Online Non-clairvoyant Scheduling to Simultaneously Minimize All Convex Functions

Transcription

1 Onlne Non-clarvoyant Schedulng to Smultaneously Mnmze All Convex Functons Kyle Fox, Sungjn Im 2, Janardhan Kulkarn 2, and enjamn Moseley 3 Department of Computer Scence, Unversty of Illnos, Urbana, IL 680. kylefox2@llnos.edu 2 Department of Computer Scence, Duke Unversty, Durham NC [sungjn,kulkarn]@cs.duke.edu 3 Toyota Technologcal Insttute at Chcago, Chcago, IL moseley@ttc.edu Abstract. We consder schedulng jobs onlne to mnmze the objectve [n] wg(c r), where w s the weght of job, r s ts release tme, C s ts completon tme and g s any non-decreasng convex functon. Prevously, t was known that the clarvoyant algorthm Hghest-Densty- Frst (HDF) s (2 + ɛ)-speed O()-compettve for ths objectve on a sngle machne for any fxed 0 < ɛ < [2]. We show the frst non-trval results for ths problem when g s not concave and the algorthm must be non-clarvoyant. More specfcally, our results nclude: A (2 + ɛ)-speed O()-compettve non-clarovyant algorthm on a sngle machne for all non-decreasng convex g, matchng the performance of HDF for any fxed 0 < ɛ <. A (3 + ɛ)-speed O()-compettve non-clarovyant algorthm on multple dentcal machnes for all non-decreasng convex g for any fxed 0 < ɛ <. Our postve result on multple machnes s the frst non-trval one even when the algorthm s clarvoyant. Interestngly, all performance guarantees above hold for all non-decreasng convex functons g smultaneously. We supplement our postve results by showng any algorthm that s oblvous to g s not O()-compettve wth speed less than 2 on a sngle machne. Further, any non-clarvoyent algorthm that knows the functon g cannot be O()-compettve wth speed less than 2 on a sngle machne or speed less than 2 on m dentcal machnes. m Introducton Schedulng a set of jobs that arrve over tme on a sngle machne s perhaps the most basc settng consdered n schedulng theory. A consderable amount of work has focused on ths fundamental problem. For examples, see [26]. In Research by ths author s supported n part by the Department of Energy Offce of Scence Graduate Fellowshp Program (DOE SCGF), made possble n part by the Amercan Recovery and Renvestment Act of 2009, admnstered by ORISE-ORAU under contract no. DE-AC05-06OR2300. Supported by NSF awards CCF and IIS

2 ths settng, there are n jobs that arrve over tme, and each job requres some processng tme to be completed on the machne. In the onlne settng, the scheduler becomes frst aware of job at tme r when job s released. Note that n the onlne settng, t s standard to assume jobs can be preempted. Generally, a clent that submts a job would lke to mnmze the flow tme of the job defned as F := C r, where C denotes the completon tme of job. The flow tme of a job measures the amount of tme the job wats to be satsfed n the system. When there are multple jobs competng for servce, the scheduler needs to make schedulng decsons to optmze a certan global objectve. One of the most popular objectves s to mnmze the total (or equvalently average) flow tme of all the jobs,.e., [n] F. It s well known that the algorthm Shortest-Remanng-Processng-Tme (SRPT) s optmal for that objectve n the sngle machne settng. The algorthm SRPT always schedules the job that has the shortest remanng processng tme at each pont n tme. Another well known result s that the algorthm Frst-In-Frst-Out (FIFO) s optmal for mnmzng the maxmum flow tme,.e., max [n] F on a sngle machne. The algorthm FIFO schedules the jobs n the order they arrve. These classc results have been extended to the case where jobs have prortes. In ths extenson, each job s assocated wth a weght denotng ts prorty; large weght mples hgher prorty. The generalzaton of the total flow tme problem s to mnmze the total weghted flow tme, [n] F. For ths problem t s known that no onlne algorthm can be O()-compettve [5]. A generalzaton of the maxmum flow tme problem s to mnmze the maxmum weghted flow tme max [n] F. It s also known for ths problem that no onlne algorthm can be O()-compettve [,5]. Due to these strong lower bounds, prevous work for these objectves has appealed to the relaxed analyss model called resource augmentaton [22]. In ths relaxaton, an algorthm A s sad to be s-speed c-compettve f A has a compettve rato of c when processng jobs s tmes faster than the adversary. The prmary goal of a resource augmentaton analyss s to fnd the mnmum speed an algorthm requres to be O()-compettve. For the total weghted flow tme objectve, t s known that the algorthm Hghest-Densty-Frst (HDF) s ( + ɛ)-speed O( ɛ )-compettve for any fxed ɛ > 0 [25,0]. The algorthm HDF always schedules the job of hghest densty, w. For the maxmum weghted flow objectve, the algorthm ggest-weght-frst (FW) s known to be ( + ɛ)- speed O( ɛ )-compettve [5]. FW always schedules the job wth the largest weght. Another wdely consdered objectve s mnmzng the l k -norms of flow tme, ( [n] F k ) /k [8,7,20,,4,23]. The lk -norm objectve s most useful for k {, 2, 3, }. Observe that total flow tme s the l -norm of flow tme, and the maxmum flow tme s the l -norm. The l 2 and l 3 norms are natural balances between the l and l norms. These objectves can be used to decrease the varance of flow tme, thereby yeldng a schedule that s far to requests. It s known that no algorthm can be n Ω() -compettve for mnmzng the l 2 -norm

3 [8]. On the postve sde, for ɛ > 0, HDF was shown to be ( + ɛ)-speed O( ɛ )- 2 compettve for any l k -norm objectve, k [8]. These objectves have also been consdered n the dentcal machne schedulng settng [24,4,3,2,9,6,3,9]. In ths settng, there are m machnes that the jobs can be scheduled on. Each job can be scheduled on any machne and job requres processng tme no matter whch machne t s assgned to. In the dentcal machne settng t s known that any randomzed onlne algorthm has compettve rato Ω(mn{ n m, log P }), where P denotes the rato between the maxmum and mnmum processng tme of a job [24]. HDF as well as several other algorthms are known to be scalable for weghted flow tme [0,4,9,3]. For the l k -norms objectve the multple machne verson of HDF s known to be scalable [3] as well as other algorthms [4,9]. For the maxmum unweghted flot s known that FIFO s (3 2/m)-compettve, and for weghted maxmum flow tme a scalable algorthm s known [,5]. The algorthms HDF and SRPT use the processng tme of a job to make schedulng decsons. An algorthm whch learns the processng tme of a job upon ts arrval s called clarvoyant. An algorthm that does not know the processng tme of a job before completng the job s sad to be non-clarvoyant. Among the aforementoned algorthms, FIFO and FW are non-clarvoyant. Non-clarvoyant schedulers are hghly desrable n many real world settngs. For example, an operatng system typcally does not know a job s processng tme. Thus, there has been extensve work done on desgnng non-clarvoyant schedulers for the problems dscussed above. Scalable non-clarvoyant algorthms are known for the maxmum weghted flow tme, average weghted flow tme, and l k -norms of flow tme objectves even on dentcal machnes [5,4]. It s common n schedulng theory that algorthms are talored for specfc schedulng settngs and objectve functons. For nstance, FIFO s consdered the best algorthm for non-clarvoyantly mnmzng the maxmum flow tme, whle HDF s consdered one of the best algorthms for mnmzng total weghted flow tme. One natural queston that arses s what to do f a system desgner wants to mnmze several objectve functons smultaneously. For nstance, a system desgner may want to optmze average qualty of servce, whle mnmzng the maxmum watng tme of a job. Dfferent algorthms have been consdered for mnmzng average flow tme and maxmum flow tme, but the system desgner would lke to have a sngle algorthm that performs well for both objectves. Motvated by ths queston, the general cost functon objectve was consdered n [2]. In the general cost functon problem, there s a functon g : R + R + gven, and the goal of the scheduler s to mnmze [n] g(f ). One can thnk of g(f ) as the penalty of makng job wat F tme steps, scaled by job s prorty (ts weght ). Ths objectve captures most schedulng metrcs. For example, ths objectve functon captures total weghted flow tme by settng g(x) = x. y settng g(x) = x k, the objectve also captures mnmzng [n] F k whch s essentally the same as the l k -norm objectve except the outer kth root s not taken. Fnally, by makng g grow very quckly the objectve can be desgned to capture mnmzng the maxmum weghted flow tme. As stated, one of the

4 reasons ths objectve was ntroduced was to fnd an algorthm that can optmze several objectves smultaneously. If one were to desgn an algorthm that optmzes the general cost functon g whle beng oblvous to g, then ths algorthm would optmze all objectve functons n ths framework smultaneously. In [2], the general cost functon objectve was consdered only assumng that g s non-decreasng. Ths s a natural assumpton snce there should be no ncentve for a job to wat longer. It was shown that n ths case, no algorthm that s oblvous to the cost functon g can be O()-compettve wth speed 2 ɛ for any fxed ɛ > 0. Surprsngly, t was also shown that HDF, an algorthm that s oblvous to g, s (2 + ɛ)-speed O(/ɛ)-compettve. Ths result shows that t s ndeed possble to desgn an algorthm that optmzes most of the reasonable schedulng objectves smultaneously on a sngle machne. Recall that HDF s clarvoyant. Ideally, we would lke to have a non-clarvoyant algorthm for general cost functons. Further, there s currently no known smlar result n the multple dentcal machnes settng. Results: In ths paper, we consder non-clarvoyant onlne schedulng to mnmze the general cost functon on a sngle machne as well as on multple dentcal machnes. In both the settngs, we gve the frst nontrval postve results when the onlne scheduler s requred to be non-clarvoyant. We concentrate on cost functons g whch are dfferentable, non-decreasng, and convex. We assume wthout loss of generalty that g(0) = 0. Note that all of the objectves dscussed prevously have these propertes. We show the followng somewhat surprsng result (Secton 4). Theorem. There exsts a non-clarvoyant algorthm that s (2+ɛ)-speed O(/ɛ)- compettve for mnmzng [n] g(c r ) on a sngle machne for any ɛ > 0, when the gven cost functon g : R + R + s dfferentable, non-decreasng, and convex (g s non-decreasng). Further, ths algorthm s oblvous to g. We then consder the general cost functon objectve on multple machnes for the frst tme, and gve a postve result. Ths algorthm s also non-clarvoyant. Theorem 2. There exsts a non-clarvoyant algorthm that s (3+ɛ)-speed O(/ɛ)- compettve for mnmzng [n] g(c r ) on multple dentcal machnes for any ɛ > 0, when the gven cost functon g : R + R + s dfferentable, nondecreasng, and convex (g s non-decreasng). Further, ths algorthm s oblvous to g. Note that we do not knof there exsts a constant compettve non-clarvoyant algorthm even for a sngle machne wth any constant speed when the cost functon s nether convex nor concave. We leave ths gap as an open problem. We complement these postve results by extendng the lower bound presented n [2]. They showed that for any ɛ > 0, no oblvous algorthm can be (2 ɛ)- speed O()-compettve on a sngle machne when the cost functon g s nondecreasng, but perhaps dscontnuous. We show the same lower bound even f g s dfferentable, non-decreasng, and convex. Thus, on a sngle machne, our

5 postve result s essentally tght up to constant factors n the compettve rato, and our algorthm acheves the same performance guarantee whle beng nonclarvoyant. Theorem 3. No randomzed clarvoyant algorthm that s oblvous to g can be (2 ɛ)-speed O()-compettve for mnmzng [n] g(c r ) on a sngle machne even f all jobs have unt weghts and g s dfferentable, non-decreasng, and convex. We go on to show that even f a non-clarvoyant algorthm knows the cost functon g, the algorthm cannot have a bounded compettve rato when gven speed less than 2. Theorem 4. Any determnstc non-clarvoyant (possbly aware of g) algorthm for mnmzng [n] g(c r ) on a sngle machne has an unbounded compettve rato when gven speed 2 ɛ for any fxed ɛ > 0 where g s dfferentable, non-decreasng, and convex.. Fnally, we show that at least 2 m speed s needed for any non-clarvoyant algorthm to be constant compettve on m dentcal machnes. Ths s the frst lower bound for the general cost functon specfcally desgned for the multple machne case. Theorem 5. Any randomzed non-clarvoyant (possbly aware of g) algorthm on m dentcal machnes has an unbounded compettve rato when gven speed less than 2 m ɛ for any fxed ɛ > 0 when g s dfferentable, non-decreasng, and convex.. Technques: To show Theorem, we consder the well-known algorthm Weghted- Shortest-Elapsed-Tme-Frst (WSETF) on a snge machne, and frst show that t s 2-speed O()-compettve for mnmzng the fractonal verson of the general cost functon objectve. Then wth a small extra amount of speed augmentaton, we convert WSETF s schedule nto the one that s (2+ɛ)-speed O()-compettve for the ntegral general cost functon. Ths converson s now a farly standard technque, and wll be further dscussed n Secton 2. Ths converson was also used n [2] when analyzng HDF. One can thnk of the fractonal objectve as convertng each job to a set of unt szed jobs of weght /. That s, the weght of the job s dstrbuted among all unt peces of the job. Notce that the resultng weght of the unt tme jobs as well as the number of them depends on the job s orgnal processng tme. Thus, to analyze a non-clarvoyant algorthm for the fractonal nstance one must consder the algorthm s decsons on the orgnal nstance and argue about the algorthm s cost on the fractonal nstance. Ths dffers from the analyss of [2], where the clarvoyant algorthm HDF can assume full knowledge of the converson. Due to ths, n [2] they can argue drectly about HDF s decsons for the fractonal nstance of the problem. Snce a non-clarvoyant algorthm does not know the fractonal nstance, t

6 seems dffcult to adapt the technques of [2] when analyzng a non-clarvoyant algorthm. If the nstance conssts of a set of unweghted jobs, WSETF always processes the job whch has been processed the least. Let q A (t) be the amount WSETF has processed job by tme t. When jobs have weghts, WSETF processes the job such that s maxmzed where s the weght n the ntegral q A (t) nstance. One can see that WSETF wll not necessarly process the jobs wth the hghest weght at each tme, whch s what the algorthm HDF wll do f all jobs are unt szed. Further, WSETF may round robn among multple jobs of the same prorty. For these reasons, our analyss of WSETF s substantally dfferent from the analyss n [2], and reles crucally on a new lower bound we develop on the optmal soluton. Ths lower bound holds for any objectve that s dfferentable, non-decreasng, and convex. Our lower bound gves a way to relate the fnal objectve of the optmal soluton to the volume of unsatsfed work the optmal soluton has at each moment n tme. We then bound the volume of unsatsfed jobs n the optmal schedule at each moment n tme and relate ths to WSETF s nstantaneous ncrease n ts objectve functon. We beleve that our new lower bound wll be useful n further analyss of schedulng problems snce t s versatle enough to be used for many schedulng objectves. Other Related Work: For mnmzng average flow tme on a sngle machne, the non-clarvoyant algorthms Shortest Elapse Tme Frst (SETF) and Latest Arrval Processor Sharng (LAPS) are known to be scalable [22,8]. Ther weghted versons Weghted Shortest Elapse Tme Frst (WSETF) and Weghted Latest Arrval Processor Sharng (WLAPS) are scalable for average weghted flow tme [8,6], and also for (weghted) l k norms of flow tme [8,7]. In [2], Im et al. showed Weghted Latest Arrval Processor Sharng (WLAPS) s scalable for concave functons g. They also showed that no onlne randomzed algorthm, even wth any constant speed-up, can have a constant compettve rato, when each job has ts own cost functon g, and the goal s to mnmze [n] g (F ). Ths more general problem was studed n the offlne settng by ansal and Pruhs [7]. They gave an O(log log np )-approxmaton (wthout speed augmentaton), where P s the rato of the maxmum to mnmum processng tme of a job. Ths s the best known approxmaton for mnmzng average weghted flow tme offlne, and a central open queston n schedulng theory s whether or not a O()-approxmaton exsts for weghted flow tme offlne. 2 Prelmnares The Fractonal Objectve: In ths secton we defne the fractonal general cost objectve and ntroduce some notaton. We wll refer to the non-fractonal general cost objectve as ntegral. For a schedule, let (t) denote the remanng processng tme of job at tme t. Let β (p) be the latest tme t such that (t) = p for any p where 0 p.

7 The fractonal objectve penalzes jobs over tme by chargng n proporton to how much of the job remans to be processed. Formally, the fractonal objectve s defned as: [n] C t=r (t) g (t r )dt () Generally when the fractonal objectve s consdered, t s stated n the form (). For our analyss t wll be useful to note that ths objectve s equvalent to: [n] p p=0 g(β (p) r )dp (2) As noted earler, consderng the fractonal objectve has proven to be qute useful for the analyss of algorthms n schedulng theory, because drectly argung about the fractonal objectve s usually easer from an analyss vewpont. A schedule whch optmzes the fractonal objectve can then be used to get a good schedule for the ntegral objectve as seen n the followng theorems. In the frst theorem (6), the algorthm s fractonal cost s compared aganst the optmal soluton for the fractonal objectve. In the second theorem (7), the algorthm s fractonal cost s compared aganst the optmal soluton for the ntegral nstance. Theorem 6 ([2]). If a (non-clarvoyant) algorthm A s s-speed c-compettve for mnmzng the fractonal general cost functon then there exsts a ( + ɛ)sspeed (+ɛ)c ɛ -compettve (non-clarvoyant) algorthm for the ntegral general cost functon objectve for any 0 ɛ. Theorem 7 ([2]). If a (non-clarvoyant) algorthm A wth s-speed has fractonal cost at most a factor c larger than the optmal soluton for the ntegral objectve then there exsts a ( + ɛ)s-speed (+ɛ)c ɛ -compettve (non-clarvoyant) algorthm for the ntegral general cost functon objectve for any 0 ɛ. These two theorems follow easly by the analyss gven n [2]. We note that the resultng algorthm that performs well for the ntegral objectve s not necessarly the algorthm A. Interestngly, [2] shows that f A s HDF then the resultng algorthm s stll HDF. However, f A s WSETF, the resultng ntegral algorthm need not be WSETF. Notaton: We nontroduce some more notaton that wll be used throughout the paper. For a schedule, let C be the completon tme of job. Let p (t) denote the remanng processng tme for job at tme t. Let q (t) = p (t) be the amount job has been processed by tme t. Let p wj,j (t) = (mn{, p j } qj (t))+. Here ( ) + denotes max{, 0}. Let,j = mn{ wj, p j } = p,j (r j). If the schedule s that produced by WSETF and t [r, C ] then p,j (t) s exactly the amount of processng tme WSETF wll devote to job j durng the nterval [t, C ]. In other words, the remanng tme job wats due to WSETF processng job j. Let Q (t) be the set of job released but unsatsfed by at tme t. Let

8 Z (t) = j Q (t) p,j (t). When the algorthm s the optmal soluton (OPT) we set to be O and f the algorthm s WSETF we set to be A. For example Q A (t) s the set of released and unsatsfed jobs for WSETF at tme t. Fnally, for a set of possbly overlappng tme ntervals I, let I denote the total length of ther unon. 3 Analyss tools In ths secton we ntroduce some useful tools that we use for our analyss. Frst we present our novel lower bound on the optmal soluton. Ths lower bound s the key to our analyss and the man techncal contrbuton of the paper. The left-hand-sde of the nequalty n the lemma has an arbtrary functon x(t) : R + R + \ {0}, whle the rght-hand-sde s smply a fractonal cost of the schedule n consderaton. Ths lower bound s nspred by one presented n [20]. However, the lower bound gven n [20] nvolves substantally dfferent terms, and s only for the l k -norms of flow tme. Our proof s consderably dfferent from [20], and perhaps smpler. Snce ths lower bound apples to any objectve that fts nto the general cost functon framework, we beleve that ths lower bound wll prove to be useful for a varety of schedulng problems. The assumpton n the lemma that g s convex s crucal; the lemma s not true otherwse. The usefulness of ths lemma wll become apparent n the followng two sectons. We prove ths lemma n Secton 6 after we show the power of the lemma. Lemma. Let σ be a set of jobs on a sngle machne wth speed s. Let be any feasble schedule and (σ) be the total weghted fractonal cost of wth objectve functon g that s dfferentable and convex (g s non-decreasng), wth g(0) = 0. Let x(t) : R + R + \{0} be any functon of t. Let p x, (t) = (mn(x(t), p (t)) q (t))+. Fnally, let Zx (t) = Q (t) p x, (t). Then, x(t) g(z x (t)/s )dt s (σ). Next we show a property of WSETF that wll be useful n relatng the volume of work of unsatsfed jobs n WSETF s schedule to that of the optmal soluton s schedule. y usng ths lemma we can bound the volume of jobs n the optmal soluton s schedule and then appeal to the lower bound shown n the prevous lemma. Ths lemma s somewhat smlar to one shown for the algorthm Shortest-Remanng-Processng-Tme (SRPT) [26,9]. However, we are able to get a stronger verson of ths lemma for WSETF. Lemma 2. Consder runnng WSETF usng s-speed for some s 2 on m dentcal machnes and the optmal schedule at unt speed on m dentcal machnes. For any job Q A (t) and tme t, t s the case that Z A(t) ZO (t) 0. Proof. For the sake of contradcton, let t be the earlest tme such that Z A (t) Z O (t) > 0. Let j be a job where pa,j (t) > po,j (t). Consder the nterval I = [r j, t].

9 Let I j be the set of ntervals where WSETF works on job j durng I and let I j be the rest of the nterval I. Knowng that pa,j (t) > po,j (t), we have that I j < s I. If ths fact were not true, then qa j (t) = s I j I, but snce OPT has speed, qj O(t) I, and therefore qa j (t) qo j (t), a contradcton of the defnton of job j. Hence, we know that I j ( s ) I. At each tme durng I ether WSETF s schedulng job j or all m machnes n WSETF s schedule are busy schedulng jobs whch contrbute to Z A (t). Thus the total amount of work done by WSETF durng I on jobs that contrbute to Z A (t) s at least qj A(t) + ms I j ms( s ) I = m(s ) I. The total amount of work OPT can do on jobs that contrbute to Z O (t) s m I. Let S denote the set of jobs that arrve durng I. The facts above mply that Z A (t) Z O (t) (Z A (r j ) + k S,k m(s ) I ) (Z O (r j ) + k S,k m I ) = (Z A (r j ) m(s ) I ) (Z O (r j ) m I ) Z A (r j ) Z O (r j ) [s 2] 0 [t s the frst tme Z A(t) ZO (t) > 0 and r j < t]. 4 Sngle machne We now show WSETF s 2-speed O()-compettve on a sngle processor for the fractonal objectve. We then derve Theorem. In Secton 5, we extend our analyss to bound the performance of WSETF on dentcal machnes as well when mgraton s allowed. Assume that WSETF s gven a speed s 2. Notce that Z A (t) always decreases at a rate of s for all jobs Q A (t) when t [r, C ]. Ths s because Z A (t) s exactly the amount of remanng processng WSETF wll do before job s completed amongst jobs that have arrved by tme t. Further, knowng that OPT has speed, we see Z O (t) decreases at a rate of at most at any tme t. We know that by Lemma 2 Z A(r ) Z O(r ) 0. Usng these facts, we derve for any tme t [r, C A], s wa(t) g(t r a(t) )dt s p a(t) Z A (t) Z O (t) (s ) (t r ). Therefore, ZO (t) s (t r ) for any t [r, C A ]. Let a(t) denote the job that WSETF works on at tme t. y the second defnton, WSETF s fractonal cost s (Za(t) O (t) ) dt w a(t) p a(t) g s s w a(t) g(za(t) O s p (t))dt a(t) The last nequalty follows snce g( ) s convex, g(0) = 0, and s. y applyng Lemma wth x(t) = p a(t) /w a(t), s = and beng OPT s schedule, we have the followng theorem.

10 Theorem 8. WSETF s s-speed (+ s )-compettve for the fractonal general cost functon when s 2. Ths theorem combned wth Theorem 6 proves Theorem. 5 Multple dentcal machnes Here we present the proof of Theorem 2. In the analyss of WSETF on a sngle machne, we bounded the cost of WSETF s schedule for the fractonal objectve to the cost of the optmal soluton for the fractonal objectve. In the multple machnes case, we wll not compare WSETF to the optmal soluton for the fractonal objectve but rather compare to the cost of the optmal soluton for the ntegral objectve. We then nvoke Theorem 7 to derve Theorem 2. We frst consder an obvous lower bound on the optmal soluton for the ntegral objectve. For each job, the best the optmal soluton can do s to process job mmedately upon ts arrval usng one of ts m unt speed machnes. We know that the total ntegral cost of the optmal soluton s at least g( ). (3) [n] Smlar to the sngle machne analyss, when a job s processed we charge the cost to the optmal soluton. However, f a job s processed at tme t where t r we charge to the ntegral lower bound on the optmal soluton above. If t r >, then we wll nvoke the lower bound on the optmal soluton shown n Lemma and use the fact that the an algorthm s fractonal objectve s always smaller than ts ntegral objectve. Assume that WSETF s gven speed s 3. If job Q A (t) s not processed by WSETF at tme t, then there must exst at least m jobs n Q A (t) processed nstead by WSETF at ths tme. Hence, for all jobs Q A (t), the quantty p A (t)+za (t)/m decreases at a rate of s durng [r, C A ]. In contrast, the quantty Z O (t)/m decreases at a rate of at most snce OPT has m unt speed machnes. Further, by Lemma 2, we know that Z A(r ) Z O(r ) 0, and p A (r )+Z A(r ) Z O(r ). Usng these facts we know for any job and t [r, C A] that p A (t) + (ZA (t) ZO (t))/m (s )(t r ). Notce that f t r, we have that p A (t)+(za (t) ZO (t))/m (s 2)(t r ). Therefore, t r ZO (t) m(s 2) when t r. Let W (t) be the set of jobs that WSETF processes at tme t. y defnton, the value of WSETF s fractonal objectve s s W (t) g(t r )dt. We dvde the set of jobs n W (t) nto two sets. The frst s the set of young jobs W y (t) whch are the set of jobs W (t) where t r. The other set s

11 W o (t) = W (t)\w y (t) whch s the set of old jobs. Let OPT denote the optmal soluton s ntegral cost. We see that WSETF s cost s at most the followng. s W (t) w g(t r )dt s W y(t) W y(t) g( ) + s [n] OPT + s w g(t r )dt + s s g( )dt + s W o(t) W o(t) g W o(t) W o(t) g(t r )dt ( Z O (t) m(s 2) ) dt [by the lower bound of (3) on OPT] OPT + s g(z O (t)/m)dt s 2 W o(t) g(t r )dt g(t r )dt The thrd nequalty holds snce a job can be n W y (t) only f s processed by WSETF at tme t, and job can be processed by at most before t s completed. More precsely, f s n W y (t), then t s processed by s dt durng tme [t, t + dt). Hence, [ W y(t)] s dt, where [ W y (t)] denotes the 0- ndcator varable such that [ W y (t)] = f and only f W y (t). The last nequalty follows snce g( ) s convex, g(0) = 0, and s 2. We know that a sngle m-speed machne s always as powerful as m unt speed machnes, because a m-speed machne can smulate m unt speed machnes. Thus, we can assume OPT has a sngle m-speed machne. We apply Lemma wth x(t) = / for each W o (t), s = m and beng OPT s schedule. Knowng that W o (t) m, we conclude that w a a W o(t) p a g(za O (t)/m) s at most the optmal soluton s fractonal cost. Knowng that any algorthm s fractonal cost s at most ts ntegral cost, we conclude that WSETF s fractonal cost wth s-speed s at most (2 + 2 s 2 ) tmes the ntegral cost of the optmal soluton when s 3. Usng Theorem 7, we derve Theorem 2. 6 Proof of the Man Lemma In ths secton we prove Lemma. Proof of [Lemma ] The ntuton behnd the lemma s that each nstance of Zx (t) s composed of several nfntesmal job slces. y ntegratng over how long these slces have left to lve, we get an upper bound on Zx (t). We then argue that the ntegraton over each slce s tme alve s actually the fractonal cost of that slce accordng to the second defnton of the fractonal objectve. Recall β (p) denotes the latest

12 tme t at whch p (t) = p. For any tme t, let Λ (t) = w p (t) g (β (p) t)dp, p=0 and let Λ(t) = Q (t) Λ (t). The proof of the lemma proceeds as follows. We frst show a lower bound on Λ(t) n terms of x(t) g(z x (t)/s ). Then we show an upper bound on Λ(t) n terms of the fractonal cost of s schedule. Ths strategy allows us to relate x(t) g(z x (t)/s ) and s cost. For the frst part of the strategy, we prove that s x(t) g(z x (t)/s ) Λ(t) at all tmes t. Consder any job Q (t) wth p x, (t) > 0. Suppose x(t). Then, Λ (t) = w p (t) g (β (p) t)dp p (t) g (β p=0 x(t) p=p (t) p x, (t) (p) t)dp. If > x(t), then by defnton of p x, (t), In ths case, p (t) p x, (t) p (t) + q (t) p x, (t) + q (t) [Snce p (t) p x,(t)] = mn( x(t), p (t)) q (t))+ + q (t) ( x(t) q (t)) + q (t) [Snce p x,(t) > 0] = x(t). Λ (t) = w p (t) g (β (p) t)dp p=0 p (t) p x, (t) x(t) p (t) p (t) p=p (t) p x, (t) g (β (p) t)dp p=p (t) p x, (t) g (β (p) t)dp [Snce g s non-decreasng, convex] [Snce p (t)/p x,(t) > /( x(t))]. In ether case, Λ (t) has a lower bound of quantty (4). y convexty of g, the lower bounds on Λ (t) are mnmzed f completes p x, (t) unts of as quckly as possble for each job. Schedule runs at speed s, so we have Λ(t) Z x (t) g (p/s )dp = s x(t) p=0 x(t) Z x (t)/s p=0 g (p)dp s x(t) g(z x (t)/s ). (4)

13 Ths proves that lower bound on Λ(t). Now we show an upper bound on Λ(t) n terms of the s fractonal cost. We show Λ(t)dt (I). Fx a job. We have Λ (t)dt = = p p=0 p (t) p=0 g(β (p))dp. g (β (p) t)dpdt = w p p=0 β (p) g (t)dtdp y summng over all jobs and usng the defnton of fractonal flow tme, we have that Λ(t)dt (I). Further, the gven lower bound and upper bounds on Λ(t)dt show us that s x(t) g(z x (t)/s )dt Λ(t)dt (I), whch proves the lemma. 7 Lower bounds We now present the proof of Theorem 3. Ths lower bound extends a lower bound gven n [2]. In [2], t was shown that no oblvous algorthm can be O()- compettve wth speed less than 2 ɛ for the general cost functon. However, they assume that the cost functon was possbly dscontnuous and not convex. We show that ther lower bound can be extended to the case where g s convex and contnuous. Ths shows that WSETF s essentally the best oblvous algorthm one can hope for. In all the proofs that follow, we wll consder a general cost functon g that s contnuous, non-decreasng, and convex. The functon s also dfferentable except at a sngle pont. The functon can be easly adapted so that t s dfferentable over all ponts n R +. Proof of [Theorem 3]: We appeal to Yao s Mn-max Prncple [2]. Let A be any determnstc onlne algorthm. Consder the cost functon g and large constant c such that g(f ) = 2c(F D) for F > D and g(f ) = 0 for 0 F D. It s easy to see that g s contnuous, non-decreasng, and convex. The constant D s hdden to A, and s set to wth probablty 2c(n+) and to n + wth probablty 2c(n+). Let E denote the event that D =. At tme 0, one bg job J b of sze n + s released. At each nteger tme t n, one unt szed job J t s released. Here n s assumed to be suffcently large. That s n > 2c ɛ. Note 2 that the event E has no effect on A s schedulng decson, snce A s gnorant of the cost functon. Suppose the onlne algorthm A fnshes the bg job J b by tme n+2. Further, say the event E occurs; that s D =. Snce 2n + volume of jobs n total are released and A can process at most (2 ɛ)(n+2) amount of work durng [0, n+2], A has at least 2n + (2 ɛ)(n + 2) = ɛ(n + 2) 3 volume of unt szed jobs unfnshed at tme n + 2. A has total cost at least 2c(ɛ(n + 2) 3) 2 /2 > c(ɛn) 2 /2., A has an expected cost greater than Ω(n). Now suppose A dd not fnsh J b by tme n+2. Condtoned on E, A has cost at least 2c. Hence A s expected cost s at least 2c( 2c(n+) ) > c. The nequalty follows snce n > 2c ɛ 2. Knowng that Pr[E] = 2c(n+)

14 We now consder the adversary s schedule. Condtoned on E (D = ), the adversary completes each unt szed job wthn one unt tme and hence has a non-zero cost only for J b. The total cost s 2c(n + ). Condtoned on E (D = n + ), the adversary schedules jobs n a frst n frst out fashon thereby havng cost 0. Hence the adversary s expected cost s 2c(n+) (2c)(n + ) =. Knowng that n s suffcently larger than c, the clam follows snce A has cost greater than c n expectaton. Next we show a lower bound for any non-clarvoyant algorthm that knows g. In [2] t was shown that no algorthm can be O()-compettve for a general cost functon wth speed less than 7/6. However, the cost functon g used n the lower bound was nether contnuous nor convex. We show that no algorthm can have a bounded compettve rato f t s gven a speed less than 2 > 7/6 even f the functon s contnuous and convex but the algorthm s requred to be non-clarvoyant. Proof of [Theorem 4]: Let A be any non-clarvoyant determnstc onlne algorthm wth speed s. Let the cost functon g be defned as g(f ) = F 0 for F > 0 and g(f ) = 0 otherwse. It s easy to verfy that g s contnuous, non-decreasng, and convex. At tme t = 0, job J of processng length 0 unts and weght w s released. At tme t = 0( 2 ), job J 2 of weght w 2 s released. Weghts of these jobs wll be set later. The processng tme of job J 2 s set based on the algorthm s decsons, whch can be done snce the algorthm A s non-clarovyant. Consder the amount of work done by A on the job J 2 by the tme t = 0. Suppose algorthm A worked on J 2 for less than 0( 2 ) unts by tme t = 0. In ths case, the adversary sets J 2 s processng tme to 0 unts. The flow tme of job J 2 n A s schedule s (0 0( 2 )) + (0 0( 2 ))/s 0 + 0( 2 )ɛ/( 2 ɛ) when s = 2 ɛ. Let ɛ = 0( 2 )ɛ/( 2 ɛ). Hence, A ncurs a weghted flow tme of ɛ w 2 towards J 2. The optmal soluton works on J 2 the moment t arrves untl ts completon, so ths job ncurs no cost. The optmal soluton processes J partally before J 2 arrves and processes t untl completon after job J 2 s completed. The largest flow tme the optmal soluton can have for J s 20, so the optmal cost s upper bounded by 0w. The compettve rato of A ɛ w 2 0w can be made arbtrarly large by settng w 2 to be much larger than w. Now consder the case where A works on J 2 for 0( 2 ) unts by tme t = 0. In ths case, the adversary sets the processng tme of job J 2 to 0( 2 ). Therefore, A completes J 2 by tme t = 0. However, A can not complete J wth flow tme of at most 0 unts, f gven a speed of at most 2 ɛ. Hence A ncurs a cost of ɛw towards flow tme of J. It s easy to verfy that for ths nput, the optmal soluton frst schedules J untl ts completon and then processes job J 2 to completon. Hence, the optmal soluton completes both the jobs wth flow tme of at most 0 unts, ncurrng a cost of 0. Agan, the compettve rato s unbounded.

15 Fnally, we show a lower bound for any non-clarvoyant algorthm that knows g on m dentcal machnes. We show that no algorthm can have a bounded compettve rato when gven speed less than 2 m. Prevously, the only prevous lower bounds for the general cost functon on dentcal machnes were lower bounds that carred over from the sngle machne settng. Proof of [Theorem 5]: We use Yao s mn-max prncple. Let A be any nonclarvoyant determnstc onlne algorthm on m parallel machnes wth the speed s = 2 ɛ, for any 0 < ɛ. Let L > be a parameter and we take m > ɛ. Let the cost functon g(f ) be defned as follows: g(f ) = F L for F > L and g(f ) = 0 otherwse. It s easy to verfy that, g s contnuous, non-decreasng, and convex. At tme t = 0, (m )L + jobs are released nto the system, out of whch (m )L jobs have unt processng tme and one job has processng tme L. The adversary sets the job wth processng tme L unformaly at random amongst all the jobs. Consder the tme t = L(m )+ sm. At the tme t, there exst a job j that has been processed to the extent of at most unt by A snce the most work A can do s smt = L(m ) +, whch s the total number of jobs. Wth probablty L(m )+, j has a processng tme of L unts. In the event that j has the processng tme of L unts, the earlest A can complete j s t + L L(m )+ sm + L s s = > L when L s suffcently large and s 2 ɛ (note that m > ɛ ). In ths case, j has a flow tme greater than L tme unts. Therefore, n expectaton A ncurs a postve cost. Let us now look at the adversary s schedule. Snce the adversary knows the processng tmes of jobs, the adversary processes the job j of length L on a dedcated machne. The rest of the unt length jobs are processed on other machnes. The adversary completes all the jobs by the tme L and hence pays cost of 0. Therefore, the expected compettve rato of the onlne algorthm A s unbounded. References. Anand, S., Garg, N., Kumar, A.: Resource augmentaton for weghted flow-tme explaned by dual fttng. In: SODA. pp (202) 2. Avraham, N., Azar, Y.: Mnmzng total flow tme and total completon tme wth mmedate dspatchng. In: SPAA 03: Proceedngs of the ffteenth annual ACM symposum on Parallel algorthms and archtectures. pp. 8 (2003) 3. Awerbuch,., Azar, Y., Leonard, S., Regev, O.: Mnmzng the flow tme wthout mgraton. SIAM J. Comput. 3(5), (2002) 4. Azar, Y., Epsten, L., Rchter, Y., Woegnger, G.J.: All-norm approxmaton algorthms. J. Algorthms 52(2), (2004) 5. ansal, N., Chan, H.L.: Weghted flow tme does not admt o()-compettve algorthms. In: SODA. pp (2009) 6. ansal, N., Krshnaswamy, R., Nagarajan, V.: etter scalable algorthms for broadcast schedulng. In: ICALP (). pp (200) 7. ansal, N., Pruhs, K.: The geometry of schedulng. In: IEE Symposum on the Foundatons of Computer Scence. pp (200)

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