Stochastic Machine Scheduling with Precedence Constraints

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1 Stochastic Machine Scheduling with Precedence Constraints Martin Skutella Fakultät II, Institut für Matheatik, Sekr. MA 6-, Technische Universität Berlin, 0623 Berlin, Gerany Marc Uetz Faculty of Econoics and Business Adinistration, Quantitative Econoics, Universiteit Maastricht, 6200 MD Maastricht, The Abstract We consider parallel, identical achine scheduling probles where the obs are subect to precedence constraints, release dates, and the processing ties of obs are governed by independent probability distributions. The obective is to iniize the expected value of the total weighted copletion tie w C, where w 0. Building upon a linear prograing relaxation by Möhring, Schulz, and Uetz (999) and an idle tie charging schee by Chekuri, Motwani, Nataraan, and Stein (200), we derive the first constant-factor approxiation algoriths for this odel. Introduction This paper addresses stochastic parallel achine scheduling probles with the obective to iniize the total weighted copletion tie in expectation. Machine scheduling probles have attracted researchers for decades since they play an iportant role in various applications fro the areas of operations research, anageent science, and coputer science. The total weighted copletion tie obective is of particular iportance in scheduling environents where any obs are to be scheduled on a liited nuber of achines, and a good average perforance is desired. Proinent exaples for such a scheduling situation are probles that arise, e.g., in copiler optiization (Chekuri et al. 996) and in parallel coputing (Chakrabarti and Muthukrishnan 996). The ain characteristic of stochastic scheduling probles is the fact that part of the input data, in this paper the processing ties of the obs, ay be subect to rando fluctuations. Hence, the effective processing ties are not known with certainty in advance. This assuption is of particular practical relevance in any applications. Denote by V n a set of obs which ust be scheduled on parallel, identical achines. Each achine can handle only one ob at a tie, and the obs can be scheduled on any of the achines. Once the processing of a ob is started on one achine, it ust be processed without preeption on this achine. Precedence constraints are given by an acyclic digraph G V A, where any arc i A restricts the start tie of ob to be not earlier than the copletion tie of ob i. We consider probles with and without release dates r for the obs, with the intended eaning that ob ust not be started earlier than r. In the classical (deterinistic) setting, the obective is to iniize the total weighted copletion tie V w C, where w is a non-negative weight and C denotes the copletion tie of ob. In the stochastic odel, it is assued that the processing tie p of a ob is not known in advance. It becoes known only upon copletion of the ob. However, the distribution of the corresponding rando variable P is given beforehand. Let P P P n denote the vector of rando variables for the processing ties, and denote by p p p n a particular realization of the processing ties. By E P we denote the expected processing tie An extended abstract of this work appeared in: Proceedings of the Twelfth Annual ACM SIAM Syposiu on Discrete Algoriths (SODA 200), Washington DC, 200, pp This work was supported in part by the EU Theatic Networks APPOL I+II, Approxiation and Online Algoriths, IST and IST , and by the DFG Research Center Matheatics for key technologies: Modelling, siulation and optiization of real-world processes. The second author was supported by the Geran-Israeli Foundation for Scientific Research and Developent (GIF) under grant I /97, and by the Deutsche Forschungsgeeinschaft (DFG) under grant Mo 446/3-4.

2 of a ob. We assue throughout that the processing ties of the obs are stochastically independent. In the classical three-field notation of Graha, Lawler, Lenstra, and Rinnooy Kan (979), the proble of iniizing the expected total weighted copletion tie can be denoted by P prec r E w C. In fact, the twist fro deterinistic to stochastic processing ties changes the nature of the scheduling proble considerably. The solution of a stochastic scheduling proble is no longer a siple schedule, but a scheduling policy. We adopt the notion of scheduling policies as proposed by Möhring et al. (984). Roughly spoken, a scheduling policy akes scheduling decisions at certain decision ties t, and these decisions are based upon the observed past up to tie t as well as the a priori knowledge of the input data of the proble. The policy, however, ust not anticipate inforation about the future, such as the actual realizations p of the processing ties of the obs which have not yet been copleted by tie t. In general, scheduling policies can be rather coplex in the sense that they exploit the stochastic inforation that evolves over tie. The siplest, yet practically attractive scheduling policies are list scheduling policies. Graha s classical list scheduling algorith (Graha 966) is perhaps the ost natural aong the: Given a priority list L of the obs, at any tie it greedily schedules the first available ob(s) fro the list. A ob is available as soon as it is released and all of its predecessors are copleted. If precedence constraints or release dates exist, it ay thus happen that the order of start ties of obs differs fro the order of the obs in the given priority list; the obs are scheduled out of order with respect to the given priority list L. For the deterinistic proble P prec C ax with akespan obective, it is well known that Graha s list scheduling algorith achieves a perforance guarantee of 2 for any priority list of the obs (Graha 966). This result can be established for stochastic processing ties as well (Chandy and Reynolds 975). However, even in the deterinistic setting there are exaples which show that the perforance of Graha s algorith can be arbitrarily bad for the total weighted copletion tie obective w C (Schulz 996). For the deterinistic proble P r prec w C, the currently best known perforance guarantee of 4 relies on another, so-called ob-based list scheduling algorith (Munier, Queyranne, and Schulz 998). For stochastic processing ties, approxiation results for parallel achines were previously known for probles without precedence constraints only (Möhring et al. 999). The results derived in this paper rely on a list scheduling algorith which generalizes both Graha s and ob-based list scheduling. It has been suggested by Chekuri et al. (200) to obtain a approxiation for the deterinistic proble P r prec w C. The basic idea is to extend Graha s list scheduling in such a way that a ob ay be scheduled out of order only if enough deliberate idle tie has accuulated before. We show that an appropriate adaption of the list scheduling algorith of Chekuri et al. (200), based upon a priority list that is derived fro an optial solution to a generalized LP-relaxation by Möhring et al. (999), leads to perforance guarantees also for the stochastic odel. The table below suarizes known perforance guarantees for stochastic parallel achine scheduling probles with the total weighted copletion tie obective. While the results in the first four rows are fro Möhring et al. (999), the reaining results are derived in this paper. In the table, is an upper bound on the coefficient of variation CV P of the processing tie distributions P, the nuber of achines is denoted by, and β is an arbitrary non-negative paraeter. The last colun shows the respective perforance bounds for processing tie distributions where the coefficient of variation is bounded by, such as exponential, unifor, or Erlang distributions. Table : Perforance bounds for stochastic achine scheduling probles. scheduling odel perforance guarantee arbitrary P CV P prec E w C 2 2 r prec E w C 3 3 P E w C 2 2 P r E w C 3 ax 4 P in-forest E w C 2 ax 3 P prec E w C β ax β P r prec E w C β β ax

3 2 Stochastic scheduling Let us specify the above entioned dynaic view on stochastic scheduling ore precisely; it is based on an ebedding of stochastic scheduling into the fraework of stochastic dynaic optiization. The state of the syste at any tie t is given by the tie t itself as well as the conditional distributions of the obs processing ties, which depend on the observed past up to tie t. The past at a tie t is given by the set of obs which have already been copleted by t, together with their start and copletion ties, and the set of obs which have been started before t but have not been copleted yet, together with their start ties. The action of a scheduling policy at tie t is given by a set of obs B t V and a tentative decision tie t tent t. The set B t is the set of obs that are scheduled at tie t. The tentative decision tie t tent is the latest point in tie when the next action of the policy takes place, subect to the condition that no other ob is released or ends before t tent. Notice that B t ay be epty, and t tent iplies that the next action of the policy takes place when the next ob is released or soe ob ends, whatever occurs first. The action of any policy at any tie t ust only depend on the state of the syste at tie t. This condition is also called the non-anticipatory constraint. Of course, the definition of B t, with respect to the state at tie t, ust respect potential release dates and precedence constraints, and the nuber of available achines. The tie instances when a policy takes its actions are called decision ties. Given an action of a policy at a decision tie t, the next decision tie is t tent, or the tie of the next ob copletion, or the tie when the next ob is released, whatever occurs first. Depending on the action of the policy, the state at the next decision tie is realized according to the probability distributions of the obs processing ties. A given policy eventually yields a feasible -achine schedule for each realization p of the processing ties. For a given policy 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. It follows fro siple exaples that, in general, a scheduling policy cannot yield the optial schedule for each possible realization of the processing ties. Hence, our obective is to find a policy Π which iniizes the obective, call it Z Π P, in expectation. But even under this ild notion of optiality, few special cases exist for which optial scheduling policies are known. One exaple is the optiality of the SEPT rule (shortest expected processing tie first) for the proble without precedence constraints or release dates, with unit weights, and with exponentially distributed processing ties P p exp λ E C (Bruno, Downey, and Frederickson 98; Weiss and Pinedo 980). This result was extended by Käpke (987) to the case where the weights w are copliant with the expected processing ties. In general, however, there exist exaples which show that optial policies can be rather coplicated in the sense that they ust utilize inforation on the conditional distributions of the obs processing ties (Uetz 200, Th ). We therefore concentrate on approxiation algoriths. In stochastic scheduling, a scheduling policy Π is said to be an α-approxiation if its expected perforance E Z Π P is within a factor of α of the expected perforance E Z Π P of an optial (non-anticipatory) scheduling policy Π. The value α is also called the perforance guarantee. Let us briefly discuss related odels. Copared to the odel described above, on-line optiization is another way of coping with the fact that the future is uncertain. We refer to Fiat and Woeginger (998) for details on on-line optiization. There is, however, a significant difference between the underlying paradigs of the above described analysis and the usual copetitive analysis that is prevailing in on-line optiization. First, copetitive analysis is based upon the ex-post coparison What was achieved under uncertainty about the future, and what could have been achieved if the future would not have been uncertain?. This is expressed by the fact that the adversary of the scheduler is generally an oracle that knows the optial solution. In contrast, stochastic scheduling addresses the ex-ante question What is the best that can be achieved under the given uncertainty about the future?. Here, the underlying adversary is uch weaker: like the scheduler, the adversary ust not anticipate future inforation. Second, in copetitive analysis the adversary is even allowed to deterine, to a certain extent, the input distribution. This is not the case in the stochastic odel considered here. It is interesting to note that two generalized on-line fraeworks were suggested by Koutsoupias and Papadiitriou (2000). They restrict the adversary s power in two ways: Its ability to choose an input distribution, and its ability to find an optial solution. To a certain extent, the above described stochastic odel incorporates both of these underlying ideas. We refer to Koutsoupias and Papadiitriou (2000) for details, and to Uetz (200) for a discussion. Another type of analysis for stochastic odels has been proposed recently by Scharbrodt, Schickinger, and Steger (2002). If Z OPT p is the optial solution value for a realization p, they analyze the expected copetitive ratio E Z Π P Z OPT P. In this type of analysis the adversary is again an oracle that knows the optial solution. We refer to Scharbrodt et al. (2002) for a ore detailed discussion. 3

4 3 List scheduling with deliberate idle ties We start with a few preliinaries and definitions that will be used later in the analysis. Assuption 3.. For any instance of P r prec, assue that r the precedence constraints. r i whenever ob i is a predecessor of ob in Obviously, this assuption can be ade without loss of generality. Additionally, we use the following definitions. Definition 3.2 (critical predecessor). Let soe realization p of the processing ties and a feasible schedule be given. For any ob, a critical predecessor of is a predecessor i of (with respect to the precedence constraints) with C i r and C i axial aong all predecessors. The following definition is illustrated in Figure. Definition 3.3 (critical chain). Let soe realization p of the processing ties and a feasible schedule be given. For a given ob, a critical chain for ob and its length p is defined backwards recursively: If has no critical predecessor, is the only ob in the critical chain, and p r p. Otherwise, p p k p, where ob k is a critical predecessor of ob. PSfrag replaceents 2 3 h r C Figure : Exaple of a critical chain for ob. Its length is p r h i p i. Notice that the critical chain and its length p depend on both the realization of the processing ties p and the underlying schedule. Moreover, since a critical predecessor is not necessarily unique, the critical chain and its length also depend on a tie-breaking rule for choosing critical predecessors. This is not relevant for our analysis, but in order to ake the above definition unique, let us suppose that soe arbitrary but fixed tie-breaking rule is used. Notice further that, independent of the realization of the processing ties, the first ob of a critical chain is available at its release date r. This follows directly fro the definition. To give a precise description of the list scheduling algorith we use, recall that a ob is called available at tie t if all predecessors have already been copleted by t, and if t r (whenever release dates are present). Like Graha s list scheduling, the algorith proceeds over tie until all obs have been scheduled. Assue a priority list L is given. Then, whenever a achine is idle and the first (not yet scheduled) ob in the list is available, the ob is scheduled. Consider the case that a achine is idle and the first ob in the list is not available, but there is soe other ob available. Then this ob is not scheduled iediately as in Graha s algorith, but it is scheduled only after a certain aount of deliberate idle tie has accuulated. Intuitively, the algorith strives to schedule the obs in the order of the list L by leaving deliberate idle ties, but if the accuulating deliberate idle tie exceeds a certain threshold, the algorith panics and schedules the first available ob. To be able to analyze the outcoe of the algorith, the deliberate idle ties are charged to the obs according to a charging schee. The idea of the charging schee is that at any tie the first available ob is responsible for the currently accuulating deliberate idle tie, hence this ob will be charged the idle tie. Algorith gives a precise description, using the following notation. Definition 3.4. For a given realization p of the processing ties and a given (partial) schedule, denote by r p r the earliest point in tie when ob becoes available. β Observe that Algorith coincides with the classical list scheduling algorith of Graha (966) if we choose 0, because then the first available ob fro the list is scheduled at any tie. We use the wildcard to denote an obective function that ay be arbitrary. 4

5 Algorith : List scheduling with idle tie charging adapted fro Chekuri et al. (200). input : Instance of P r prec and a priority list L of the obs output : A feasible schedule initialize t 0; while there are unscheduled obs in list L do initialize t tent ; let be the first unscheduled ob in list L; if ob is available and a achine is idle at tie t then schedule ob at t on any of the idle achines; charge all uncharged deliberate idle tie (fro all achines) in r p t to ob ; else let i be the first unscheduled ob in list L which is available at tie t (if any); if such a ob i exists and a achine is idle at tie t then if there is at least βe P i uncharged deliberate idle tie in the tie interval r i p t then schedule ob i at t on any of the idle achines; charge all uncharged deliberate idle tie (fro all achines) in r i p t to ob i ; else denote by I be the total aount of uncharged deliberate idle tie in r i p t, let be the nuber of idle achines at tie t, and define t tent such that I t tent t βe P i ; augent t to t tent or the next tie when a achine falls idle or a ob is released, whatever occurs first; else augent t to the next tie when a achine falls idle or a ob is released, whatever occurs first; Like Chekuri et al. (200), we prove soe properties of the schedules constructed by Algorith. To this end, let us first fix soe additional notation. For a given ob, denote by B and A the sets of obs that coe before and after ob in the priority list L, respectively. By convention, B also includes ob. For a given realization p of the processing ties and the schedule constructed by Algorith, consider a critical chain, 2,..., h for ob. Let us abuse notation and define B p : B h i i. For a given realization p and a given schedule, B p contains all obs that coe before ob in the priority list L, except for those which belong to the given critical chain for. Moreover, for a given realization p of the processing ties, let O p A be the obs in A that are started out of order, that is, before. The following lea is the analogue to the results for the deterinistic setting by Chekuri et al. (200). Lea 3.5. For any realization p of the processing ties and any ob : (i) ob is charged no ore than βe P deliberate idle tie; (ii) the deliberate idle tie in r p S p is charged only to obs in B ; (iii) there is no uncharged deliberate idle tie. Proof. Any ob gets charged the deliberate idle tie that accuulates (fro tie r p on) during tie intervals when is the first available ob in the list. In these tie intervals, however, is scheduled as soon as the uncharged deliberate idle tie (fro tie r p on) accuulates to βe P ; this proves (i). In the tie interval r p S p no ob fro A is the first available ob fro the list, since ob is available fro r p on, and has higher priority than any ob fro A ; this yields (ii). Finally, whenever deliberate idle tie accuulates, there is soe ob which is the first unscheduled and available ob in the list L; this proves (iii). We next derive an upper bound on the copletion tie of any ob for a given realization p. Lea 3.6. Consider the schedule constructed by Algorith for any β 0, any realization p of the processing ties, and any priority list L which is a linear extension of the precedence constraints. Let C p denote the 5

6 resulting copletion tie of any ob, and let p be the length of a critical chain for ob. Then C p p r p i βe p i p i () i B i O p Proof. The basic idea is analogous to Graha s analysis for the akespan obective (Graha 966). Consider a critical chain for ob with total length p, consisting of obs 2 h. Now partition the interval r C p into tie intervals where soe ob fro the critical chain is in process and the reaining tie intervals. The latter are exactly r i p S i p, i h. (Recall that r r p due to the definition of a critical chain). By definition, C p p h S i p r i p (2) i To bound the total length of the intervals r i p S i p, i h, observe that in each of these intervals there is no idle tie except (possibly) deliberate idle tie, since ob i is available in r i p S i p. Hence, the total processing in these intervals can be partitioned into three categories: processing of obs fro B which do not belong to the critical chain for, that is, obs in B p, deliberate idle tie, processing of obs fro A which are scheduled out of order, that is, obs in O p. Due to Lea 3.5 (ii), all deliberate idle tie in the interval r i p S i p is charged only to obs in B i, i h. Since the priority list L is a linear extension of the precedence constraints, we have B B 2 B h B. Hence, all deliberate idle tie in the intervals r i p S i p, i h, is charged only to obs in B. Since there is no uncharged deliberate idle tie (Lea 3.5 (iii)), and since each ob i B gets charged no ore than βe P i idle tie (Lea 3.5 (i)), the total aount of deliberate idle tie in the intervals r i p S i p, i h, is bounded fro above by β i B E P i. This yields h S i p r i p i Finally, due to Assuption 3. we have r r, thus p i i B p p i p i i B p i B Now put (4) into (3), and then (3) into (2), and the clai follows. i B βe P i p i (3) i O p p r (4) Before we take expectations in (), we concentrate on the ter i O p p i. The following lea shows that the expected total processing of the obs in O p the obs that are scheduled out of order with respect to (and p) is independent of their actual processing ties. Lea 3.7. E P i E E P i i O P i O P Proof. We can write i O P P i equivalently as i A δ i P P i, where δ i P is a binary rando variable which is if and only if i O p. Linearity of expectation yields E P i E δ i P P i E δ i P P i i O P i A i A But δ i P is stochastically independent of the processing tie P i : when ob i is started, it is already decided whether i O p, and this decision is independent of the actual processing tie of ob i. (Here we require that the processing ties are stochastically independent, and that policies are non-anticipatory.) Hence, i A E δ i P P i This concludes the proof of the lea. i A E δ i P E P i E δ i P E P i E E P i (5) i A i O P 6

7 Finally, we obtain an upper bound on the expected copletion tie of any ob under Algorith. Lea 3.8. For any instance of a stochastic scheduling proble P r prec and any priority list L which is a linear extension of the precedence constraints, the expected copletion tie of any ob under Algorith (with paraeter β 0) fulfills E C P β E P β E P i i B Proof. Let be arbitrary. First, taking expectations in () together with Lea 3.7 yields E C P E P r β E P i E E P i i B i O P r (6) Next, consider any realization p of the processing ties. If soe ob i A is scheduled out of order, i gets charged exactly βe P i idle tie. Hence, the total aount of deliberate idle tie in 0 S p that is charged to obs in A is β i O p E P i. Now consider a critical chain for ob, consisting of obs 2 h, with total length p. Fro the proof of Lea 3.6, we know that all deliberate idle tie in the intervals r i p S i p, i h, is charged only to obs in B. In other words, all deliberate idle tie in 0 S p that is charged to obs in A lies in the copleentary intervals 0 r and S i p C i p, i h. (Recall that r r p due to the definition of a critical chain.) The total length of these intervals is exactly p p. Hence, the total aount of deliberate idle tie in 0 S p that is charged to obs in A is at ost p. (In fact, it is at ost p p, but this is not essential.) Hence, we obtain β i O p E P i p, for any realization of the processing ties. Taking expectations now yields and (6) follows. E E P i i O P β E P 4 Linear prograing relaxation To obtain a priority list L as input for Algorith, Chekuri et al. (200) use a single achine relaxation. This approach does not help in the stochastic setting, since the single achine proble does not necessarily provide a lower bound for the parallel achine proble; see Möhring et al. (999) for an exaple. Instead, we use LP-relaxations which extend those used by Möhring et al. (999) by adding inequalities which represent the precedence constraints. First, define f : 2 V IR by f W 2 W E P 2 2 E P W 2 2 E P W W V (7) Here, 0 is an upper bound on Var P E P 2 for all obs, where Var P E P 2 E P 2 is the variance of P. In other words, the coefficient of variation CV P : Var P E P of the distribution P is bounded by. The following load inequalities are crucial for the derivation of our results. Theore 4. (Möhring et al. 999). If CV P for all P and soe E P E C Π P W are valid for all W V and any non-anticipatory scheduling policy Π. 0, the load inequalities f W (8) In fact, an upper bound on the coefficients of variation of the processing tie distributions P sees to be a reasonable assuption for any scheduling probles. For instance, assue that ob processing ties follow so-called NBUE distributions. 7

8 Definition 4.2 (NBUE). A non-negative rando variable X is NBUE, new better than used in expectation, if E X t X t E X for all t 0. Here, E X t X t is the conditional expectation of X t under the assuption that X t. Roughly spoken, when processing ties are NBUE, on average it is not disadvantageous to process a ob. Exaples for NBUE distributions are, aong others, exponential, unifor, and Erlang distributions. A result of Hall and Wellner (98) states that the coefficient of variation CV X of any NBUE distribution X is bounded by. Hence, by choosing the second ter of the right hand side of (8) can be neglected for NBUE distributions, which leads to siplified perforance guarantees in Section 5. Observe that under any scheduling policy Π the trivial inequalities and E C Π P E C Π P E C Π i P E P i A E P are valid, since they even hold point-wise for any realization of the processing ties. Due to Theore 4., the following is thus a linear prograing relaxation for the proble P r prec E w C. iniize subect to w C LP V E P C LP W C LP C LP V f W W V Ci LP E P i A E P V where f : 2 V IR is the set function defined in (7). The load inequalities W E P C LP f W, W V, can be separated in tie O nlogn (Möhring et al. 999; Uetz 200). Hence, due to the fact that the reaining nuber of inequalities is polynoial in ters of n, this LP-relaxation can be solved in tie polynoial in n by the equivalence of separation and optiization (Grötschel, Lovász, and Schriver 988). The following technical lea of Möhring et al. (999) is required later in the analysis. Lea 4.3 (Möhring et al. 999). Let C LP IR n be any point that satisfies the first and the last set of inequalities we then have fro (9). Assuing C LP C2 LP Cn LP for all V. 5 Results E P k ax k C LP We are now ready to prove our results for stochastic achine scheduling probles with precedence constraints. General precedence constraints. We consider the general proble with precedence constraints and release dates, P r prec E w C. Fro an optial solution to the LP-relaxation (9), we define a priority list L according to non-decreasing LP copletion ties C LP. It is perhaps interesting to note that inequalities C LP Ci LP E P, i A, are only required to ensure that the order according to nondecreasing LP copletion ties C LP is a linear extension of the precedence constraints. They are not required elsewhere in the analysis. Moreover, instead of the weaker inequalities C LP E P we could as well use C LP r E P, but this does not yield an iproveent of our results. Theore 5.. Consider an instance of the scheduling proble P r prec E w C with CV P for all P and soe 0. Let L be a priority list according to an optial solution C LP of the linear prograing relaxation (9). Then Algorith (with paraeter β 0) is an α-approxiation with α : β β ax (9) 8

9 Proof. Since L is a linear extension of the precedence constraints, Lea 3.8 yields E C P β E P β i B E P i r for any ob V. (Recall that B denotes the obs that coe before ob in the priority list L.) Lea 4.3 yields for all V. Hence, w E C P V β i B E P i ax C LP w E P β ax V w C LP V w r V Now, for any realization p of the processing ties, p C p by definition of a critical chain. Hence, the value E P is a lower bound on the expected copletion tie E C P of any ob, for any scheduling policy. Thus V w E P is a lower bound on the expected perforance of an optial scheduling policy. Moreover, both ters V w C LP and V w r are lower bounds on the expected perforance of an optial scheduling policy as well. This gives a perforance bound of β β ax Rearranging the ters yields the desired result. Notice that Theore 5. iplies a perforance bound of if β 2 and if the obs processing ties are distributed according to NBUE distributions (see Definition 4.2). In fact, the perforance bound in Theore 5. can be slightly iproved if release dates are absent. Theore 5.2. Consider an instance of the scheduling proble P prec E w C with CV P for all V and soe 0. Let L be a priority list according to an optial solution of the linear prograing relaxation (9). Then Algorith (with paraeter β 0) is an α-approxiation with α β β ax The tighter bound follows fro two odifications in the proof of Theore 5.. On the one hand, in the proof of Lea 3.8, one can show that E E P i i O P β E P The reason is that there are only achines available for the deliberate idle tie that is charged to obs which are scheduled out of order: Siultaneous to the deliberate idle tie, at least a ob fro the critical chain 2 h is in process. (This arguent does not hold if release dates are present, since deliberate idle tie could possibly accuulate before r.) On the other hand, it is iediate that the last ter r on the right hand side of (6) disappears. With these odifications, the clai follows exactly as in Theore 5.. In-forest precedence constraints. Let us now turn to the special case P in-forest E w C. In-forest precedence constraints are characterized by the fact that each ob has at ost one successor. Moreover, we assue that there are no release dates. For this proble, the results of the preceding section can be further iproved. We start with the following observation which is also contained in (Chekuri et al. 200, Lea 0); we give a short proof for the sake of copleteness. Lea 5.3. Consider the schedule constructed by Graha s list scheduling for any priority list L which is a linear extension of the (in-forest) precedence constraints, and any realization p of the processing ties. Then, in the interval r p S p there is no processing of obs in A. 9

10 Proof. Suppose the clai is false and aong all obs which violate it, let ob be one that is scheduled earliest. Obviously, S p r p, otherwise the clai is trivially true. In the interval r p S p no ob fro A is started, since is available fro tie r p on. Hence, there ust be soe ob k A that has been started before r p and that is still in process at r p. Thus r p 0. Denote by h the nuber of obs that are started at tie r p. All of these obs i have higher priority than, and the fact that is the first ob that violates the clai yields r i p r p. (At this point it is crucial that the priority list extends the precedence constraints.) In other words, for each of these obs a critical predecessor ends at tie r p, and due to the fact that the precedence constraints for an in-forest, all of these predecessors are different. Hence, including s critical predecessor, h different obs end at tie r p, but only h are started. This is a contradiction since ob is available at tie r p. Lea 5.4. For any (stochastic) instance of P in-forest and any priority list L which is a linear extension of the precedence constraints, the expected copletion tie of any ob under Graha s list scheduling fulfills E C P E P i B E P i (0) Proof. Consider any realization p of the processing ties. Given any ob, consider a critical chain for, consisting of obs 2 h and with total length p. The tie interval 0 C p can be partitioned into tie intervals where a ob fro a critical chain for is in process, and the reaining tie intervals. Due to Lea 5.3, in each tie interval r i p S i p there is no ob fro A i in process, for all i h. Moreover, there is no idle tie on any of the achines in these tie intervals (we consider Graha s list scheduling, and there are no release dates). Since A A 2 A h A, it follows that the only processing in these tie intervals is the obs in B, or ore precisely, in B p. In other words, the total processing in these tie intervals is at ost i B p i p. Hence, C p p p i i B for any realization p. Taking expectations, the clai follows. 0. Let L be a priority list according to an optial solution of the linear prograing relaxation (9). Then Graha s list scheduling is an α-approxiation with Theore 5.5. Consider an instance of P in-forest E w C with CV P for all V and soe α 2 ax Proof. Graha s list scheduling coincides with Algorith for β 0. The proof is therefore exactly the sae as the one of Theore 5., except that Lea 5.4 is used instead of Lea 3.8. For NBUE distributions (see Definition 4.2), Theore 5.5 yields a perforance guarantee of 3. Single achine probles. Theore 5.2 iplies a 2-approxiation for the special case of a single achine: In this case the ter β disappears, and we can choose β 0 to obtain perforance guarantee 2. (For β 0, the algorith corresponds to Graha s list scheduling.) This holds for arbitrarily distributed, independent processing ties. In fact, this atches the best bound currently known in the deterinistic setting; see Open Proble 9 in the collection of Schuuran and Woeginger (999). 6 Further Rearks A scheduling policy defines a apping of processing ties to start ties of obs. This apping has to be universally easurable in order to grant existence of the expected obective function value (Möhring et al. 984). Without going into further details we ust ention that the scheduling policies discussed in this paper fulfill this requireent (Uetz 200, Corollary 3.6.5). We point out that, apart fro the expected processing ties of the obs, a unifor upper bound on their coefficients of variation is the sole stochastic inforation required as input for the presented scheduling policy. Nevertheless, in our analysis we copare its perforance to a lower bound on the perforance of any non-anticipatory scheduling policy. This refers to the broadest possible sense of scheduling policies as defined by Möhring et al. 0

11 (984). In general, a policy is allowed to take advantage of the coplete knowledge of the conditional distributions of the processing ties, at any tie. Our analysis, however, does not take anticipatory scheduling policies into account since the linear prograing lower bound (9) does not hold in this case. This can be seen fro the observation that such a scheduling policy could, for instance, copute an optial schedule for any realization of the processing ties, and Theore 4. is no longer true in this case (Uetz 200). In other words, our analysis is based upon an adversary that is ust as powerful as the algorith itself. This constitutes a aor difference copared to the rather unfair copetitive analysis known fro on-line optiization. References Bruno, J. L., P. J. Downey, and G. N. Frederickson (98). Sequencing tasks with exponential service ties to iniize the expected flowtie or akespan. Journal of the Association for Coputing Machinery 28, Chakrabarti, S. and S. Muthukrishnan (996). Resource scheduling for parallel database and scientific applications. In Proceedings of the 8th Annual ACM Syposiu on Parallel Algoriths and Architectures, Padua, Italy, pp Chandy, K. M. and P. F. Reynolds (975). Scheduling partially ordered tasks with probabilistic execution ties. Operating Systes Review 9, Chekuri, C., R. Johnson, R. Motwani, B. Nataraan, B. Rau, and M. Schlansker (996). An analysis of profiledriven instruction level parallel scheduling with application to super blocks. In Proceedings of the 29th Annual IEEE/ACM International Syposiu on Microarchitecture, Paris (France), pp Chekuri, C., R. Motwani, B. Nataraan, and C. Stein (200). Approxiation techniques for average copletion tie scheduling. SIAM Journal on Coputing 3, Fiat, A. and G. J. Woeginger (Eds.) (998). Online Algoriths: The State of the Art, Volue 442 of Lecture Notes in Coputer Science. Berlin: Springer. Graha, R. L. (966). Bounds for certain ultiprocessing anoalies. Bell Syste Technical Journal 45, published in (Graha 969). Graha, R. L. (969). Bounds on ultiprocessing tiing anoalies. SIAM Journal on Applied Matheatics 7, Graha, R. L., E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan (979). Optiization and approxiation in deterinistic sequencing and scheduling: A survey. Annals of Discrete Matheatics 5, Grötschel, M., L. Lovász, and A. Schriver (988). Geoetric algoriths and cobinatorial optiization, Volue 2 of Algoriths and Cobinatorics. Berlin: Springer. Hall, W. J. and J. A. Wellner (98). Mean residual life. In M. Csörgö, D. A. Dawson, J. N. K. Rao, and A. K. Md. E. Saleh (Eds.), Proceedings of the International Syposiu on Statistics and Related Topics, Ottawa, Ontario, pp Asterda: North-Holland. Käpke, T. (987). On the optiality of static priority policies in stochastic scheduling on parallel achines. Journal of Applied Probability 24, Koutsoupias, E. and C. H. Papadiitriou (2000). Beyond copetitive analysis. SIAM Journal on Coputing 30, Möhring, R. H., F. J. Raderacher, and G. Weiss (984). Stochastic scheduling probles I: General strategies. ZOR - Zeitschrift für Operations Research 28, Möhring, R. H., A. S. Schulz, and M. Uetz (999). Approxiation in stochastic scheduling: The power of LP-based priority policies. Journal of the Association for Coputing Machinery 46, Munier, A., M. Queyranne, and A. S. Schulz (998). Approxiation bounds for a general class of precedence constrained parallel achine scheduling probles. In R. E. Bixby, E. A. Boyd, and R. Z. Ríos-Mercado (Eds.), Proceedings of the 6th International Conference on Integer Prograing and Cobinatorial Optiization, Houston (TX), Volue 42 of Lecture Notes in Coputer Science, pp Berlin: Springer. Journal version: SIAM Journal on Coputing, to appear.

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