Models and Algorithms for Stochastic Online Scheduling 1

Models and Algoriths for Stochastic Online Scheduling 1 Nicole Megow Technische Universität Berlin, Institut für Matheatik, Strasse des 17. Juni 136, 10623 Berlin, Gerany. eail: negow@ath.tu-berlin.de Marc Uetz Maastricht University, Departent of Quantitative Econoics, P.O. Box 616, 6200 MD Maastricht, The Netherlands. eail:.uetz@ke.uniaas.nl Tark Vredeveld 2 Maastricht University, Departent of Quantitative Econoics, P.O. Box 616, 6200 MD Maastricht, The Netherlands. eail: t.vredeveld@ke.uniaas.nl We consider a odel for scheduling under uncertainty. In this odel, we cobine the ain characteristics of online and stochastic scheduling in a siple and natural way. Job processing ties are assued to be stochastic, but in contrast to traditional stochastic scheduling odels, we assue that obs arrive online, and there is no knowledge about the obs that will arrive in the future. The odel incorporates both, stochastic scheduling and online scheduling as a special case. The particular setting we consider is non-preeptive parallel achine scheduling, with the obective to iniize the total weighted copletion ties of obs. We analyze siple, cobinatorial online scheduling policies for that odel, and derive perforance guarantees that atch perforance guarantees previously known for stochastic and online parallel achine scheduling, respectively. For processing ties that follow NBUE distributions, we iprove upon previously best known perforance bounds fro stochastic scheduling, even though we consider a ore general setting. Key words: scheduling; stochastic; online; total weighted copletion tie; approxiation MSC2000 Subect Classification: Priary: 90B36 ; Secondary: 68W40, 68W25, 68M20 OR/MS subect classification: Priary: Production/scheduling ; Secondary: approxiation/heuristic 1. Introduction. Scheduling on identical parallel achines to iniize the total weighted copletion tie of obs, P w C in the three-field notation of Graha et al. [12], is one of the classical probles in cobinatorial optiization. The proble plays a role whenever any obs ust be processed on a liited nuber of achines or processors, with applications in anufacturing, parallel coputing [5] or copiler optiization [6]. The literature has witnessed any papers on this proble, as well as its variant where the obs have individual release dates before which they ust not be processed, P r w C. In the off-line deterinistic) setting, where the set of obs and their characteristics are known in advance, the coplexity status of both probles is solved; both are strongly NP-hard [17], and they adit a polynoial tie approxiation schee [1, 28]. In order to cope with scenarios where there is uncertainty about the future, there are two aor fraeworks in the theory of scheduling, one is stochastic scheduling, the other is online scheduling. In stochastic scheduling the population of obs is assued to be known beforehand, but in contrast to deterinistic odels, the processing ties of obs are rando variables. The actual processing ties becoe known only upon copletion of the obs. The distribution functions of the respective rando variables, or at least their first oents, are assued to be known beforehand. In online scheduling, the assuption is that the instance is presented to the scheduler only piecewise. Jobs are either arriving one-by-one in the online-list odel), or over tie in the online-tie odel) [22]. The actual processing ties are usually disclosed upon arrival of a ob, and decisions ust be ade without any knowledge of the obs to coe. We consider a odel that generalizes both, stochastic scheduling and online scheduling. Like in online scheduling, we assue that the instance is presented to the scheduler piecewise, and nothing is known about obs that ight arrive in the future. Once a ob arrives, like in stochastic scheduling, we assue 1 Research partially supported by the DFG Research Center Matheon Matheatics for key technologies. An extended abstract with parts of this work appeared in the Proc. of the 2nd Workshop on Approxiation and Online Algoriths [19]. 2 Parts of this work were done while the author was with the Konrad-Zuse-Zentru für Inforationstechnik Berlin, Gerany. 1

2 Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS that its expected processing tie is disclosed, but the actual processing tie reains unknown until the ob copletes. Before we discuss the odel and related work in ore detail, let us fix soe basic notation and definitions. Model and definitions. Given is a set J = {1,..., n} of obs, with nonnegative weights w, J. In the odel with release dates, r denotes the earliest point in tie when ob can be started. Each ob ust be processed non-preeptively on any of the identical achines, and each achine can only handle one ob at a tie. The goal is to find a schedule that iniizes the total weighted copletion tie w C, where C denotes the copletion tie of ob. By P we denote the rando variable for the processing tie of ob, by E [P ] its expected processing tie, and by p a particular realization of P. The processing tie distributions P are assued to be independent. We assue that the obs are arriving over tie upon their respective release dates r in the order 1,..., n. Therefore, we can assue w.l.o.g. that r r k for < k. Notice that the nuber of obs n is not known in advance. When a ob arrives at tie r, the scheduler is infored about its weight w and its expected processing tie E [P ]. The goal is to find a stochastic online scheduling policy that iniizes the expected value of the weighted copletion ties of obs, E [ w C ]. Our definition of a stochastic online scheduling policy extends the traditional definition of stochastic scheduling policies by Möhring et al. [20] to the setting where obs arrive online. A scheduling policy specifies actions at decision ties t. An action is a set of obs that is started at tie t, and a next decision tie t > t at which the next action is taken, unless soe ob is released or ends at tie t < t. In that case, t becoes the next decision tie. In order to decide, the policy ay utilize the coplete inforation contained in the partial schedule up to tie t, as well as inforation about unscheduled obs that have arrived before t. However, a policy is required to be online, thus at any tie, it ust not utilize any inforation about obs that will be released in the future. Moreover, it needs to be non-anticipatory, thus at any tie, it ust not utilize the actual processing ties of obs that are scheduled or unscheduled) but not yet copleted. An optial scheduling policy is defined as a non-anticipatory scheduling policy that iniizes the obective function value in expectation. Notice that we do not assue that an optial policy needs to be online. Notice also that even an optial scheduling policy generally fails to yield an optial solution for all realizations of the processing ties; this because it is non-anticipatory. For an instance I, consisting of the nuber of achines, the set of obs J together with their release dates r, weights w and processing tie distributions P, let S ΠI) and CΠ I) denote the rando variables for start- and copletion ties of obs under policy Π. We also write S Π and C Π, for short. We let [ ] E [ΠI)] = E w C Π I) = w E [ C Π I) ] J J denote the expected perforance of a scheduling policy Π on instance I. Let us denote the above defined odel as SoS, for stochastic online scheduling. Generalizing the definitions by Möhring et al. in [21] for traditional stochastic scheduling, we define the perforance guarantee of a stochastic online scheduling policy as follows. Definition 1.1 A stochastic online scheduling policy SoS policy) Π is a ρ approxiation if, for soe ρ 1, and all instances I of the given proble, E [ΠI)] ρ E [OPTI)]. Here, OPTI) denotes an optial stochastic scheduling policy on the given instance I, assuing a priori knowledge of the set of obs J, their weights w, release dates r, and processing tie distributions P. The value ρ is called the perforance guarantee of policy Π. Notice that the stochastic online scheduling policy Π in this definition does not have a priori knowledge of the set of obs J, their weights w, release dates r, and processing tie distributions P. Policy Π is an online policy, and only learns about the existence of any ob upon its release date r. Hence, the policy has to copete with an adversary that knows the online sequence of obs in advance. However, with respect to the processing ties P of the obs, the adversary is ust as powerful as the policy Π itself, since it does not foresee their actual realizations p either.

Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS 3 Probably the best known scheduling policy in stochastic scheduling is the WSEPT rule. Its online version will also play a proinent role in this paper, and is defined next. To this end, call a ob available at a given tie t if it has not yet been scheduled, and if its release date has passed, that is, if r t. Definition 1.2 Online WSEPT, weighted shortest expected processing tie first) At any point in tie when a achine is idle, aong all obs that are available, schedule the ob with the highest ratio of weight over expected processing tie, w /E [P ]. Whenever release dates are absent, it reduces to the traditional WSEPT rule known fro stochastic scheduling, and the obs appear in the schedule in order of non-increasing ratios w /E [P ]. For unit weights, it reduces to SEPT, shortest expected processing tie first. For single achine stochastic) scheduling without release dates, 1 w C, the WSEPT rule is optial; this follows by a siple ob interchange arguent [23, 30]. Related work. Stochastic achine scheduling odels have been addressed ainly since the 1980s [9]. In the traditional stochastic setting, Weiss [34, 35] analyzes the perforance of the WSEPT rule. He derives additive perforance bounds for the stochastic parallel achine odel without release dates, P E [ w C ]. His bounds yield asyptotic optiality of the WSEPT rule for a certain class of processing tie distributions. More recently, approxiation algoriths for stochastic achine scheduling have been derived by Möhring et al. [21] and Skutella and Uetz [27]. In these papers, the expected perforance of the WSEPT rule, and linear prograing based stochastic scheduling policies, are copared against the expected perforance of an optial stochastic scheduling policy. The results are constant-factor approxiations for odels without or with release dates [21], and also with precedence constraints [27]. However, these papers do not address the situation where obs arrive online. In contrast to traditional stochastic scheduling, in online scheduling it is assued that nothing is known about obs that are about to arrive in the future. However, once a ob becoes known, its weight w and its actual processing tie p are disclosed. The quality of online algoriths is assessed by their copetitive ratio [15, 29]. An algorith is called ρ-copetitive if, for any instance, a solution is achieved with value not worse than ρ ties the value of an optial off-line solution. In the online-tie odel obs becoe known upon their release dates r. In the online-list odel, obs are presented one-by-one, all at tie 0. Upon presentation, a ob has to be scheduled iediately before the next ob can be seen at soe tie that is feasible with respect to release dates and the already scheduled obs). We oit further details and refer to Borodin and El-Yaniv [3] or Pruhs et al. [22]. In the online-list odel, Fiat and Woeginger [11] show that the single achine proble 1 r C does not allow a deterinistic or randoized online algorith with a copetitive ratio of log n. In the online-tie odel, Anderson and Potts [2] provide a 2-copetitive online algorith for the sae proble. This result is best possible, since Hoogeveen and Vestens [14] prove a lower bound of 2 on the copetitive ratio of any deterinistic online algorith. For settings with parallel achines, Vestens [33] proves a lower bound of 1.309 for the copetitive ratio of any deterinistic online algorith for P r w C, even for unit weights. The currently best known deterinistic algorith for P r w C is 2.62- copetitive, proposed by Correa and Wagner [8]. The currently best known randoized algorith for the proble has an expected copetitive ratio of 2; see Schulz and Skutella [26]. A odel that cobines features of stochastic and online scheduling has also been considered by Chou et al. [7]. They prove asyptotic optiality of the online WSEPT rule for the single achine proble 1 r w C, assuing that the weights w and processing tie p can be bounded fro above and below by constants. The definition of the adversary in their paper coincides with our definition. Hence, asyptotic optiality eans that the ratio of the expected perforance of the WSEPT rule over the expected perforance of an optial stochastic scheduling policy tends to 1 as the nuber of obs tends to infinity. A different type of analysis for stochastic scheduling has been proposed by Steger et al. [24, 31]. Both papers address the parallel achine odel without release dates. They copare the perforance of the W)SEPT rule to the optial solution per realization, and take the expectation of this ratio on the basis of the given processing tie distributions. Their analysis is thus different fro the traditional stochastic scheduling odel; in particular is it not based upon an coparison to the optial stochastic scheduling

4 Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS policy. The underlying adversary is stronger than in traditional stochastic scheduling, yet they derive constant bounds on what they call the expected copetitive ratio. Subsequently to our work and inspired by a recent paper [8], Schulz [25] gave a randoized 3- approxiation policy for the stochastic online version of P r E [ w C ], under the assuption of a certain class of processing tie distributions. As stated in his paper, a derandoized version of this policy atches our perforance guarantee for this special class of processing tie distributions. Results and ethodology. We propose siple, cobinatorial online scheduling policies for stochastic online scheduling odels on parallel achines with and without release dates, and derive constant perforance bounds for these policies. For identical parallel achine scheduling without release dates, P E [ w C ], the perforance guarantee is 1) + 1) ρ = 1 +. Here, is an upper bound on the squared coefficients of variation of the processing tie distributions P, that is, Var [P ]/E [P ] 2 for all obs. This perforance guarantee atches the previously best known perforance guarantee of Möhring et al. [21]; they obtain the sae bound for the perforance of the WSEPT rule in the traditional stochastic scheduling odel. However, we derive this bound in a ore restricted setting than traditional stochastic scheduling. We consider a stochastic online odel where the obs are presented to the scheduler sequentially, and the scheduler ust iediately and irrevocably assign obs to achines, without knowledge of the obs to coe. Once the obs are assigned to the achines, the obs on each achine can be sequenced in any order. We thus show that there exists a stochastic online policy in this restricted setting that achieves the sae perforance guarantee as the above entioned bound for the WSEPT rule. The traditional WSEPT rule is not a feasible policy in this setting, since the assignent of obs to achines depends on the realizations of processing ties.) For the odel with release dates we prove a slightly ore coplicated perforance guarantee that is valid for a class of processing tie distributions that we call δ-nbue, generalizing the well known class of NBUE distributions new better than used in expectation). Definition 1.3 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. For identical parallel achine scheduling with release dates P r w C, and δ-nbue distributions, we obtain a perforance guarantee of { ρ = 1 + ax 1 + δ } 1) + 1), + δ +. Here, > 0 is an arbitrary paraeter, and again, is an upper bound on the squared coefficients of variation Var[P ]/E [P ] 2 of the processing tie distributions P. For exaple for ordinary NBUE distributions, where = δ = 1, we obtain a perforance guarantee ρ < 3.62 1/). Thereby, we iprove upon the previously best known perforance guarantee of 4 1/ for traditional stochastic scheduling, which was derived for an LP based list scheduling policy [21]. Again, this iproved bound holds even though we consider an online odel in which the obs arrive over tie, and the scheduler does not know anything about the obs that are about to arrive in the future. Moreover, for deterinistic processing ties, where = 0 and δ = 1, we obtain a perforance guarantee ρ < 3.281, atching the bound fro deterinistic online scheduling by Megow and Schulz [18]. For both odels, our results are in fact achieved by fixed-assignent policies. That is, whenever a ob is presented to the scheduler, it is iediately and irrevocably assigned to a achine. The sequencing of obs on the individual achines is in both cases an online version of the traditional WSEPT rule. 2. Discussion and further preliinaries. As a atter of fact, results in stochastic scheduling either rely on the traditional W)SEPT rule [21, 24, 31, 34, 35] for odels without release dates, or they use linear prograing relaxations to define list scheduling policies other than WSEPT [21, 27]. As soon as we assue that obs arrive online, the approaches of these papers fail: The traditional off-line W)SEPT rule cannot be ipleented since it requires an a priori ordering of obs in order of ratios w /E [P ]. To

Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS 5 obtain the results for odels with release dates, Möhring et al. [21] use optial LP solutions not only for the purpose of analysis, but also to define the corresponding list scheduling policies. Although we still use the sae linear prograing relaxation as [21] within our analysis, the ain difference lies in the fact that the algoriths we propose are cobinatorial, and do not require the solution of linear progras. As in traditional online optiization, the adversary in the proposed stochastic online SoS) odel ay choose an arbitrary sequence of obs. These obs, however, are stochastic, with corresponding processing tie distributions. The actual processing ties are realized according to exogenous probability distributions. Thus, the best the adversary can do is indeed to use an optial stochastic scheduling policy in the traditional definition of stochastic scheduling policies by Möhring et al. [20]. In this view, our odel soewhat copares to the idea of a diffuse adversary as defined by Koutsoupias and Papadiitriou [16]. Since deterinistic processing ties are contained as a special case, however, all lower bounds on the approxiability known fro deterinistic online scheduling also hold for the SoS odel of the present paper. Hence, no SoS policy can exist with a perforance bound better than 1.309 [33]. Observe that the expected perforance of any stochastic online policy is by definition no less than the expected perforance of an optial policy for a corresponding traditional stochastic proble, where the set of obs J, their release dates r, weights w, and processing tie distributions P are given at the outset. Hence, lower bounds on the expected obective value of an optial stochastic scheduling policy carry over to the stochastic online setting that we consider in this paper. Therefore, we have the following lower bound on the perforance of any stochastic online scheduling policy; it is a generalization of a lower bound by Eastan et al. [10] to stochastic processing ties. Lea 2.1 Möhring et al. [21]) For any instance I of P r E [ w C ], we have that E [OPTI)] 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. Here, we have used a piece of notation that coes handy also later. For a given ob J, H) denotes the obs that have a higher priority in the order of ratios w /E [P ], that is { w k H) = k J E [P k ] > w } { w k k E [P ] E [P k ] = w }. E [P ] Accordingly, we define L) = J\H) as those obs that have lower priority in the order of ratios w /E [P ]. As a tie-breaking rule for obs k with equal ratio w k /E [P k ] = w /E [P ] we decide depending on the position in the online sequence relative to. That is, if k then k belongs to set H), otherwise it is included in set L). Notice that, by convention, we assue that H) contains ob, too. 3. Stochastic online scheduling on a single achine. In this section we consider the stochastic online scheduling proble on a single achine. When release dates are absent, it is well known that the WSEPT rule is optial [23, 30]. For the proble of scheduling obs with nontrivial release dates on a single achine, the currently best known result fro traditional stochastic scheduling is an LP-based list scheduling algorith with a perforance bound of 3 [21]. Inspired by a corresponding algorith for the deterinistic online setting on parallel achines fro [18], we propose the following scheduling policy. Algorith 1: -Shift-WSEPT Modify the release date r of each ob to r = ax{r, E [P ]}, for soe fixed > 0. At any tie t, when the achine is idle, start the ob with highest ratio w /E [P ] aong all available obs, respecting the odified release dates. In case of ties, sallest index first.)

6 Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS We first derive an upper bound on the expected copletion tie of a ob, E [ ] C, when scheduling obs on a single achine according to the -Shift-WSEPT policy. Lea 3.1 Let all processing ties be δ-nbue. Then the expected copletion tie of ob under - Shift-WSEPT on a single achine can be bounded by E [ C ] 1 + δ/) r + E [P k ]. k H) Proof. We consider soe ob. Let X denote a rando variable easuring the reaining processing tie of a ob being processed at tie r, if such a ob exists. Otherwise X has value 0. Moreover, for any ob k, let ξ k be an indicator rando variable that equals 1 if and only if ob k starts processing at the earliest at tie r, i.e., ξ k = 1 if and only if Sk r. The start of ob will be postponed beyond r by X, and until there are no ore higher priority obs available. Hence, the expected start tie of ob can be bounded by E [ S ] [ ] E r + X + P k ξ k = r + E [X] + r + E [X] + k H)\{} k H)\{} k H)\{} E [P k ξ k ] E [P k ], where the last inequality follows fro the fact that P k ξ k P k for any ob k. Next, we show that E [X] δ/)r. If the achine ust finishes a ob at tie r or is idle at that tie, X has value 0. Otherwise, soe ob l is in process at tie r. Note that this ob ight have lower or higher priority than ob. Such ob l was available at tie r l < r, and by definition of the odified release dates, we therefore know that E [P l ] 1/)r l < 1/)r for any such ob l. Moreover, letting t = r S l, the expected reaining processing tie of such ob l, given that it is indeed in process at tie r, is E [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/)r for any ob l that could be in process at tie r. Hence, we obtain E [X] δ/)r. Finally, the fact that E [ ] [ ] C = E S + E [P ] concludes the proof. In fact, it is quite straightforward to use Lea 3.1 in order to show the following. Theore 3.1 The -Shift-WSEPT algorith is a δ+2)-approxiation for the stochastic online single achine proble 1 r E [ w C ], for δ-nbue processing ties. The best choice for is = 1. Proof. With Lea 3.1 and the definition of odified release dates r = ax{r, E [P ]} we can bound the expected value of a schedule obtained by -Shift-WSEPT. w E [ C ] 1 + δ/) w ax {r, E [P ]} + w E [P k ] k H) = { } w ax 1 + δ/) r, + δ) E [P ] + w E [P k ] k H) { } ax 1 + δ/), + δ) w r + E [P ]) + w E [P k ]. k H) We can now apply the trivial lower bound w r + E [P ]) E [OPTI)], and exploit the fact that w k H) E [P k] E [OPTI)] by Lea 2.1 for = 1), and obtain w E [ C ] { 1 + ax 1 + δ } ), + δ E [OPTI)]. Now, = 1 iniizes this expression, independently of δ, and the theore follows.

Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS 7 Note that for NBUE processing ties, the result atches the currently best known perforance bound of 3 derived by Möhring et. al in [21] for the traditional stochastic scheduling odel. Their LP-based policy, however, requires an a priori knowledge of the set of obs J, their weights w, and their expected processing ties E [P ]. Moreover, in the deterinistic online setting, the best possible algorith is 2-copetitive [14], hence the corresponding lower bound of 2 holds for the stochastic online setting too. 4. Stochastic online scheduling on parallel achines. In this section, we define stochastic online scheduling policies for the proble on parallel achines. We first consider the proble without nontrivial release dates, and later generalize to the proble with nontrivial release dates. 4.1 Scheduling obs without release dates. In the case that all obs arrive at tie 0, the proble effectively turns into a traditional stochastic scheduling proble, P E [ w C ]. For that proble, it is known that the WSEPT rule yields a 1 + 1) + 1)/))-approxiation, being an upper bound on the squared coefficients of variation of the processing tie distributions [21]. Nevertheless, we consider an online variant of the proble P E [ w C ] that resebles the onlinelist odel fro online optiization. We assue that the obs are presented to the scheduler sequentially, and each ob ust iediately and irrevocably be assigned to a achine: a fixed-assignent policy. In particular, during this assignent phase, the scheduler does not know anything about the obs that are still about to coe. Once the obs are assigned to the achines, the obs on each achine ay be sequenced in any order. We show that an intuitive and siple fixed-assignent policy exists that eventually yields the sae perforance bound as the one proved in [21] for the WSEPT rule. In this context, recall that the WSEPT rule is not a feasible online policy in the considered online odel. We introduce the following notation: If a ob is assigned to achine i, this is denoted by i. Now we can define the MinIncrease policy as follows. Algorith 2: MinIncrease MI) When a ob is presented to the scheduler, it is assigned to the achine i that iniizes the expression z, i) = w E [P k ] + E [P ] w k + w E [P ]. k H), k<, k i k L), k<, k i Once all obs are assigned to achines, the obs on each achine are sequenced in order of non-increasing ratios w /E [P ]. Since WSEPT is known to be optial on a single achine, MinIncrease in fact assigns each ob to that achine where it causes the least increase in the expected obective value, given the previously assigned obs. This is expressed in the following lea. Lea 4.1 The expected obective value E [MII)] of the MinIncrease policy equals in i z, i). Proof. The assignent of obs to achines is independent of the realization of processing ties. Hence, the expected copletion tie E [C ] for soe ob that has been assigned to achine i by MinIncrease is E [C ] = E [P k ], k H), k i because all obs that are assigned to the sae achine i are sequenced in order of non-increasing ratios w /E [P ]. Now, weighted suation over all obs gives by linearity of expectation ] E [MII)] = E [ w C = w E [P k ] k H), k i = w E [P k ] + w E [P k ] + w E [P ]. k H), k i,k< k H), k i,k>

8 Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS This allows us to apply the following index rearrangeent. E [P ] Thus, we have E [MII)] = = = w w k H), k> E [P k ] = E [P ] E [P k ] + k H), k i,k< w E [P k ] + E [P ] k H), k i,k< in i z, i), k L), k< w k. 1) w E [P ] w k + k L), k i,k< ) w k + w E [P ] k L), k i,k< where the second equality akes use of 1) applied to each individual achine i, and the last equality holds since i is the achine iniizing z, i) over all achines i. Now, we can derive the following perforance guarantee for the MinIncrease policy. Theore 4.1 Consider the stochastic online scheduling proble on parallel achines, P E [ w C ], as described above. Given that Var[P ]/E [P ] 2 for all obs and soe constant 0, the MinIncrease policy is a ρ approxiation, where 1 ρ = 1 + 1) + 1) Proof. Fro Lea 4.1 we know that E [MII)] = in i z, i) and, thus, E [MII)] = { } in w E [P k ] + E [P ] w k + w E [P ] i k H), k<, k i k L), k<, k i w E [P k ] + E [P ] w E [P ], k H), k<. k L), k< w k ) + where the inequality holds because the least expected increase is not ore than the average expected increase over all achines. Now, we first apply the index rearrangeent 1) as in Lea 4.1, and then plug in the inequality of Lea 2.1. Using the trivial fact that w E [P ] is a lower bound for the expected perforance E [OPTI)] of an optial policy, we thus obtain E [MII)] 1 ) w E [P k ] + w E [P k ] + w E [P ] k H), k< k H), k> = 1 w E [P k ] + 1 w E [P ] k H) 1) 1) E [OPTI)] + ) 1 + E [OPTI)]. 1) + 1) w E [P ] + 1 w E [P ] As entioned above, this perforance guarantee atches the currently best known perforance guarantee for the traditional stochastic setting, which was derived for the perforance of the WSEPT rule in [21]. 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, notice that these two policies are indeed different; this follows fro siple exaples.

Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS 9 Lower bound for fixed assignent policies. The requireent of a fixed assignent of obs to achines beforehand ay be interpreted as ignoring the additional inforation that evolves over tie in the for of the actual realization of processing ties. In the following, we therefore give a lower bound on the expected perforance E [FIXI)] 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 4.2 For stochastic parallel achine scheduling with unit weights and i.i.d. exponential processing ties, P p exp1) E [ C ], there exist instances I such that E [FIXI)] 3 2 1) E [OPTI)] ε, for any ε > 0. Here, 3 2 1) 1.24. Hence, no policy that uses fixed 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 can be obtained for arbitrary distributions. However, since our perforance guarantees, as in [21], 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 on the perforance of any fixed-assignent policy, and for that case the perforance bound of MinIncrease equals 2 1/ = 1.5. Proof of Theore 4.2. Let us consider an instance with achines and n = +k exponentially distributed obs, P exp1), where k 1 is an integer. The optial stochastic scheduling policy is SEPT, shortest expected processing tie first [4, 36], and the expected perforance is see, e.g., [32, Cor. 3.5.17]) E [OPTI)] = E [SEPTI)] = E [ C SEPT ] n kk + 1) = + = + k +. =+1 When, in a fixed assignent, one achine has to process at least two obs ore than another achine, the assignent can be iproved by oving one ob fro the ost loaded achine to the least loaded achine. Therefore, the best fixed assignent policy tries to distribute the obs evenly over the achines. That is, it assigns 1 + k obs to k + k achines and 1 + k obs to k k achines. Hence, there are obs with E [C ] = l for each l in the range 1,..., 1 + k, and k k obs with E [C ]=2 + k. The expected perforance for the best fixed assignent policy FIX is E [FIXI)] = E [ C FIX ] = + 2k + k 2 ) k k. 2 For < k, the value of E [FIXI)] is equal to 3k. Hence, for < k, the ratio E [FIXI)]/E [OPTI)] is E [FIXI)] E [OPTI)] = 3k + k + kk + 1)/), which is axiized for k) {, }. With this choice of k, the ratio E [FIXI)]/E [OPTI)] tends to 3 2/2 + 2) = 3 2 1) 1.24 as tends to infinity. Notice that the lower bound of 1.24 holds whenever, the nuber of achines, tends to infinity. For saller nubers of achines, e.g. = 2, 3, or 4, we use saller nubers k = k), naely k2) = 1, k3) = 2, and k4) = 2, and obtain the lower bounds 8/7 1.14, 7/6 1.16, and 32/27 1.18. Lower bound for MinIncrease. The lower bound on the perforance ratio for any fixed assignent policy given in Theore 4.2 holds for the MinIncrease policy, too. Hence, MinIncrease cannot be better than 1.24-approxiative. For general i.e., non-exponential) probability distributions we obtain a lower bound of 3/2 on the expected perforance of MinIncrease relative to the expected perforance of an optial scheduling policy, as shown by the following instance. Exaple 4.1 The instance consists of n 1 deterinistic unit length obs and one ob with a stochastic two-point distributed processing tie. There are = 2 achines, and we assue that n, the nuber

10 Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS of obs is even. The n 1 deterinistic obs have unit weight w = 1; they appear first in the online sequence. The final ob in the online sequence is the stochastic ob. It has processing tie p = n 2 /4 with probability 2/n, and p = 1 with probability 1 2/n. The weight w of the stochastic ob equals the value of its expected processing tie, i.e. 1 2/n + n/2. The MinIncrease policy assigns n/2 1 deterinistic obs to one achine, and n/2 deterinistic obs to the other. The stochastic ob is assigned to the achine with n/2 1 deterinistic obs. Hence, the expected obective value of the schedule under MinIncrease is E [ w C ] = 3n 2 /4 + on 2 ). An optial stochastic policy would start the uncertain ob and one deterinistic ob at tie 0. At tie t = 1 it is known if the stochastic ob has copleted, or if it blocks the achine for another n 2 /4 1 tie units. If the stochastic ob has copleted then the reaining unit obs are distributed equally on both achines, otherwise all deterinistic obs are scheduled on the sae achine. Thus, the expected obective value of an optial schedule is E [OPTI)] = n 2 /2 + on 2 ). The ratio of both values tends to 3/2 if the nuber of obs tends to infinity. Notice, however, that this result is less eaningful in coparison to the perforance bound of Theore 4.1, which depends on an upper bound on the squared coefficient of variation. 4.2 Scheduling obs with nontrivial release dates. In this section, we consider the setting where obs arrive over tie, that is, the stochastic online version of P r E [ w C ]. The ain idea is to adopt the MinIncrease policy to this setting. However, the difference is that we are no longer equipped with an optial policy as it was WSEPT in the previous section) to schedule the obs that are assigned to a single achine. In addition, even if we knew such a policy for a single achine, it would not be straightforward how to use it in the setting with parallel achines to define a feasible online scheduling policy. However, we propose to use the -Shift-WSEPT rule as introduced in Section 3 to sequence the obs that we have assigned to the sae achine. The assignent of obs to achines, on the other hand, reains the sae as before in the case without release dates. In a sense, when assigning obs to achines, we thus ignore the possible gain of inforation that occurs over tie in the online-tie odel. Algorith 3: Modified MinIncrease When a ob is presented to the scheduler at its release date r, it is assigned to the achine i that iniizes the expression z, i) = w E [P k ] + E [P ] w k + w E [P ]. k H), k<, k i k L), k<, k i On each achine, the obs assigned to this achine are sequenced according to the -Shift-WSEPT rule. The crucial observation is that the -Shift-WSEPT policy on achine i learns about ob s existence iediately at tie r. Hence, for each single achine, it is indeed feasible to use the -Shift-WSEPT rule, and the so-defined policy is a feasible SoS policy. Theore 4.3 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 odified MinIncrease policy running -Shift-WSEPT on each single achine is a ρ approxiation, where ρ = 1 + ax { 1 + δ, + δ + 1) + 1) Here, is such that Var[P ]/E [P ] 2 for all obs. In particular, since all processing ties P are δ-nbue, Lea A.1 yields that 2δ 1, hence ρ 1 + ax {1 + δ/, + δ 1)/}. }, Proof. Let i be the achine to which ob is assigned. Then, by Lea 3.1 we know that E [C ] 1 + δ ) r + E [P k ], 2) k H), k i

Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS 11 and the expected value of MinIncrease can be bounded by E [MII)] 1 + δ ) w r + w E [P k ]. 3) k H), k i For the second part of the right hand side of 3), we can use the sae index rearrangeent as in the proof of Lea 4.1; see 1). We thus obtain w E [P k ] = w E [P k ] + E [P ] w k + w E [P ]. k H), k i k H), k i, k< k L), k i, k< By definition of the odified MinIncrease algorith, we know that any ob is assigned to the achine which iniizes the ter in parenthesis. Hence, by the sae averaging arguent as before, we know that w E [P k ] E [P k ] w + E [P w k ] + w E [P ] k H), k i = w k H), k< k H) E [P k ] + 1 k L), k< w E [P ], where the last equality again follows fro index rearrangeent. Plugging this into 3), leads to the following bound on the expected perforance of MinIncrease. E [MII)] 1 + δ ) w r + E [P k ] w + 1 w E [P ]. k H) Applying the bound of Lea 2.1 into the above inequality, we obtain E [MII)] 1 + δ ) w r 1) + 1) + E [OPTI)] + w E [P ] = E [OPTI)] + w 1 + δ ) ) r 1) + 1) + E [P ], 4) where again, is an upper bound on the squared coefficient of variation of the processing tie distributions P. By bounding r by r + E [P ], we obtain the following bound on the ter in parenthesis of the right hand side of 4). 1 + δ ) r + 1) + 1) E [P ] 1 + δ ) r + + δ + ) r + E [P ] ax { 1 + δ ) 1) + 1) E [P ], + δ + 1) + 1) By using this inequality in equation 4), and applying the trivial lower bound w r + E [P ]) E [OPTI)] on the expected optiu perforance, we get the claied perforance bound of { ρ = 1 + ax 1 + δ } 1) + 1), + δ +. Since all processing ties are δ-nbue, we know by Lea A.1 in the appendix that 2δ 1 and thus the second clai of the theore follows, naely ρ 1 + ax{1 + δ/, + δ 1)/}. For NBUE processing ties, where = δ = 1, Theore 4.3 yields a perforance bound of { 1 ρ = 2 + ax, + 1 }. This ter is inial for = 5 2 + 1 + 1)/), which yields a ratio of ρ = 2 + 5 2 + 1 + 1)/). This is less than 5 + 5)/2 1/) 3.62 1/) iproving }.

12 Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS upon the previously best known bound of 4 1/ fro [21] for the traditional stochastic proble. More generally, for δ-nbue processing ties, optiizing the ter ax{1+δ/, +δ 1)/} for yields ρ < 3/2 + δ 1)/ + 4δ 2 + 1/2. Moreover, for deterinistic processing ties, where = 0 and δ = 1, Theore 4.3 yields a perforance bound of { 1 ρ = 2 + ax, + 1 }. Optiizing for yields = 17 2 + 1 + 1)/4), which yields a ratio of ρ = 2 + 17 2 + 1 + 1)/4). This is less than 3.281 for any value of, leaving only a sall gap to the currently best known bound of 2.62 for deterinistic online scheduling [8]. In fact, it atches the copetitive ratio of the deterinistic parallel achine version of -Shift-WSEPT fro [18]. 4.3 Randoized ob assignent. As a atter of fact, the MinIncrease policy can be interpreted as the derandoized version of a policy that assign obs uniforly at rando to the achines. Even though randoly assigning obs to achines ignores uch inforation, it is nevertheless known to be quite powerful as has been observed already by, e.g., Schulz and Skutella [26]. They apply a rando assignent strategy, based on the solution of an LP-relaxation, for scheduling obs with deterinistic processing ties on unrelated achines. For the special case of identical achines, their approach corresponds to assigning obs uniforly at rando to the achines. The rando assignent strategy for the stochastic online scheduling proble at hand is as follows. Algorith 4: RandAssign When a ob is presented to the scheduler, it is assigned to achine i with probability 1/ for all i = 1,...,. The obs assigned to achine i are scheduled according to the -Shift-WSEPT policy. Theore 4.4 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 RandAssign policy running -Shift- WSEPT on each single achine is a ρ approxiation, where ρ = 1 + ax { 1 + δ, + δ + 1) + 1) Here, is such that Var[P ]/E [P ] 2 for all obs. In particular, since all processing ties P are δ-nbue, Lea A.1 yields that 2δ 1, hence ρ 1 + ax {1 + δ/, + δ 1)/}. Proof. Consider a ob and let i denote the achine to which it has been assigned. Let Pr [ i] be the probability for ob being assigned to achine i. Then, by Lea 3.1 we know that E [C i] 1 + δ )r + Pr [k i i] E [P k i] = k H) 1 + δ )r + k H) }, Pr [k i i] E [P k ]. The probability that a ob is assigned to a certain achine is equal for all achines, i.e., Pr [ i] = 1/ for all i = 1,...,, and for any ob. Unconditioning the expected value of ob s copletion tie yields E [C ] = Pr [ i] E [C i] = i=1 1 + δ ) r + Pr [ i] i=1 1 + δ )r + k H) k H) E [P k ] + 1 E [P ], Pr [k i i] E [P k ]

Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS 13 where the last equality is due to the independence of the ob assignents to the achines. Then, the expected obective value of RandAssign, E [RAI)], can be bounded by E [RAI)] = w E [C ] 1 + δ ) w r + w k H) E [P k ] + 1 w E [P ]. This bound equals the upper bound that we achieved on the expected perforance of the odified) MinIncrease policy in the proof of Theore 4.3. Hence, we conclude the proof in the sae way and with the sae result as for MinIncrease. Acknowledgents. The authors thank the two anonyous referees for valuable coents, and Martin Skutella for pointing out that the odified) MinIncrease algorith can be interpreted as the derandoization of a randoized algorith. Thanks to Rolf H. Möhring for helpful discussions, and Andreas S. Schulz and Sven O. Kruke for pointing out soe flaws in an earlier version of this paper. References [1] F. N. Afrati, E. Bapis, C. Chekuri, D. R. Karger, C. Kenyon, S. Khanna, I. Milis, M. Queyranne, M. Skutella, C. Stein, and M. Sviridenko, Approxiation schees for iniizing average weighted copletion tie with release dates, Proc. 40th Ann. Syp. on Foundations of Coputer Science, 1999, pp. 32 43. [2] E. J. Anderson and C. N. Potts, On-line scheduling of a single achine to iniize total weighted copletion tie, Matheatics of Operations Research 29 2004), 686 697. [3] A. Borodin and R. El-Yaniv, Online coputation and copetitive analysis, Cabridge University Press, 1998. [4] J. L. Bruno, P. J. Downey, and G. N. Frederickson, Sequencing tasks with exponential service ties to iniize the expected flowtie or akespan, Journal of the ACM 28 1981), 100 113. [5] S. Chakrabarti and S. Muthukrishnan, Resource scheduling for parallel database and scientific applications, Proc. 8th Ann. ACM Syp. on Parallel Algoriths and Architectures, 1996, pp. 329 335. [6] C. Chekuri, R. Johnson, R. Motwani, B. Nataraan, B. Rau, and M. Schlansker, An analysis of profile-driven instruction level parallel scheduling with application to super blocks, Proc. 29th IEEE/ACM Int. Syp. on Microarchitecture, 1996, pp. 58 69. [7] M.C. Chou, H. Liu, M. Queyranne, and D. Sichi-Levi, On the asyptotic optiality of a siple online algorith for the stochastic single achine weighted copletion tie proble and its extensions, 2005, Operations Research, to appear. [8] J. Correa and M. Wagner, LP-based online scheduling: Fro single to parallel achines, Integer Prograing and Cobinatorial Optiization M. Jünger and V. Kaibel, eds.), Lecture Notes in Coputer Science, vol. 3509, Springer, 2005, pp. 196 209. [9] M. A. H. Depster, J. K. Lenstra, and A. H. G. Rinnooy Kan editors), Deterinistic and stochastic scheduling, D. Reidel Publishing Copany, Dordrecht, 1982. [10] W. Eastan, S. Even, and I. Isaacs, Bounds for the optial scheduling of n obs on processors, Manageent Science 11 1964), 268 279. [11] A. Fiat and G. J. Woeginger, On-line scheduling on a single achine: Miniizing the total copletion tie, Acta Inforatica 36 1999), 287 293. [12] 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, Annals of Discrete Matheatics 5 1979), 287 326. [13] W. J. Hall and J. A. Wellner, Mean residual life, Proc. Int. Syp. on Statistics and Related Topics Ottawa, ON, Canada) M. Csörgö, D. A. Dawson, J. N. K. Rao, and A. K. Md. E. Saleh, eds.), North-Holland, Asterda, The Netherlands, 1981, pp. 169 184. [14] H. Hoogeveen and A. P. A. Vestens, Optial on-line algoriths for single-achine scheduling, Integer Prograing and Cobinatorial Optiization W. H. Cunningha, S. T. McCorick, and M. Queyranne, eds.), Lecture Notes in Coputer Science, vol. 1084, Springer, 1996, pp. 404 414.

14 Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS [15] A. Karlin, M. Manasse, L. Rudolph, and D. Sleator, Copetitive snoopy paging, Algorithica 3 1988), 70 119. [16] E. Koutsoupias and C. H. Papadiitriou, Beyond copetitive analysis, SIAM Journal on Coputing 30 2000), 300 317. [17] E. L. Lenstra, A. H. G. Rinooy Kan, and P. Brucker, Coplexity of achine scheduling probles, Annals of Discrete Matheatics 1 1977), 243 362. [18] N. Megow and A. S. Schulz, On-line scheduling to iniize average copletion tie revisited, Operations Research Letters 32 2004), 485 490. [19] N. Megow, M. Uetz, and T. Vredeveld, Stochastic online scheduling on parallel achines, Approxiation and Online Algoriths G. Persiano and R. Solis-Oba, eds.), Lecture Notes in Coputer Science, vol. 3351, Springer, 2005, pp. 167 180. [20] R. H. Möhring, F. J. Raderacher, and G. Weiss, Stochastic scheduling probles I: General strategies, ZOR - Zeitschrift für Operations Research 28 1984), 193 260. [21] R. H. Möhring, A. S. Schulz, and M. Uetz, Approxiation in stochastic scheduling: the power of LP-based priority policies, Journal of the ACM 46 1999), 924 942. [22] K. Pruhs, J. Sgall, and E. Torng, Online scheduling, Handbook of Scheduling: Algoriths, Models, and Perforance Analysis J. Leung, ed.), CRC Press, 2004. [23] M. H. Rothkopf, Scheduling with rando service ties, Manageent Science 12 1966), 703 713. [24] M. Scharbrodt, T. Schickinger, and A. Steger, A new average case analysis for copletion tie scheduling, Proc. 34th Ann. ACM Syp. on the Theory of Coputing, 2002, pp. 170 178. [25] A. S. Schulz, New old algoriths for stochastic scheduling, Algoriths for Optiization with Incoplete Inforation S. Albers, R. H. Möhring, G. Ch. Pflug, and R. Schultz, eds.), Dagstuhl Seinar Proceedings, no. 05031, Internationales Begegnungs- und Forschungszentru IBFI), Schloss Dagstuhl, Gerany, 2005. [26] A. S. Schulz and M. Skutella, Scheduling unrelated achines by randoized rounding, SIAM Journal on Discrete Matheatics 15 2002), 450 469. [27] M. Skutella and M. Uetz, Stochastic achine scheduling with precedence constraints, SIAM Journal on Coputing 34 2005), 788 802. [28] M. Skutella and G.J. Woeginger, A PTAS for iniizing the total weighted copletion tie on identical parallel achines, Matheatics of Operations Research 25 2000), 63 75. [29] D. Sleator and R. Taran, Aortized efficiency of list update and paging rules, Counications of the ACM 28 1985), 202 208. [30] W. Sith, Various optiizers for single-stage production, Naval Research Logistics Quarterly 3 1956), 59 66. [31] A. Souza and A. Steger, The expected copetitive ratio for weighted copletion tie scheduling, Proc. 21st Syp. on Theoretical Aspects of Coputer Science V. Diekert and M. Habib, eds.), Lecture Notes in Coputer Science, vol. 2996, Springer, 2004, pp. 620 631. [32] M. Uetz, Algoriths for deterinistic and stochastic scheduling, Cuvillier Verlag, Göttingen, Gerany, 2002. [33] A. P. A. Vestens, On-line achine scheduling, Ph.D. thesis, Eindhoven University of Technology, Netherlands, 1997. [34] G. Weiss, Approxiation results in parallel achines stochastic scheduling, Annals of Operations Research 26 1990), 195 242. [35], Turnpike optiality of Sith s rule in parallel achines stochastic scheduling, Matheatics of Operations Research 17 1992), 255 270. [36] G. Weiss and M. Pinedo, Scheduling tasks with exponential service ties on non-identical processors to iniize various cost functions, Journal of Applied Probability 17 1980), 187 202.

Megow, Uetz, Vredeveld: Models and Algoriths for Stochastic Online Scheduling Matheatics of Operations Research xxx), pp. xxx xxx, c 200x INFORMS 15 Appendix A. Coefficient of variation for δ-nbue rando variables In this section, we show the relation between the value of δ and the squared) coefficient of variation Var [X] /E [X] 2 for a δ-nbue rando variable X. Lea A.1 Let X be a δ-nbue rando variable and let CV[X] = Var [X]/E [X] denote the coefficient of variation of X. Then CV[X] 2 2δ 1. Proof. We prove the lea for continuous rando variables X; the proof for discrete rando variables goes along the sae lines. Let X be a non-negative δ-nbue rando variable, with cuulative distribution function F and density f. By definition of conditional expectation, we know that [ ] E X t X > t x t)fx)dx t =. 5) 1 F t) As x t = x t dy, we can write the noinator of the right hand side as 0 x=t x t)fx)dx = = x t x=t x=t y=0 fx)dydx = y=0 x=y+t fx)dxdy 1 F x)dx, 6) where the second equality is obtained by changing the order of integration. As X is δ-nbue, i.e., E [X t X > t] δe [X], it follows fro 5) and 6) that [ ] 1 F x)dx = E X t X > t 1 F t)) δe [X] 1 F t)). 7) x=t By integrating the right hand side of the above inequality over t, we obtain δe [X] t=0 1 F t)dt = δe [X] 2. 8) Hall and Wellner [13, Equality 4.1)] showed that integrating the left hand side of 7) over t yields t=0 Hence, using 8) and 9) in 7), we have x=t 1 F x)dxdt = 1 2 E [ X 2]. 9) E [ X 2] 2δE [X] 2. Rearranging ters yields the desired bound on the squared coefficient of variation: CV[X] 2 = E [ X 2] E [X] 2 E [X] 2 2δ 1.