Models and Algorithms for Stochastic Online Scheduling

Transcription

1 MATHEMATICS OF OPERATIONS RESEARCH Vol. 31, No. 3, August 2006, pp issn X eissn infors doi /oor INFORMS Models and Algoriths for Stochastic Online Scheduling Nicole Mego Institut für Matheatik, Technische Universität Berlin, Strasse des 17. Juni 136, Berlin, Gerany, Marc Uetz, Tark Vredeveld Departent of Quantitative Econoics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands We consider a odel for scheduling under uncertainty. In this odel, e cobine the ain characteristics of online and stochastic scheduling in a siple and natural ay. Job processing ties are assued to be stochastic, but in contrast to traditional stochastic scheduling odels, e assue that obs arrive online, and there is no knoledge about the obs that ill arrive in the future. The odel incorporates both stochastic scheduling and online scheduling as a special case. The particular setting e consider is nonpreeptive parallel achine scheduling, ith the obective to iniize the total eighted copletion ties of obs. We analyze siple, cobinatorial online scheduling policies for that odel, and derive perforance guarantees that atch perforance guarantees previously knon for stochastic and online parallel achine scheduling, respectively. For processing ties that follo ne better than used in expectation NBUE distributions, e iprove upon previously best-knon perforance bounds fro stochastic scheduling, even though e consider a ore general setting. Key ords: scheduling; stochastic dynaic optiization; online optiization; total eighted copletion tie; approxiation MSC2000 subect classification: Priary: 90B36; secondary: 68W40, 68W25, 68M20 OR/MS subect classification: Priary: production/scheduling; secondary: approxiation/heuristic History: Received January 6, 2005; revised Septeber 7, Introduction. Scheduling on identical parallel achines to iniize the total eighted copletion tie of obs, P 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 henever any obs ust be processed on a liited nuber of achines or processors, ith applications in anufacturing, parallel coputing Chakrabarti and Muthukrishnan [5], or copiler optiization Chekuri et al. [6]. The literature has itnessed any papers on this proble as ell as its variant, here the obs have individual release dates before hich they ust not be processed, Pr C. In the offline deterinistic setting, here the set of obs and their characteristics are knon in advance, the coplexity status of both probles is solved; both are strongly NP-hard Lenstra et al. [17] and they adit a polynoial tie approxiation schee Afrati et al. [1], Skutella and Woeginger [28]. To cope ith scenarios here there is uncertainty about the future, there are to aor fraeorks in the theory of scheduling: 1 stochastic scheduling and 2 online scheduling. In stochastic scheduling, the population of obs is assued to be knon beforehand, but in contrast to deterinistic odels, the processing ties of obs are rando variables. The actual processing ties becoe knon only upon copletion of the obs. The distribution functions of the respective rando variables, or at least their first oents, are assued to be knon beforehand. In online scheduling, the assuption is that the instance is presented to the scheduler only pieceise. Jobs are either arriving one by one in the online list odel or over tie in the online tie odel Pruhs et al. [22]. The actual processing ties are usually disclosed upon arrival of a ob, and decisions ust be ade ithout any knoledge of the obs to coe. We consider a odel that generalizes both stochastic scheduling and online scheduling. Like in online scheduling, e assue that the instance is presented to the scheduler pieceise, and nothing is knon about obs that ight arrive in the future. Once a ob arrives, like in stochastic scheduling, e assue that its expected processing tie is disclosed, but the actual processing tie reains unknon until the ob copletes. Before e discuss the odel and related ork in ore detail, let us fix soe basic notation and definitions. Model and definitions. Given is a set J = 1n of obs, ith nonnegative eights, J. In the odel ith release dates, r denotes the earliest point in tie hen ob can be started. Each ob ust be processed nonpreeptively 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 eighted copletion tie C, here C denotes the copletion tie of ob. ByP, e denote the rando variable for the processing tie of ob, by ƐP, 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 1n. Therefore, e can assue.l.o.g. that r r k for <k. Note that the nuber 513

2 514 Matheatics of Operations Research 313, pp , 2006 INFORMS of obs n is not knon in advance. When a ob arrives at tie r, the scheduler is infored about its eight and its expected processing tie, ƐP. The goal is to find a stochastic online scheduling SOS policy that iniizes the expected value of the eighted copletion ties of obs, Ɛ C. Our definition of an SOS policy extends the traditional definition of stochastic scheduling policies by Möhring et al. [20] to the setting here 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 hich 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. To decide, the policy ay utilize the coplete inforation contained in the partial schedule up to tie t, as ell as inforation about unscheduled obs that have arrived before t. Hoever, a policy is required to be online, thus at any tie, it ust not utilize any inforation about obs that ill be released in the future. Moreover, it needs to be nonanticipatory, 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 nonanticipatory scheduling policy that iniizes the obective function value in expectation. Note that e do not assue that an optial policy needs to be online. Note also that even an optial scheduling policy generally fails to yield an optial solution for all realizations of the processing ties; this is because it is nonanticipatory. For an instance I, consisting of the nuber of achines, the set of obs J together ith their release dates r, eights, and processing tie distributions P, let S I and C I denote the rando variables for start and copletion ties of obs under policy. We also rite S and C for short. We let [ ] ƐI = Ɛ C I = ƐC I J J denote the expected perforance of a scheduling policy on instance I. Let us denote the above-defined odel as stochastic online scheduling SOS. Generalizing the definitions by Möhring et al. in [21] for traditional stochastic scheduling, e define the perforance guarantee of an SOS policy as follos. Definition 1.1. An SOS policy is a -approxiation if, for soe 1, and all instances I of the given proble, ƐI ƐOPTI Here, OPTI denotes an optial stochastic scheduling policy on the given instance I, assuing a priori knoledge of the set of obs J, their eights, release dates r, and processing tie distributions P. The value is called the perforance guarantee of policy. Note that the SOS policy in this definition does not have a priori knoledge of the set of obs J, their eights, release dates r, and processing tie distributions P. Policy is an online policy and only learns about the existence of any ob upon its release date r. Hence, the policy has to copete ith an adversary that knos the online sequence of obs in advance. Hoever, ith respect to the processing ties P of the obs, the adversary is ust as poerful as policy itself because it does not foresee their actual realizations p either. Probably the best-knon scheduling policy in stochastic scheduling is the rule eighted shortest expected processing tie WSEPT first. Its online version ill 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. At any point in tie hen a achine is idle, aong all obs that are available, schedule the ob ith the highest ratio of eight over expected processing tie, /ƐP. Whenever release dates are absent, it reduces to the traditional WSEPT rule knon fro stochastic scheduling and the obs appear in the schedule in order of nonincreasing ratios /ƐP. For unit eights, it reduces to shortest expected processing tie SEPT first. For single achine stochastic scheduling ithout release dates, 1 C, the WSEPT rule is optial; this follos by a siple ob interchange arguent Rothkopf [23], Sith [30]. Related ork. Stochastic achine scheduling odels have been addressed ainly since the 1980s Depster et al. [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 ithout release dates, P Ɛ 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

3 Matheatics of Operations Research 313, pp , 2006 INFORMS 515 and linear prograing LP-based stochastic scheduling policies are copared against the expected perforance of an optial stochastic scheduling policy. The results are constant-factor approxiations for odels ithout or ith release dates Möhring et al. [21] and also ith precedence constraints Skutella and Uetz [27]. Hoever, these papers do not address the situation here obs arrive online. In contrast to traditional stochastic scheduling, in online scheduling it is assued that nothing is knon about obs that are about to arrive in the future. Hoever, once a ob becoes knon, its eight and its actual processing tie p are disclosed. The quality of online algoriths is assessed by their copetitive ratio Karlin et al. [15], Sleator and Taran [29]. An algorith is called -copetitive if, for any instance, a solution is achieved ith value not orse than ties the value of an optial offline solution. In the online tie odel, obs becoe knon 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 ith 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] sho that the single-achine proble 1r C does not allo a deterinistic or randoized online algorith ith 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 because Hoogeveen and Vestens [14] prove a loer bound of 2 on the copetitive ratio of any deterinistic online algorith. For settings ith parallel achines, Vestens [33] proves a loer bound of for the copetitive ratio of any deterinistic online algorith for Pr C, even for unit eights. The currently best-knon deterinistic algorith for Pr C is 2.62-copetitive, proposed by Correa and Wagner [8]. The currently best-knon 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 1r C, assuing that the eights and processing tie p can be bounded fro above and belo by constants. The definition of the adversary in their paper coincides ith 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 Scharbrodt et al. [24] and Souza and Steger [31]. Both papers address the parallel achine odel ithout release dates. They copare the perforance of the WSEPT 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, it is not based on a coparison to the optial stochastic scheduling policy. The underlying adversary is stronger than in traditional stochastic scheduling, yet they derive constant bounds on hat they call the expected copetitive ratio. Subsequently to our ork and inspired by a recent paper Correa and Wagner [8], Schulz [25] gavea randoized 3-approxiation policy for the stochastic online version of Pr Ɛ 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 SOS odels on parallel achines ith and ithout release dates, and derive constant perforance bounds for these policies. For identical parallel achine scheduling ithout release dates, P Ɛ C, the perforance guarantee is = Here, is an upper bound on the squared coefficients of variation of the processing tie distributions P, that is, VarP /ƐP 2 for all obs. This perforance guarantee atches the previously best-knon 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. Hoever, e derive this bound in a ore restricted setting than traditional stochastic scheduling. We consider a stochastic online odel here the obs are presented to the scheduler sequentially, and the scheduler ust iediately and irrevocably assign obs to achines, ithout knoledge 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 sho 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

4 516 Matheatics of Operations Research 313, pp , 2006 INFORMS WSEPT rule is not a feasible policy in this setting because the assignent of obs to achines depends on the realizations of processing ties. For the odel ith release dates, e prove a slightly ore coplicated perforance guarantee that is valid for a class of processing tie distributions that e call -NBUE, generalizing the ell-knon class of NBUE distributions ne better than used in expectation. Definition 1.3 -NBUE. A nonnegative rando variable X is -NBUE if, for 1, ƐX t X>t ƐX for all t 0. For identical parallel achine scheduling ith release dates Pr C and -NBUE distributions, e obtain a perforance guarantee of { = 1 + ax 1 + } Here, >0 is an arbitrary paraeter, and again, is an upper bound on the squared coefficients of variation VarP /ƐP 2 of the processing tie distributions P. For exaple, for ordinary NBUE distributions, here = = 1, e obtain a perforance guarantee <362 1/. Thereby, e iprove upon the previously best-knon perforance guarantee of 4 1/ for traditional stochastic scheduling, hich as derived for an LP-based list scheduling policy Möhring et al. [21]. Again, this iproved bound holds even though e consider an online odel in hich the obs arrive over tie, and the scheduler does not kno anything about the obs that are about to arrive in the future. Moreover, for deterinistic processing ties, here = 0 and = 1, e obtain a perforance guarantee <3281, atching the bound fro deterinistic online scheduling by Mego and Schulz [18]. For both odels, our results are, in fact, achieved by fixed assignent policies. That is, henever 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 WSEPT rule Möhring et al. [21], Scharbrodt et al. [24], Souza and Steger [31], Weiss [34, 35] for odels ithout release dates, or they use LP-relaxations to define list scheduling policies other than WSEPT Möhring et al. [21], Skutella and Uetz [27]. As soon as e assue that obs arrive online, the approaches of these papers fail: The traditional offline WSEPT rule cannot be ipleented because it requires an a priori ordering of obs in order of ratios /ƐP. To obtain the results for odels ith 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 e still use the sae LP-relaxation as Möhring et al. [21] ithin our analysis, the ain difference lies in the fact that the algoriths e 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, hoever, are stochastic ith 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 vie, our odel soehat copares to the idea of a diffuse adversary as defined by Koutsoupias and Papadiitriou [16]. Because deterinistic processing ties are contained as a special case, hoever, all loer bounds on the approxiability knon fro deterinistic online scheduling also hold for the SOS odel of this paper. Hence, no SOS policy can exist ith a perforance bound better than Vestens [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, here the set of obs J, their release dates r, eights, and processing tie distributions P are given at the outset. Hence, loer bounds on the expected obective value of an optial stochastic scheduling policy carry over to the stochastic online setting that e consider in this paper. Therefore, e have the folloing loer bound on the perforance of any SOS policy; it is a generalization of a loer bound by Eastan et al. [10] to stochastic processing ties. Lea 2.1 Möhring et al. [21]. For any instance I of Pr Ɛ C, e have that ƐOPTI k H ƐP k 1 1 ƐP here bounds the squared coefficient of variation of the processing ties, that is, VarP /ƐP 2 for all obs = 1nand soe 0.

5 Matheatics of Operations Research 313, pp , 2006 INFORMS 517 Here, e 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 /ƐP, that is, { H = k J k ƐP k > } { k k ƐP ƐP k = } ƐP Accordingly, e define L = J \H as those obs that have loer priority in the order of ratios /ƐP. As a tie-breaking rule for obs k ith equal ratio k /ƐP k = /ƐP, e decide depending on the position in the online sequence relative to. That is, if k, then k belongs to set H, otherise it is included in set L. Note that, by convention, e assue that H also contains ob. 3. Stochastic online scheduling on a single achine. In this section, e consider the SOS proble on a single achine. When release dates are absent, it is ell knon that the WSEPT rule is optial Rothkopf [23], Sith [30]. For the proble of scheduling obs ith nontrivial release dates on a single achine, the currently best-knon result fro traditional stochastic scheduling is an LP-based list scheduling algorith ith a perforance bound of 3 Möhring et al. [21]. Inspired by a corresponding algorith for the deterinistic online setting on parallel achines fro Mego and Schulz [18], e propose the folloing scheduling policy. Algorith 1 -Shift-WSEPT. Modify the release date r of each ob to r = axr ƐP for soe fixed >0. At any tie t, hen the achine is idle, start the ob ith the highest ratio /ƐP aong all available obs, respecting the odified release dates. In case of ties, sallest index first. We first derive an upper bound on the expected copletion tie of a ob, ƐC, hen scheduling obs on a single achine according to the -Shift-WSEPT policy. Lea 3.1. Let all processing ties be -NBUE. Then, the expected copletion tie of ob under -Shift-WSEPT on a single achine can be bounded by ƐC 1 + /r + ƐP k 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. Otherise, X has value 0. Moreover, for any ob k, let 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., k = 1 if and only if Sk r. The start of ob ill 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 [ ƐS Ɛ r + X + ] P k k k H k H\ = r + ƐX + r + ƐX + k H\ k H\ ƐP k k ƐP k here the last inequality follos fro the fact that P k k P k for any ob k. Next, e sho that ƐX /r. If the achine ust finishes a ob at tie r or is idle at that tie, X has value 0. Otherise, soe ob l is in process at tie r. Note that this ob ight have loer or higher priority than ob. Such ob l as available at tie r l <r, and by definition of the odified release dates, e therefore kno that ƐP 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,isɛp l t P l >t. Because of the assuption of -NBUE processing ties, e thus kno that ƐP l t P l >t ƐP l /r for any ob l that could be in process at tie r. Hence, e obtain ƐX /r. Finally, the fact that ƐC = ƐS + ƐP concludes the proof.

6 518 Matheatics of Operations Research 313, pp , 2006 INFORMS In fact, it is quite straightforard to use Lea 3.1 to sho the folloing. Theore 3.1. The -Shift-WSEPT algorith is a + 2-approxiation for the stochastic online single achine proble 1r Ɛ C for -NBUE processing ties. The best choice for is = 1. Proof. With Lea 3.1 and the definition of odified release dates r = axr ƐP, e can bound the expected value of a schedule obtained by -Shift-WSEPT: ƐC 1 + / axr ƐP + ƐP k k H = ax1 + /r + ƐP + ƐP k k H ax1 + / + r + ƐP + ƐP k k H We can no apply the trivial loer bound r + ƐP ƐOPTI, and exploit the fact that k H ƐP k ƐOPTI by Lea 2.1 for = 1, and obtain ƐC 1 + ax {1 + } + ƐOPTI No, = 1 iniizes this expression, independently of, and the theore follos. Note that for NBUE processing ties, the result atches the currently best-knon perforance bound of 3 derived by Möhring et al. in [21] for the traditional stochastic scheduling odel. Their LP-based policy, hoever, requires an a priori knoledge of the set of obs J, their eights, and their expected processing ties ƐP. Moreover, in the deterinistic online setting, the best-possible algorith is 2-copetitive Hoogeveen and Vestens [14], hence the corresponding loer bound of 2 holds for the stochastic online setting too. 4. Stochastic online scheduling on parallel achines. In this section, e define SOS policies for the proble on parallel achines. We first consider the proble ithout nontrivial release dates, and later generalize to the proble ith nontrivial release dates Scheduling obs ithout release dates. In the case that all obs arrive at tie 0, the proble effectively turns into a traditional stochastic scheduling proble, P Ɛ C. For that proble, it is knon that the WSEPT rule yields a /-approxiation, being an upper bound on the squared coefficients of variation of the processing tie distributions Möhring et al. [21]. Nevertheless, e consider an online variant of the proble P Ɛ C that resebles the online list 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 kno 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 sho that an intuitive and siple fixed assignent policy exists that eventually yields the sae perforance bound as the one proved in Möhring et al. [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 folloing notation: If a ob is assigned to achine i, this is denoted by i. No,e can define the MinIncrease policy as follos. 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 = ƐP k + ƐP k + ƐP 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 nonincreasing ratios /ƐP. Because WSEPT is knon to be optial on a single achine, MinIncrease in fact assigns each ob to that achine here it causes the least increase in the expected obective value, given the previously assigned obs. This is expressed in the folloing lea. Lea 4.1. The expected obective value ƐMII of the MinIncrease policy equals in i z i.

7 Matheatics of Operations Research 313, pp , 2006 INFORMS 519 Proof. The assignent of obs to achines is independent of the realization of processing ties. Hence, the expected copletion tie ƐC for soe ob that has been assigned to achine i by MinIncrease is ƐC = ƐP k k H k i because all obs that are assigned to the sae achine i are sequenced in order of nonincreasing ratios /ƐP. No, eighted suation over all obs gives by linearity of expectation [ ] ƐMII = Ɛ C = = ƐP k k H k i k + k H k i k<ɛp k + k H k i k>ɛp ƐP This allos us to apply the folloing index rearrangeent: ƐP k = ƐP k H k> Thus, e have ƐMII = = k 1 k L k< k + k H k i k<ɛp ƐP k + k L k i k< ƐP ƐP k + ƐP k + ƐP k H k i k< k L k i k< = in z i i here the second equality akes use of 1 applied to each individual achine i, and the last equality holds because i is the achine iniizing z i over all achines i. No, e can derive the folloing perforance guarantee for the MinIncrease policy. Theore 4.1. Consider the stochastic online scheduling SOS proble on parallel achines, P Ɛ C, as described above. Given that VarP /ƐP 2 for all obs and soe constant 0, the MinIncrease policy is a -approxiation, here = 1 + Proof. Fro Lea 4.1, e kno that ƐMII = in i z i, and thus, ƐMII = { } in ƐP k + ƐP k + ƐP i k H k< k i k L k< k i 1 ƐP k + ƐP k + ƐP k H k< k L k< here the inequality holds because the least expected increase is not ore than the average expected increase over all achines. No, e 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 ƐP is a loer bound for the expected perforance ƐOPTI of an optial policy, e thus obtain ƐMII 1 = 1 k H k< k H ƐP k + ƐP k + 1 k H k> ƐP ƐP k + ƐP

8 520 Matheatics of Operations Research 313, pp , 2006 INFORMS 1 1 ƐOPTI ƐOPTI ƐP + 1 ƐP As entioned above, this perforance guarantee atches the currently best-knon perforance guarantee for the traditional stochastic setting, hich as derived for the perforance of the WSEPT rule in Möehring et al. [21]. The WSEPT rule, hoever, requires the knoledge of all obs ith their eights and expected processing ties ƐP at the outset. In contrast, the MinIncrease policy decides on achine assignents online, ithout any knoledge of the obs to coe. Finally, note that these to policies are indeed different; this follos fro siple exaples. Loer 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 folloing, e therefore give a loer bound on the expected perforance ƐFIXI of an optial stochastic scheduling policy FIX that assigns obs to achines beforehand. A fortiori, this loer bound holds for the best-possible SOS policy too. Theore 4.2. For stochastic parallel achine scheduling ith unit eights and i.i.d. exponential processing ties, Pp exp1ɛ C, there exist instances I such that ƐFIXI ƐOPTI for any >0. Here, Hence, no policy that uses fixed assignents of obs to achines can perfor better in general. Note that the theore is forulated for the special case of exponentially distributed processing ties. Stronger bounds can be obtained for arbitrary distributions. Hoever, because our perforance guarantees, as in Möhring et al. [21], depend on the coefficient of variation of the processing ties, e are particularly interested in loer bounds for classes of distributions here 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, e get a loer bound of 8/7 114 on the perforance of any fixed assignent policy, and for that case the perforance bound of MinIncrease equals 2 1/ = 15. Proof of Theore 4.2. Let us consider an instance ith achines and n = + k exponentially distributed obs, P exp1, here k 1 is an integer. The optial stochastic scheduling policy is SEPT, shortest expected processing tie first Bruno et al. [4], Weiss and Pinedo [36], and the expected perforance is see, e.g., Uetz [32, Corollary ] ƐOPTI = ƐSEPTI = ƐC SEPT = + n =+1 kk + 1 = + k + When in a fixed assignent, one achine has to process at least to 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 ith ƐC = l for each l in the range 11 +k/, and k k/ obs ith ƐC = 2 +k/. The expected perforance for the best fixed assignent policy FIX is ƐFIXI = ƐC FIX = + 2k + k 2 k k 2 For <k, the value of ƐFIXI is equal to 3k. Hence, for <k, the ratio ƐFIXI/ƐOPTI is ƐFIXI ƐOPTI = 3k + k + kk + 1/ hich is axiized for k. With this choice of k, the ratio ƐFIXI/ƐOPTI tends to 3 2/2 + 2 = as tends to infinity. Note that the loer bound of 1.24 holds henever, the nuber of achines, tends to infinity. For saller nubers of achines, e.g., = 2, 3, or 4, e use saller nubers k = k; naely, k2 = 1, k3 = 2, and k4 = 2, and obtain the loer bounds 8/7 114, 7/6 116, and 32/

9 Matheatics of Operations Research 313, pp , 2006 INFORMS 521 Loer bound for MinIncrease. The loer 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 approxiative. For general i.e., nonexponential probability distributions, e obtain a loer bound of 3/2 on the expected perforance of MinIncrease relative to the expected perforance of an optial scheduling policy, as shon by the folloing instance. Exaple 4.1. The instance consists of n 1 deterinistic unit length obs and one ob ith a stochastic to-point distributed processing tie. There are = 2 achines, and e assue that n, the nuber of obs is even. The n 1 deterinistic obs have unit eight = 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 ith probability 2/n, and p = 1 ith probability 1 2/n. The eight 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 ith n/2 1 deterinistic obs. Hence, the expected obective value of the schedule under MinIncrease is Ɛ C = 3n 2 /4 + on 2. An optial stochastic policy ould start the uncertain ob and one deterinistic ob at tie 0. At tie t = 1, it is knon 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, otherise all deterinistic obs are scheduled on the sae achine. Thus, the expected obective value of an optial schedule is ƐOPTI = n 2 /2 + on 2. The ratio of both values tends to 3/2 if the nuber of obs tends to infinity. Note, hoever, that this result is less eaningful in coparison to the perforance bound of Theore 4.1, hich depends on an upper bound on the squared coefficient of variation Scheduling obs ith nontrivial release dates. In this section, e consider the setting here obs arrive over tie, that is, the stochastic online version of Pr Ɛ C. The ain idea is to adopt the MinIncrease policy to this setting. Hoever, the difference is that e are no longer equipped ith an optial policy as it as WSEPT in the previous section to schedule the obs that are assigned to a single achine. In addition, even if e kne such a policy for a single achine, it ould not be straightforard ho to use it in the setting ith parallel achines to define a feasible online scheduling policy. Hoever, e propose to use the -Shift-WSEPT rule as introduced in 3 to sequence the obs that e have assigned to the sae achine. The assignent of obs to achines, on the other hand, reains the sae as before in the case ithout release dates. In a sense, hen assigning obs to achines, e 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 = ƐP k + ƐP k + ƐP 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 SOS proble on parallel achines ith release dates, Pr Ɛ C. Given that all processing ties are -NBUE, the odified MinIncrease policy running -Shift-WSEPT on each single achine is a -approxiation, here { = 1 + ax 1 + } Here, is such that VarP /ƐP 2 for all obs. In particular, because all processing ties P are -NBUE, Lea A.1 yields that 2 1, hence 1 + ax1 + / + 1/. Proof. Let i be the achine to hich ob is assigned. Then, by Lea 3.1, e kno that ƐC 1 + r + ƐP k 2 k H k i and the expected value of MinIncrease can be bounded by ƐMII 1 + r + ƐP k 3 k H k i

10 522 Matheatics of Operations Research 313, pp , 2006 INFORMS For the second part of the right-hand side of 3, e can use the sae index rearrangeent as in the proof of Lea 4.1 see 1. We thus obtain ƐP k = ƐP k + ƐP k + ƐP k H k i k H k i k< k L k i k< By definition of the odified MinIncrease algorith, e kno that any ob is assigned to the achine, hich iniizes the ter in parentheses. Hence, by the sae averaging arguent as before, e kno that ƐP k ƐP k k H k i + ƐP k + ƐP k H k< k L k< = ƐP k + 1 ƐP k H here the last equality again follos fro index rearrangeent. Plugging this into 3 leads to the folloing bound on the expected perforance of MinIncrease: ƐMII 1 + r + ƐP k + 1 ƐP k H Applying the bound of Lea 2.1 into the above inequality, e obtain ƐMII 1 + r ƐOPTI + ƐP = ƐOPTI r ƐP 4 here again is an upper bound on the squared coefficient of variation of the processing tie distributions P. By bounding r by r + ƐP, e obtain the folloing bound on the ter in parentheses of the right-hand side of 4: 1 + r ƐP r ƐP { r + ƐP ax By using this inequality in Equation 4, and applying the trivial loer bound r + ƐP ƐOPTI on the expected optiu perforance, e get the claied perforance bound of { = 1 + ax 1 + } Because all processing ties are -NBUE, e kno by Lea A.1 in the appendix that 2 1, and thus the second clai of the theore follos; naely, 1 + ax1 + / + 1/. For NBUE processing ties, here = = 1, Theore 4.3 yields a perforance bound of { 1 = 2 + ax + 1 } This ter is inial for = /, hich yields a ratio of = /. This is less than 5 + 5/2 1/ 362 1/, iproving upon the previously best-knon bound of 4 1/ fro Möhring et al. [21] for the traditional stochastic proble. More generally, for -NBUE processing ties, optiizing the ter ax1 + / + 1/ for yields <3/2 + 1/ /2. Moreover, for deterinistic processing ties, here = 0 and = 1, Theore 4.3 yields a perforance bound of { 1 = 2 + ax + 1 } Optiizing for yields = /4, hich yields a ratio of = /4. This is less than for any value of, leaving only a sall gap to the currently best-knon bound of 2.62 for deterinistic online scheduling Correa and Wagner [8]. In fact, it atches the copetitive ratio of the deterinistic parallel achine version of -Shift-WSEPT fro Mego and Schulz [18]. }

11 Matheatics of Operations Research 313, pp , 2006 INFORMS Randoized ob assignent. As a atter of fact, the MinIncrease policy can be interpreted as the derandoized version of a policy that assigns obs uniforly at rando to the achines. Even though randoly assigning obs to achines ignores uch inforation, it is nevertheless knon to be quite poerful 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 ith 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 SOS proble at hand is as follos. Algorith 4 RandAssign. When a ob is presented to the scheduler, it is assigned to achine i ith 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 SOS proble on parallel achines ith release dates, Pr Ɛ C. Given that all processing ties are -NBUE, the RandAssign policy running -Shift- WSEPT on each single achine is a -approxiation, here { = 1 + ax 1 + } Here, is such that VarP /ƐP 2 for all obs. In particular, because all processing ties P are -NBUE, Lea A.1 yields that 2 1, hence, 1 + ax1 + / + 1/. Proof. Consider a ob and let i denote the achine to hich it has been assigned. Let Pr i be the probability for ob being assigned to achine i. Then, by Lea 3.1, e kno that ƐC i 1 + r + Pr k i i ƐP k i k H = 1 + r + Pr k i i ƐP k k H 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 ƐC = Pr i ƐC i i=1 1 + r + Pr i Pr k i i ƐP k i=1 k H = 1 + r + ƐP k + 1 ƐP k H here the last equality is because of the independence of the ob assignents to the achines. Then, the expected obective value of RandAssign, ƐRAI, can be bounded by ƐRAI = ƐC 1 + r + ƐP k + 1 ƐP k H This bound equals the upper bound that e achieved on the expected perforance of the odified MinIncrease policy in the proof of Theore 4.3. Hence, e conclude the proof in the sae ay and ith the sae result as for MinIncrease. Appendix. Coefficient of variation for -NBUE rando variables. In this section, e sho the relation beteen the value of and the squared coefficient of variation VarX/ƐX 2 for a -NBUE rando variable X. Lea A.1. Let X be a -NBUE rando variable, and let CVX = VarX/ƐX denote the coefficient of variation of X. Then, CVX

12 524 Matheatics of Operations Research 313, pp , 2006 INFORMS 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 nonnegative -NBUE rando variable, ith cuulative distribution function F and density f. By definition of conditional expectation, e kno that x tfxdx t ƐX t X>t= 5 1 Ft As x t = x t dy, e can rite the noinator of the right-hand side as 0 x=t x tfxdx = = = x t x=t y=0 y=0 x=y+t x=t f x dy dx f x dx dy here the second equality is obtained by changing the order of integration. As X is -NBUE, i.e., ƐX t X>t ƐX, it follos fro 5 and 6 that x=t 1 Fxdx 6 1 Fxdx= ƐX t X>t1 Ft ƐX1 F t 7 By integrating the right-hand side of the above inequality over t, e obtain ƐX t=0 1 Ftdt= ƐX 2 8 Hall and Wellner [13, Equality 4.1] shoed that integrating the left-hand side of 7 over t yields t=0 x=t 1 Fxdxdt= 1 2 ƐX2 9 Hence, using 8 and 9 in7, e have ƐX 2 2ƐX 2 Rearranging ters yields the desired bound on the squared coefficient of variation CVX 2 = ƐX2 ƐX ƐX 2 Acknoledgents. The authors thank the to 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 to Andreas S. Schulz and Sven O. Kruke for pointing out soe flas in an earlier version of this paper. Research as partially supported by the DFG Research Center, Matheon, Matheatics for Key Technologies in Berlin. An extended abstract ith parts of this ork appeared in the WAOA 2004 Conference Proceedings on Approxiation and Online Algoriths Mego et al. [19]. Parts of this ork ere done hile Tark Vredeveld as ith the Konrad-Zuse-Zentru für Inforationstechnik, Berlin, Gerany. References [1] Afrati, F. N., E. Bapis, C. Chekuri, D. R. Karger, C. Kenyon, S. Khanna, I. Milis, M. Queyranne, M. Skutella, C. Stein, M. Sviridenko Approxiation schees for iniizing average eighted copletion tie ith release dates. Proc. 40th Annual Sypos. on Foundations of Coput. Sci., Ne York. IEEE Coputer Society, Los Alaitos, CA, [2] Anderson, E. J., C. N. Potts On-line scheduling of a single achine to iniize total eighted copletion tie. Math. Oper. Res [3] Borodin, A., R. El-Yaniv Online Coputation and Copetitive Analysis. Cabridge University Press, Cabridge, UK. [4] Bruno, J. L., P. J. Doney, G. N. Frederickson Sequencing tasks ith exponential service ties to iniize the expected flotie or akespan. J. ACM [5] Chakrabarti, S., S. Muthukrishnan Resource scheduling for parallel database and scientific applications. Proc. 8th Annual ACM Sypos. on Parallel Algoriths and Architectures, Padua, Italy. ACM Press, Ne York, [6] Chekuri, C., R. Johnson, R. Motani, B. Nataraan, B. Rau, M. Schlansker An analysis of profile-driven instruction level parallel scheduling ith application to super blocks. Proc. 29th IEEE/ACM Internat. Sypos. on Microarchitecture, Paris, France. IEEE Coputer Society, Los Alaitos, CA, [7] Chou, M. C., H. Liu, M. Queyranne, D. Sichi-Levi On the asyptotic optiality of a siple on-line algorith for the stochastic single achine eighted copletion tie proble and its extensions. Oper. Res

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