get a lower bound of 16/ Now the clai of the theore follows along the sae lines if we rede ne the nuber of obs as n = +. 4 Stochastic Online Scheduling We next de ne a stochastic online scheduling policy for the proble without release dates, P E [ w C ]. The basic idea is siple: any ob, once it appears, will be assigned to the achine where it causes the inial increase in expected obective value (given that obs 1,..., 1 have been assigned already). In order to be able to do that, we rst need to specify how the obs will be scheduled on every single achine. We introduce a nal bit of notation, letting M i denote all obs that are assigned to achine i, and letting M i () = k < k M i denote the subset of obs assigned to a achine i before soe ob appears 4. WSEPT (weighted shortest expected processing tie first) On each achine i, schedule the obs M i in non-decreasing order of their ratios of weight over expected processing tie w /E [P ]. This policy is known to be optial for (stochastic) achine probles on a single achine, 1 E [ w C ], by results of Sith and Rothkopf, respectively [15, 12]. Now we can de ne the MinIncrease policy as follows. 4 Recall that we assue a nubering of the obs in the order in which they appear in the online sequence; hence k<denotes obs k that appeared earlier than ob.

8 174 N. Megow, M. Uetz, and T. Vredeveld MinIncrease When a ob is presented to the scheduler, it is assigned to the achine i that iniizes the expression incr (,i) =w k H() M i() E[P k ] + E[P ] k L() M i() w k +w E[P ]. In fact, given that WSEPT is used on each achine, MinIncrease ust chooses the achine where ob causes the least increase in expected perforance. Theore 2. Consider the stochastic online scheduling proble on parallel achines, P E [ w C ]. Given that Var[P ]/E[P ] 2 for all obs and soe constant 0, the MinIncrease policy is a ρ approxiation, where ρ =1+ ( + 1)( 1) Proof. Denote by E[incr () ] the increase in the expected obective value caused by xing the assignent of a ob using MinIncrease. Since MinIncrease chooses the achine i on which causes the least expected increase, the expected increase is E[incr () ] = in E[incr (,i)] i { } = in w E[P k ]+E[P ] w k + w E[P ] i k H() M i() k L() M i() 1 ( ) w E[P k ]+E[P ] w k + w E[P ], k H(),k<. k L(),k< where the inequality holds because the least expected increase is not ore than the average expected increase over all achines. By suing up these quantities over all obs we obtain the expected perforance E [ Z MI] of the MinIncrease policy. E [ Z MI] = E[incr ()] 1 ( w = 1 w k H(),k< k H() E[P k ] + E[P ] E[P k ]+ 1 k L(),k< w E[P ], w k ) + w E[P ] where the last equality holds by index rearrangeent, since E[P ] k L(),k< w k = w E[P k ]. k H(),k>

9 Stochastic Online Scheduling on Parallel Machines 175 Now, we plug in the inequality of Lea 1, and using the trivial fact that w E[P ] is a lower bound for the expected perforance E [ Z OPT] of an optial policy, we obtain E [ Z MI] E [ Z OPT] ( 1)( 1) + ( 1+ ( + 1)( 1) w E[P ]+ 1 w E[P ] ) E [ Z OPT]. This perforance guarantee atches the currently best known perforance guarantee for the classical stochastic setting, which was derived for the perforance of the WSEPT rule in [11]. The WSEPT rule, however, requires the knowledge of all obs with their weights w and expected processing ties E[P ] at the outset. In contrast, the MinIncrease policy decides on achine assignents online, without any knowledge of the obs to coe. Finally, it is worthy to note that siple instances show that these two policies are indeed different. Lower bounds for MinIncrease. The lower bound on the perforance ratio for any xed assignent policy given in Theore 1 holds for the MinIncrease policy, too. Hence, in general, MinIncrease cannot be better than approxiative. We can strengthen the lower bound via ore sophisticated instances, but the coputations of the optial values becoe unpleasant. We next give an instance for = 2 achines. Exaple 1. We are given 6 obs with exponentially distributed processing ties such that E[P 1 ]=E[P 4 ]=1, E[P 2 ]=E[P 5 ]=k and E[P 3 ]=E[P 6 ]=2k, for soe xed k. The obs appear in order of their indices in the online sequence. Without going into further details, it turns out that the expected perforance of the MinIncrease policy is 6 + 9k, and the expected perforance of an optial scheduling policy is 5 + k(167/24+7/(1 + k)+1/(2+4k)). For k, this yields a lower bound of 216/ , whereas Theore 2 yields a perforance guarantee of 1.5. For less restricted probability distributions, i.e., non-exponential and with larger coefficients of variation, we obtain a lower bound of 3/2 on the expected perforance of MinIncrease relative to an optial scheduling policy. However, this is less eaningful copared to the perforance bound of Theore 2, which depends on an upper bound on the squared coefficient of variation. We skip the details. 5 Stochastic Online Scheduling with Release Dates In this section we consider the proble of stochastic online scheduling on parallel achines where obs have release dates. As the optial single achine scheduling policy is unknown to date for this proble, we analyze the expected perforance of the MinIncrease policy which runs the following single achine scheduling policy.

10 176 N. Megow, M. Uetz, and T. Vredeveld α-shift-wsept Modify the release date r of each ob such that r = ax r,αe[p ], for soe xed 0 <α 1. At any tie t, when the achine is idle, start the ob with the highest priority in the WSEPT order aong all available obs (respecting the odi ed release dates). In the deterinistic (online) setting, this policy was proposed for parallel achines in [9]. For the analysis of this policy, we restrict ourselves to rando variables that we call δ-nbue. This is a generalization of NBUE rando variables. Definition 1 (δ-nbue). A non-negative rando variable X is δ-nbue if, for δ 1, E[X t X >t] δ E[X] for all t 0. Ordinary NBUE distributions are by de nition 1-NBUE. For a NBUE rando variable X, Hall and Wellner [7] showed that the (squared) coefficient of variation is bounded by 1, that is, Var[X]/E[X] 2 1. Fro their work, it also follows that, if X is δ-nbue, then Var[X]/E[X] 2 2δ 1. Exaples of ordinary NBUE (or 1-NBUE) distributions are exponential, Erlang, unifor, or Weibull distributions (with shape paraeter at least 1). We next derive an upper bound on the expected copletion tie of a ob, E[C ], when scheduling obs on a single achine according to the α-shift-wsept policy. This bound is used later to analyze the expected perforance of MinIncrease. Lea 2. Let all processing ties be δ-nbue. Then the expected copletion tie of ob for α-shift-wsept on a single achine can be bounded by E[C ] (1 + δ/α) r + E[P k ]. k H() Proof. We consider soe ob. Let us denote by B the event that the achine is busy processing soe ob at tie r, and let us denote by I the copleent of B, naely that the achine is idle (or ust nished processing soe ob) at tie r. Under the condition I it could still be that there are higher priority obs k H() \ available at tie r, but in any case the expected start tie of ob can be postponed by at ost E[P k I]. k H()\ However, due to independence of processing ties, we have that E[P k I]= E[P k ], and therefore E[S I] r + E[P k ]. k H()\ Consider condition B and let us denote by E[x(B)] the expected length of the tie period until the achine becoes idle for the rst tie after r. Under the

11 Stochastic Online Scheduling on Parallel Machines 177 condition B, for any realization p of the processing ties, conditioned on B, soe ob l(p) is in process at tie r (in fact, l(p) ight have lower or higher priority than ). Any such ob l was available at tie r l <r, and by de nition of the odi ed release dates, we therefore know that E[P l ] (1/α)r l < (1/α)r for any such ob l. Moreover, letting t = r S l, the expected reaining processing tie of such ob l, conditioned on the fact that it is indeed in process at tie r,ise[p l t P l >t]. Due to the assuption of δ-nbue processing ties, we thus know that E[P l t P l >t] δ E[P l ] (δ/α)r. Therefore, the expected reaining processing tie of any ob l that ight be in process at tie r is bounded by (δ/α)r, and thus E[x(B)] (δ/α)r. Repeating the sae arguent as above, we can now conclude that E[S B] (1 + δ/α)r + E[P k ]. (1) k H()\ As each of the two conditional expectations E[S I] and E[S B] is bounded by the right hand side of (1), we obtain that E[S ] (1 + δ/α)r + E[P k ], k H()\ and the fact that E[C ]=E[S ]+E[P ] concludes the proof. In fact, it is quite straightforward to use Lea 2 in order to show the following. Corollary 1. The α-shift-wsept algorith is a 3-approxiation for the single achine proble 1 r E [ w C ], for NBUE processing ties. We ust use that δ = 1, and we choose α = 1. We skip further details, and note that this atches the best known LP based perforance bound derived in [11], which even holds for arbitrary processing tie distributions. The MinIncrease policy for the proble with release dates is now the following. In order to decide on which achine a ob should go, we ust ignore the release dates, and use the sae policy for assigning obs to achines that we used before in the setting without release dates. Theore 3. Consider the stochastic online scheduling proble on parallel achines with release dates, P r E [ w C ]. Given that all processing ties are δ-nbue, the odi ed MinIncrease policy is a ρ approxiation, where ρ = 1 + ax 1+δ/α, α + δ +( 1)( +1)/().

12 178 N. Megow, M. Uetz, and T. Vredeveld Here, is such that Var[P ]/E[P ] 2 for all obs. In particular, since all processing ties are δ-nbue, we know that 2δ 1 in the above perforance bound. Proof. Let i be the achine to which ob is assigned. Then, by Lea 2 we know that E[C ] (1 + δ α )r + E[P k ], (2) k H() M i and the expected value of MinIncrease can be bounded by E [ Z MI] ( 1+ δ ) w r + α w k H() M i E[P k ] (3) Using an index rearrangeent arguent as in the proof of Theore 2, we can write w E [P k ]= w E[P k ]+E [P ] w k +w E[P ]. k H() M i k H() M i () k L() M i () By de nition of MinIncrease, we know that ob is assigned to the achine which iniizes the sus in parenthesis of the right hand side of this equation. Hence, by an averaging arguent, we know that w E[P k ] w k H() M i = w k H(),k< k H() E [P k ] E [P k ] E [P ] k L(),k< w E [P ], w k + we[p ] where the last equality follows fro index rearrangeent. Plugging this into (2), leads to the following bound on the expected perforance of MinIncrease. E [ Z MI] ( 1+ δ ) w r + α w k H() E[P k ] + 1 w E[P ]. As entioned before, the relaxed proble without release dates provides a lower bound on the expected optiu with release dates. We therefore can plug into the above inequality the bound of Lea 1, and obtain E [ Z MI] ( 1+ δ ) w r + E [ Z OPT] + α = E [ Z OPT] + ( (1+ δ ) w r α + ( 1)( +1) w E[P ] ( 1)( +1) E[P ] ). (4)

13 Stochastic Online Scheduling on Parallel Machines 179 By bounding r by r + αe[p ], we obtain the following bound on the ter in parenthesis of the su in the right hand side of inequality (4). ( δ ) 1+ r α + ( 1)( +1) E[P ] ( 1+ δ ) r + ( α + δ + α ( r + E[P ] ) ax { 1+ δ α ( 1)( +1) ) E[P ] ( 1)( +1),α+ δ + The proof is copleted by using this inequality in equation (4), and applying the trivial lower bound w (r + E[P ]) E [ Z OPT] on the expected optiu perforance. }. For NBUE processing ties, where we can choose = δ = 1, the approxiation ratio is inial for α =( )/(), obtaining a ratio of 2+( )/(), which is less than (5+ 5)/2 1/() /(), iproving upon the previously best know approxiation ratio of 4 1/ fro [11]. Acknowledgeent. The authors would like to thank Rolf H. M ohring for helpful discussions, and Andreas S. Schulz for pointing out a isinterpretation in an earlier version of this paper. References 1. J. L. Bruno, P. J. Downey, and G. N. Frederickson. Sequencing tasks with exponential service ties to iniize the expected owtie or akespan. J. ACM, 28: , S. Chakrabarti and S. Muthukrishnan. Resource scheduling for parallel database and scienti c applications. In Proc. 8th Ann. ACM Syp. on Parallel Algoriths and Architectures, pages , Padua, Italy, C. Chekuri, R. Johnson, R. Motwani, B. Nataraan, B. Rau, and M. Schlansker. An analysis of pro le-driven instruction level parallel scheduling with application to super blocks. In Proc. 29th IEEE/ACM Int. Syp. on Microarchitecture, Paris, France, pages 58 69, M. A. H. Depster, J. K. Lenstra, and A. H. G. Rinnooy Kan, editors. Deterinistic and Stochastic Scheduling. D. Reidel Publishing Copany, Dordrecht, W. Eastan, S. Even, and I. Isaacs. Bounds for the optial scheduling of n obs on processors. Mgt. Sci., 11: , R. L. Graha, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan. Optiization and approxiation in deterinistic sequencing and scheduling: A survey. Ann. Discr. Math., 5: , W. J. Hall and J. A. Wellner. Mean residual life. In M. Cs org o, D. A. Dawson, J. N. K. Rao, and A. K. Md. E. Saleh, editors, Proc. Int. Syp. on Statistics and Related Topics, pages , Ottawa, ON, Asterda: North-Holland.

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