Primal Dual Gives Almost Optimal Energy Efficient Online Algorithms

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1 Prmal Dual Gves Almost Optmal Energy Effcent Onlne Algorthms Nkhl R. Devanur Zhy Huang Abstract We consder the problem of onlne schedulng of obs on unrelated machnes wth dynamc speed scalng to mnmze the sum of energy and weghted flow tme. We gve an algorthm wth an almost optmal compettve rato for arbtrary power functons. No earler results handled arbtrary power functons for mnmzng flow tme plus energy wth unrelated machnes.) For power functons of the form fs) = s α for some constant α >, we get a compettve rato of O α ), mprovng upon a prevous log α compettve rato of Oα 2 ) by Anand et al. [3], along wth a matchng lower bound of Ω α ). Further, n the resource log α augmentaton model, wth a + ɛ speed up, we gve a 2 + ) compettve algorthm, wth essentally the same ɛ technques, mprovng the bound of + O ) by Gupta ɛ 2 et al. [5] and matchng the bound of Anand et al. [3] for the specal case of fxed speed unrelated machnes. Unlke the prevous results most of whch used an amortzed local compettveness argument or dual fttng methods, we use a prmal-dual method, whch s useful not only to analyze the algorthms but also to desgn the algorthm tself. Introducton The desgn of onlne algorthms for schedulng problems has been an actve area of research. Typcally n such problems obs arrve onlne, over tme, and n order to complete a ob t must be assgned a certan amount of processng, ts processng volume. The algorthm has to schedule the obs on one or more machnes so as to complete them as soon as possble. A standard obectve s a weghted sum of flow tmes; the flow tme of a ob s the duraton of tme between ts release and completon. In the unrelated machnes verson of the problem, each ob can have a dfferent volume and a dfferent weght for each machne. Preempton/resumpton s allowed, but mgraton of a ob from one machne to another s not. Mcrosoft Research, Redmond. Emal: nkdev@mcrosoft.com. Stanford Unversty. Emal: hzhy@stanford.edu. Ths work was done whle the author was a graduate student at Unversty of Pennsylvana and an ntern at Mcrosoft Research, Redmond. Supported n part by an ONR MURI Grant N4797 and a Smons Graduate Fellowshp for Theoretcal Computer Scence No ). Of late, an mportant consderaton n such problems has been the energy consumpton. A popular approach to model the energy consumpton s va the dynamc speed scalng model: a machne can run at many dfferent speeds, hgher speeds process obs faster but consume more energy. The rate of consumpton of energy w.r.t. tme s the power consumpton whch s gven as a functon of speed. Typcal power functons are of the form s α where s s the speed and α s some constant, commonly equal to 2 or 3. The obectve now s to mnmze the sum of energy consumed and weghted flow tme. In ths paper we show that a natural and prncpled prmal-dual approach gves almost optmal compettve ratos for the most general verson of the problem, wth unrelated machnes and arbtrary monotone non-decreasng and convex power functons. See Secton 2 and Secton 3 for formal problem defntons.) We summarze our contrbutons below. For power functons fs) = s α, we gve an algorthm wth compettve rato 8α/ log 2 α. We also gve a correspondng lower bound of α/4 log 2 α. Ths mproves upon a prevous Oα 2 ) compettve rato of Anand et al. [3]. We also show bounds for specfc values of α, for α = 2 and 3 we show compettve ratos of 4 and 5.58 respectvely. It s worth notng that these bounds mprove upon the prevous best bounds known for these values of α, even for a sngle machne whch were 5.24 and 8 respectvely for α = 2 and 3, due to Bansal et al. [7]). 2 For an arbtrary power functon f, we defne a quantty Γ f such that for the power functon fs) = s α, Γ f = α. Therefore Γ f can be thought of as a generalzaton of the quantty α. We show analogous bounds for arbtrary power functons, an upper bound of 8Γ f / log 2 Γ f and a lower bound of Γ f /4 log 2 Γ f. No prevous bounds were known for arbtrary power functons. 3 These results are summarzed n Table. An alternate obectve functon often consdered because t s sometmes easer to deal wth, s the Ths s w.l.o.g. as one can reduce problems wth arbtrary power functons to those wth non-decreasng and convex power functons e.g., Secton 3. of [6]). 2 For the unweghted case, however, a compettve rato of 2 s known, due to Andrew et al. [4]. 3 The power functons can dffer by machne, and the compettve rato s determned by the worst one.

2 fractonal flow tme. For the fractonal flow tme, magne that a ob s broken nto nfntesmally small peces and each pece has an ndependent flow tme whch s equal to the tme between ts own completon tme and ts release tme. The fractonal flow tme s the average flow tme of all the peces put together. Our guarantees for the fractonal flow tme are essentally the same as those for the ntegral flow tme and they are summarzed n Table 2. We also consder the resource augmentaton model, where for the same gven power, machnes used by the algorthm run + ɛ tmes faster than the machnes used by the offlne optmum. Our technques wth mnor modfcatons) extend to ths model as well. We show a compettve rato of 2 ɛ + ) whch mproves upon the prevous best known bound of + O ɛ ) by Gupta et 2 al. [5]. As a specal case when the power functon s a - step functon, ths bound matches the best known compettve rato for mnmzng the weghted flow tme on fxed speed unrelated machnes obtaned n Anand et al. [3] and Chadha et al. []). These results are summarzed n Table 3. Technques and proof overvew The most common and successful technque for analyzng onlne algorthms for such schedulng problems has been the amortzed local compettveness argument. The technque calls for a potental functon that stores the excess cost ncurred by the optmum soluton and uses t as needed to pay for the algorthm s soluton. More recently Anand et al. [3] used the dual fttng method to gve several mproved compettve ratos. We use a prmal dual approach, whch s a prncpled approach that s used to gude the desgn of the algorthm tself n addton to beng a tool for the analyss. Our algorthms are based on a convex programmng relaxaton. Most of the work done s n understandng the structure of the convex program and the propertes of the optmal prmal and dual solutons; the algorthm and the analyss follow naturally after that. In other words, we derve the algorthm from the structure of the convex program. Almost all the specal cases of our problem, such as related machnes, a sngle machne, unweghted flow tme, etc. use the followng speed scalng rule ntroduced by Albers and Fuwara [2]: set the speed so that the power consumed s equal to the remanng weght PERW). The power s the rate of energy consumed and the remanng weght can be thought of as the rate of accrual of the weghted flow tme. What the PERW rule therefore ensures s that the energy consumed and the weghted flow tme are both equal, whch s convenent for the analyss. At frst glance, t may also appear that PERW s the greedy choce,.e., t s the optmal choce f no more obs are released. A more careful analyss shows ths to be false, that the optmal speed to set, f no more obs are released, s s so that f f s)) equals the remanng weght. 4 We frst analyze the natural greedy algorthm that follows from ths speed scalng rule coupled wth a greedy ob assgnment polcy: a ob s assgned to the machne for whch the ncrease n the energy plus flow tme s the smallest. Ths gves an Oα) compettve rato, whch already beats the prevous bound of Oα 2 ) whch uses the greedy ob assgnment polcy along wth the PERW speed scalng rule. Settng the speed so that no other ob arrves n the future s really a conservatve approach. The algorthm should antcpate the arrval of some obs n the future and run faster. Such an aggressve algorthm faces a trade-off: the mprovement obtaned when there are more obs n the future versus the degradaton when there aren t. The algorthm should hedge aganst both the cases and balance the compettve rato. A systematc way to do ths s to set the speed s so that f f s/c)) equals remanng weght, for some constant c, obtan a compettve rato as a functon of c and then set c to mnmze the compettve rato. For nstance, for the power functon fs) = s α, the best ln α choce of c we have turns out to be + α, gvng a 8α compettve rato of log 2 α. Moreover ths framework allows us to also analyze the PERW rule for power functons fs) = s α, whch corresponds to a choce of c = α ) /α. We show that the compettve rato wth the PERW speed scalng rule s stll O α log α ), thus gvng some ustfcaton for ths rule as well. We start our analyss by consderng the obectve of weghted fractonal flow tme plus energy Secton 2). We consder a convex programmng relaxaton and ts dual usng Fenchel conugates. We analyze the smple case of schedulng a sngle ob on a sngle machne and derve the speed scalng rule f f s)) = remanng fractonal) weght as optmal. We then consder many obs on a sngle machne, all released at tme, and show that the same speed scalng rule s optmal, along wth the ob selecton rule of hghest densty frst HDF, densty = weght/volume). We characterze the optmal dual soluton for the same and show several structural propertes of the optmal prmal and dual solutons. Fnally we consder the unrelated machnes case, whch calls for a ob assgnment rule whch assgns obs to machnes). At any tme, gven the ob assgnments, the algorthm uses the optmal 4 f s the Fenchel conugate of f, defned as f µ) := sup x {µx fx)}, and f f s)) has the followng geometrc nterpretaton: f you draw a tangent to f at s, then ths s the length of the y-ntercept of the tangent.

3 Table : Compettve ratos for mnmzng ntegral weghted flow tme plus energy Secton 3) Sngle machne, fs) = s α 5.24 [7] Best known Ths paper α = 2 α = 3 General α = 2 α = 3 General 8 [7] Unrelated machnes, fs) = s α Oα 2 ) [3] Unrelated machnes, any fs) α O log 2 α ) [6, 7] c new) O α log 2 α )a Thm. 3.2) Θ α log 2 α )a Thm. 3.2, 5.) Θ ) Γf ab log 2 Γ f Thm. 3.2, 5.) Table 2: Compettve ratos for mnmzng fractonal weghted flow tme plus energy Secton 2) Best known Ths paper General < α < 2 α = 2 α = 3 General Sngle machne, arbtrary fs) 2 [6] α O α log 2 α ) Unrelated machnes, fs) = s α Oα) [3] 2α Unrelated machnes, any fs) new) Θ α log 2 α )a Thm. 2.3, 5.) Θ ) Γf a log 2 Γ f Thm. 2.3, 5.) Table 3: Compettve ratos for mnmzng fractonal/ntegral weghted flow tme plus energy any power functon fs)) wth resource augmentaton + ɛ)-speed) Secton 4) Best known Ths paper Integral flow tme plus energy, arbtrary power functon + O ɛ 2 ) [5] ɛ Thm. 4.) Fractonal flow tme plus energy, arbtrary power functon + 5 ɛ [5] Integral flow tme, fxed speed ɛ [3] ɛ Thm. 4.2) Fractonal flow tme, fxed speed ɛ [] a 8Γ Specfcally, we show an upper bound of f 8α log 2 Γ f log 2 α for fs) = sα Γ ) and a lower bound of f α 4 log 2 Γ f 4 log 2 α for fs) = sα ) on the compettve ratos. b Γ f s defned to be max s f f s))/fs) +. c To acheve ths compettve rato, one need to combne the technques n [6, 7]. E.g., see the dscussons by Anand et al. [3] for more detals.

4 schedule for each machne assumng no future obs arrve and the correspondng dual. The ob assgnment rule follows from the complementary slackness condtons usng these duals. 5 The compettve rato of Oα) follows from a smple local chargng of the prmal to the dual cost, along wth some of the structural propertes establshed earler. We then consder a systematcally aggressve speed scalng rule, f f s/c)) = remanng weght for some constant c Secton 2.5). Wth the rest of the algorthm/proof more or less dentcal, we derve a compettve rato as a functon of c. On optmzng, we get a compettve rato of Oα/ log α). Typcally algorthms desgned for the fractonal flow tme also work for the ntegral flow tme wth some loss n the compettve rato. However, our analyss for the fractonal flow tme also goes through for the ntegral flow tme wth small modfcatons) wthout any loss n the compettve rato! Almost the same proof structure works for the ntegral case and we outlne what mnor modfcatons are requred Secton 3). Once agan essentally the same proof goes through for the resource augmentaton model as well Secton 4). Fnally we show a smple example wth 2 machnes that gves almost tght lower bounds Secton 5). Related work We summarze here a selecton of the most relevant related work. The fact that energy costs are a substantal part of the overall cost n data centers see Barroso [9]) motvates energy consderatons n schedulng problems. Early work on energy effcent algorthms was for the case of a sngle machne. Albers and Fuwara [2] ntroduced the obectve of energy plus flow tme n the dynamc speed scalng model, and the PERW speed scalng rule. Generalzatons to weghted flow tme followed [7, 5, 7] wth the current best compettve ratos gven by Bansal et al. [6] and Andrew et al. [4]. For other varants n the dynamc speed scalng model, and other models such as the power down model and the mportance of energy effcent algorthms, see the survey by Albers []. Motvated by the desgn of archtectures wth heterogenous cores/processors, Gupta et al. [5] consdered the case of related machnes wth dfferent power functons. See the references theren for more reasons to consder heterogenous machnes.) All these algorthms use the PERW speed scalng rule and use potental functons wth amortzed local compettveness arguments. Anand et al. [3] used a dual fttng based argument 5 Ths assgnment rule s almost the same as the greedy assgnment rule, but s slghtly dfferent. We state the ob assgnment rule derved from the complementary slackness condtons n the algorthm snce that seems more prncpled. In any case, the analyss remans the same for both ob assgnment rules. to generalze ths to unrelated machnes and the fxed speed case n the resource augmentaton model mprovng upon a prevous algorthm by Chadha et al. []). The maor dfference between dual fttng and prmal dual s that dual fttng s only used as an analyss tool for a gven algorthm whle prmal dual gudes the desgn of the algorthm tself. Also n recent work Gupta et al. [4] showed that the natural extensons of several well known algorthms that work for homogeneous machnes fal for heterogeneous machnes, thus ustfyng the use of non-standard algorthms of Gupta et al. and Anand et al. [5, 3]. As summarzed n Table - 3, our work unfes and mproves upon several of these results, most promnently [7, 6, 3, 5, ]. Buchbnder and Naor [] establshed the prmaldual approach for packng and coverng problems, unfyng several prevous potental functon based analyss. Gupta et al. [6] gave a prmal-dual algorthm for a general class of schedulng problems wth cost functons of the form fs) = s α. A dual and equvalent) problem of onlne concave matchng was consdered by Devanur and Jan [3] who also used the prmal-dual approach to gve optmal compettve ratos for arbtrary cost functons. Usng ether of these results for the problems we consder only gves a compettve rato of α α whch s sgnfcantly far from the ratos we obtan. Recent related work In concurrent and ndependent work, Nguyen [8] shows an Oα/ log α)-compettve algorthm for mnmzng flow tme plus energy on unrelated machnes when the power functon s fs) = s α. The analyss uses dual fttng based on a non-convex program and Lagrangan dualty. 2 Weghted fractonal flow-tme plus energy In ths secton we consder the obectve of fractonal flow tme plus energy and obtan compettve algorthms for t. Suppose that we are gven a power functon, f : R + R +, whch s monotoncally non-decreasng, convex and s at. Further, we assume that f s also dfferentable. For ease of presentaton we assume that all the machnes have the same power functon although all of our results go through easly wth dfferent power functons for dfferent machnes. offlne verson of the problem. Frst, we defne the Input: A set of obs J and a set of machnes M. For each ob J ts release tme r R +. For each ob J and each machne M, the volume v and the weght w of ob f scheduled on machne. Let the densty of ob on machne be ρ = w /v.

5 Output: An assgnment of each ob to a sngle machne no mgraton). For each machne and tme t R +, the ob scheduled at tme t on machne, denoted by t) and the speed at whch the machne s run, denoted by s t. Constrants: Job must be scheduled only after ts release tme. It must receve a total amount of v unts of computaton f t s assgned to machne,.e., t [r s,+ ]: t)= t dt = v. Obectves: The obectve has two components, energy and fractonal flow-tme. Recall that f s the power functon, whch gves the power consumpton as a functon of the speed. The energy consumed by machne s therefore E = fs t )dt. The fractonal flow-tme s an aggregated measure of the watng tme of a ob. Suppose ob s scheduled on machne. Let ˆv t) be the remanng volume of ob at tme t,.e., ˆv t) = v t [r,t]: t )= s t )dt. The fractonal flow-tme of ob s then defned to be F := v r ˆv dt. The obectve s to mnmze the total energy consumed by all the machnes and a weghted sum of the flow-tmes of all the obs: E + : w F ) In the onlne verson of the problem the detals of ob are gven only at tme r. The algorthm has to make decsons at tme t wthout knowng anythng about the obs released n the future. 2. Convex programmng relaxaton and the dual The algorthms we desgn are based on a convex programmng relaxaton of the problem and ts dual, whch are shown n Fgure. The dual convex program s obtaned usng Fenchel dualty. The Fenchel conugate of f s the functon f µ) := sup x {µx fx)}. See Appendx A or [2] for a detaled explanaton.) The varables s t denote the speed at whch ob s scheduled on machne at tme t. s t = s t s the total speed of machne at tme t. For each ob, constrant 2.) enforces that the schedulng must complete ob. We wll not state trval constrants such as s t throughout the paper.) In the obectve functon, the frst summaton corresponds to. 2.) P frac ) mnmze, ρ, t : : r + fs t )dt +, r s t w s t = :r t s t r s t v dt t r )s t dt w f ) w)dw ) dt D frac ) maxmze α f β t )dt,, t r : α v ρ t r ) + β t + w w f ) w)dw Fgure : Convex Programmng relaxaton of mnmzng fractonal flow tme plus energy. the fractonal flow-tme: s t dt unts of ob s processed between t and t + dt, all of whch wated for a duraton of t r resultng n t r ) st v dt amount of fractonal flow-tme. The second summaton s the total energy consumed. The thrd summaton s requred because the convex program allows a ob to be splt among many machnes and even have dfferent parts run n parallel. Wthout the thrd term, the convex program fals to provde a good lower bound on the cost of the optmal soluton. We show that the optmum of the convex program wth an addtonal factor of 2 s a lower bound on opt, the optmum offlne soluton to the problem. We note ths n the followng theorem. Theorem 2.. The optmum value of the convex crogram P frac ) s at most 2opt. Proof. Consder an nstance wth only one ob released at tme and a large number of machnes. The optmal soluton to the convex program schedules the ob smultaneously on all the machnes and the total cost w.r.t. the frst two terms wll tend to zero as the number of machnes tends to nfnty. The optmal algorthm has to schedule the ob on a sngle machne and hence pays a fxed non-zero cost. The thrd term fxes ths problem: consder a modfed nstance where we have multple copes of each machne as many as the number of obs); the cost of the optmal soluton to ths nstance s only lower. In ths modfed nstance, w.l.o.g., no two obs are ever scheduled on the same machne. It can be shown Lemma 2.2) that f ob s scheduled on a copy of machne all by tself, then the optmal cost energy + flow-tme) due to ob s v w w f ) w)dw. Now, stll allowng a ob to be splt among dfferent machnes, an r s t v dt fracton of ob s scheduled on machne.

6 Thus s t w r w f ) w)dw ) dt s a lower bound on the cost of schedulng ob. The algorthm heavly uses the structure of the optmal solutons to the prmal and the dual programs. We explan ths structure n stages. There s a natural decomposton of the problem tself: at the hghest level s the decson to allocate a ob to one of the machnes. Gven these choces, the rest of the problem decomposes nto a separate one for each machne. For each machne, gven the set of obs that have to be scheduled on t and the volumes and denstes, there s the problem of pckng the ob to schedule and the speed to set at any tme, n order to mnmze the total energy and flow-tme on that machne. Further, gven the choce of the ob to schedule on a machne, there s an even smpler problem of settng the speed. We start wth the smplest problem of all, gven ust a sngle ob and a sngle machne, what s the optmal speed schedule that mnmzes the total energy and fractonal flow-tme Optmal schedulng for sngle ob We obtan a smpler convex program and ts dual for the problem of schedulng a sngle ob on a sngle machne. We drop the thrd term n the obectve snce that deals wth nonntegral assgnment of obs to machnes. Snce there s only one ob n ths subsecton, we assume w.l.o.g. that r =. 2.2) mnmze t : maxmze t : ρ ts t dt + fs t )dt s t = s t s t v dt α f β t )dt ρ t + β t α v Recall that the conugate functon f s defned as f β) := sup s {βs fs)}. The functon f s also convex and monotoncally non-decreasng. If f s strctly convex, then so s f. The most mportant property we use about f s the noton of a complementary par. β and s are sad to be a complementary par f any one of the followng condtons hold. It can be shown that f one of them holds, then so do the others.). f s) = β; 2. f ) β) = s; 3. fs) + f β) = sβ. 6 A characterzaton of the optmal schedulng can be found, e.g., n [8], but we beleve t s llustratve to rederve the optmal schedulng usng our prmal dual approach. The optmal solutons to these programs are characterzed by the generalzed) complementary slackness or KKT condtons. These are:. t, s t > α = v ρ t + β t ); 2. α > s t v dt = ; 3. β t and s t are a complementary par for all t. The frst condton mples that for the entre duraton that the machne s runnng wth non-zero speed), the quantty ρ t+β t remans the same, snce t must always equal α. In other words, β t must lnearly decrease wth tme, at the rate of ρ,.e., dβt dt = ρ. The man result n ths secton s that the optmum soluton has a closed form expresson where s t and β t are set as a functon of the remanng weght of the ob at tme t, whch we denote by ŵ t. More generally, ŵ t wll denote the total remanng weght of all the obs on machne.) Also the remanng volume at tme t s denoted ˆv t. Lemma 2.. The optmum soluton to the convex program Eqn. 2.2) s such that f f s t )) = f β t ) = ŵ t, and α = v f w ). Proof. Snce β t and s t form a complementary par, we have that df β t) dβ t = s t. We start by multplyng the LHS by dβt dt and the RHS by ρ, whch s vald snce we showed these are equal. Ths gves Therefore df β t ) dβ t = s t ρ. dβ t dt df β t ) dt = ρ dˆv t dt = dŵ t dt When ŵ t =, then s t =, fs t ) = and therefore f β t ) = by the thrd property of complementary pars). Therefore at any tme f β t ) = ŵ t. The rest of the assertons n the lemma follow mmedately. Remark 2.. By f β t ) = ŵ t and that β t and s t are a complementary par, we get a closed form of s t = α ) /α ŵ /α t when fs) = s α. Contrast ths wth the PERW rule for whch s t = ŵ /α t. We also gve a closed form for the value of the optmum. Ths form ustfes the ncluson of the thrd term n the obectve for the convex program P frac ). We wll also use ths lemma later to analyze how the cost of the optmum soluton changes as we add new obs..

7 Lemma 2.2. The cost of the optmal soluton s w ρ f ) w)dw. Proof. Recall that the total weghted flow-tme s equal to ŵ t dt and the total energy s equal to fs t )dt. From Lemma 2., f β t ) = ŵ t. Usng ths and the propertes of complementary pars, we get the followng sequence of equaltes. ŵ t + fs t )) dt = f β t ) + fs t )) dt = β t s t dt. Further, by the defnton of s t and β t, the above equals β t dˆv t = ρ β t dŵ t = ρ f ŵ t )dŵ t. The lemma follows from observng that as t goes from to, ŵ t goes from w to. 2.3 Optmal schedulng for a sngle machne We now consder the next stage where there are multple obs to be scheduled on a sngle machne, and the correspondng convex programs. We wll contnue to assume that r = for all obs. 2.3) mnmze t : : maxmze t, : r s t = s t ρ ts t dt + fs t )dt s t v dt α f β t )dt ρ t + β t α v The complementary slackness condtons for these par of programs are more or less as before. To begn wth, s t > α v = ρ t + β t. As before, ths mples that β t decreases at rate ρ whenever ob s scheduled, but the man new ssue s the choce of obs to schedule. The above complementary slackness condton mples that ob must be scheduled when the term ρ t+β t attans ts mnmum. The frst part, ρ t, always ncreases at rate ρ, whle the second part, β t decreases at rate ρ t) where t) s the ob scheduled at tme t. So f ρ < ρ t) then ρ t+β t s decreasng and vce-versa, f ρ > ρ t) then ρ t+β t s ncreasng. Ths mples that the hghest densty frst HDF) rule s optmal,.e., schedule the obs n the decreasng order of the densty. For any, ρ t + β t frst decreases when hgher densty obs are scheduled, then remans constant as ob s scheduled and then ncreases as lower densty obs are scheduled. Gven the choce of obs scheduled, the choce of speed s very smlar to the sngle-ob case. We state the followng generalzaton of Lemma 2. wthout proof, snce t ether follows from the dscusson above or s very smlar to the proof of Lemma 2.. Lemma 2.3. The optmum soluton to the convex program 2.3) s such that. Jobs are scheduled n the decreasng order of densty. 2. f f s t )) = f β t ) = ŵ t. 3. α = w t +v f ŵ t ) where t s the frst tme ob s scheduled. Unlke the sngle-ob case, we no longer have a closed form expresson for the optmal cost. Instead, we consder the margnal ncrease n the optmal cost due to a sngle ob, and show the followng generalzaton of Lemma 2.2. Lemma 2.4. The ncrease n the cost of the optmal soluton on ntroducng a new ob s at most w t + ρ w f ŵ t + w)dw where t s the frst tme ob s scheduled and ŵ t s the remanng weght at tme t n the orgnal nstance, wthout ob. Proof. It suffces to show that even f we use a suboptmal speed-scalng after ntroducng the new ob, the ncrease n the cost s only the amount as stated n the lemma. In partcular consder the sub-optmal schedulng n whch we keep the schedule tll tme t unchanged, and then start to schedule ob and use the optmal speed scalng. The entre ob wats tll tme t contrbutng w t to the total flow-tme. For the rest of the ncrease, consder ntroducng an nfntesmal weght dw at tme t. Ths causes the followng nfntesmal ncrease n the flow-tme and energy: df = ŵ t dt and de = fs t )dt. The rest of the proof s along the same lnes as that of Lemma 2.2 and s omtted here. Fnally, we note a couple of smple observatons that follow almost mmedately from Lemma 2.3. Let Γ f = max s f f s)) fs) +. Lemma 2.5. f β t )dt equals the total flow-tme. Lemma 2.6. The total flow-tme s at most Γ f tmes the total energy consumed. Proof. The total energy used s fs t )dt. The total flow-tme s ŵ t dt. Recall that by Lemma 2.3, s t s chosen such that ŵ t = f f s t )). So the rato between the fractonal flow tme and the energy s f f s t ))dt dvded by fs t )dt, whch s at most max s f f s)) fs).

8 2.4 Conservatve greedy algorthm In ths secton, we analyze a prmal-dual algorthm whch we call conservatve-greedy. The basc dea s that gven the choce of ob assgnments to machnes, the algorthm schedules the obs as f no other obs wll be released n the future. That s, t schedules the obs as per the optmal schedule for the current set of obs, as detaled n the prevous secton. The choce of ob assgnments to machnes s done va a natural prmal-dual method, the one dctated by the complementary slackness condtons. Concretely, at any pont, gven the obs already released and ther assgnment to machnes, the algorthm pcks the optmal schedulng on each machne, assumng no future obs are released. Ths also gves dual solutons, n partcular the varables β t for all and t n the future. When a new ob s released, ts assgnment to a machne s naturally drven by the followng dual constrants and the correspondng complementary slackness condtons. For all, t, α ρ t r ) + β t + v w w f ) w)dw. For a gven machne, we saw earler that the rght hand sde RHS) s mnmzed over all t) at t where t would be the frst tme ob s scheduled on gven the HDF rule. That stll holds true snce the thrd term above s ndependent of t. Now we need to mnmze over all as well and the algorthm does exactly ths. It assgns ob to the machne that mnmzes the RHS of the nequalty above wth t = t multpled by v. It sets the dual α so that the correspondng constrant s tght. It then updates the schedule and β t s for machne. Note that as we add more obs, the β t s can only ncrease, thus preservng dual feasblty. The entre algorthm s summarzed n Fgure 2. We wll show that, surprsngly, such a conservatve approach already acheves a meanngful compettve rato for arbtrary power functons and a near optmal compettve rato for polynomal power functons. Formally, we wll show the followng theorem n ths secton. Theorem 2.2. The fractonal conservatve greedy algorthm s 2Γ f -compettve for mnmzng weghted fractonal flow tme plus energy. The above compettve rato mght be unbounded f the functon s hghly skewed. Theorem 2.2 does not contradct known lower bounds for fxed-speed onlne schedulng, whch s a specal case when the power functon s a - step functon. For nce power functons such as polynomal power functons, the above theorem gves meanngful compettve ratos. In partcular, the fractonal conservatve greedy algorthm s 2α-compettve for mnmzng weghted fractonal flow tme plus energy wth fs) = s α. In the remanng of ths secton, we wll present the prmal-dual analyss of Theorem 2.2. It s easy to see that the algorthm constructs feasble prmal and dual solutons and the rato s obtaned by relatng the cost of the prmal to that of the dual. In fact, for every ob released, we relate the ncrease n the cost of the prmal to the ncrease n the cost of the dual. Lemma 2.7. When ob s released, the ncrease n the total cost of the algorthm s at most α. Proof. Suppose ob s assgned to machne and wll be scheduled at t droppng subscrpt ) accordng to the HDF rule on the current set of obs assgned to. Then α = w t r ) + v β t + ρ w f ) w)dw. The ncrease n the total cost of the algorthm s only the ncrease n that for machne. From Lemma 2.4 ths s at most w t r ) + ρ w f ) w + ŵ t )dw. Comparng the two, t suffces to show that 2.4) β t + w w f ) w)dw w w f ) w + ŵ t )dw. By Lemma 2.3, we have β t = f ) ŵ t ). Further, f ) s concave because f s convex. So f ) ŵ t ) + f ) w) f ) w + ŵ t ) for all w w. Integrate ths nequalty for w from to w and we get Eqn. 2.4) as desred. Proof. [Theorem 2.2] Consder the release of a new ob and suppose t s assgned to machne. The change n the dual cost, D, equals α plus the change n the contrbuton of β t s,.e., D = α f β t )dt ). Let the change n the total energy and the total flowtme of the algorthm be E and F respectvely. From Lemma 2.7, α E + F. Earler n Lemma 2.5 we showed that the total flow-tme s always equal to f β t )dt. Thus the same holds for the dfference. f β t )dt ) = F. From the three nequaltes above, the change n the dual cost s at least the change n the energy cost of the algorthm. Snce ths holds for

9 Fractonal conservatve greedy algorthm Speed scalng: Choose speed s t s.t. f f s t )) equals the fractonal remanng weght on machne. Set duals β t = f s t ), also for future tmes based on the planned schedule currently. Job selecton: Schedule the ob wth the hghest densty HDF). Job assgnment: Assgn ob to machne that mnmzes ρ t r ) + β t + w w f ) w)dw where t would be the frst tme ob s scheduled on gven the HDF rule. Set α so that the correspondng constrant s tght. Update the β t s for machne. Fgure 2: The conservatve greedy onlne schedulng algorthm for mnmzng fractonal flow tme plus energy wth arbtrary power functons every change and both the dual cost and the energy cost of the algorthm are zero to begn wth, t follows that the fnal dual cost s at least the fnal energy cost of the algorthm,.e., D E D E. Further, by Lemma 2.6, the total flow-tme and the energy are wthn a factor of Γ f. Even though that was for a sngle machne wth no future obs, t s easy to see that the same holds true for the conservatve-greedy algorthm as well. Therefore F Γ f )E. The total cost of the algorthm, alg can now be bounded n terms of the energy cost alone, whch s bound by the dual as above,.e., alg = E + F Γ f E. Also by Theorem 2. the dual s a lower bound on 2opt,.e., D 2opt. Puttng t all together, we get alg 2Γ f opt. An alternate algorthm wth essentally the same analyss s the followng: assgn ob to machne for whch the ncrease n the total cost s the mnmum. The dual α must however be set as we do currently, so there mght be a dsconnect between whch machne the ob s assgned to and whch machne dctates the dual soluton. The analyss stll s pretty much the same however. 2.5 Aggressve greedy algorthm In the conservatve greedy algorthm, the speed s scaled to the conservatve extreme as the speed s optmal assumng no future obs arrve. However, n an onlne nstance there mght be obs n the future, some of whch wll be effectvely delayed by the current ob. Therefore, a good onlne algorthm should take ths nto account when choosng the speed. In ths secton, we consder a famly of algorthms wth the aggressveness n terms of speed scalng parameterzed by any constant C, as gven n Fgure 3. The followng property of the β t s mples that our choce of β t s can be derved from the prmal dual analyss and ensures dual feasblty. Lemma 2.8. For any tme t at whch no new obs are released, β t lnearly decrease wth tme, at the rate of ρ, where s the ob beng processed on at tme t,.e., dβt dt = ρ. Proof. [Sketch] Note that the clamed equaton s satsfed by the conservatve greedy algorthm. Further, the C-aggressve greedy algorthm s runnng at exactly C tmes the speed of the conservatve greedy algorthm gven the same remanng weght of obs, and the β t s are set to be C fracton of those n the conservatve greedy gven the same remanng weght of obs. Puttng together these observatons and our defnton of β t we can easly verfy the lemma. We wll show that by choosng the optmal aggressveness we can mprove the compettve rato by a logarthmc factor. Concretely, we show the followng. Theorem 2.3. The fractonal C-aggressve greedy algorthm s 2Γ f,c -compettve for mnmzng weghted fractonal flow tme plus energy wth power functon f, where Γ f,c = CCΓ f + Γ f ). Γ f C Γ f + Recall that Γ f = max s f f s)) fs) +.) Proof of Theorem 2.3 In the rest of ths secton, we wll present of proof of Theorem 2.3. Gven Lemma 2.8, t s easy to check that the way we are settng the prmal and dual varables guarantees feasblty. The compettve rato comes from analyzng the rato between the

10 Fractonal C-aggressve greedy algorthm Speed scalng: Choose speed s t s.t. f f st C )) equals the total remanng weght on machne. Set duals β t = C f ) ŵ t ) s.t. f Cβ t ) equals the total remanng weght on machne, also for future tmes based on the planned schedule currently. Job selecton: Schedule the ob wth hghest densty HDF). Job assgnment: Assgn ob to machne that mnmzes ρ t r ) + β t + w w f ) w)dw where t would be the frst tme ob s scheduled on gven the HDF rule. Set α so that the correspondng constrant s tght. Update the β t s for machne. Fgure 3: The aggressve greedy onlne schedulng algorthm for mnmzng weghted fractonal flow tme plus energy wth arbtrary power functons ncremental costs of the algorthm and the dual due to the arrval of new obs. We wll frst develop a few lemmas that are needed n the analyss. We start by showng that the parameter Γ f for arbtrary functon f plays a smlar role as the degree of polynomal functons as the followng lemma holds. Recall Γ f = α for fs) = s α.) Lemma 2.9. For any power functon f, any s > and any C >, we have fcs) C Γ f fs). Proof. Note that for any s >, we have that f s )s fs ) = f f s )) + fs ) fs ) = f f s )) fs ) + Γ f, where the frst equalty s because s and f s ) are a complementary par and the nequalty s by defnton of Γ f. So we have that lnfcs)) lnfs)) = C d lnfxs)) The lemma follows. = C = C f xs)s fxs) dx f xs)xs fxs) x dx C Γ f x dx = Γ f lnc). Alternatve dual obectve: For the convenence of analyss, we wll consder an alternatve dual obectve, max α C f Cβ t )dt, subect to the same constrants. We let D denote the value of the above dual obectve. By the convexty of f, we have the followng lemma. Lemma 2.. For any values of the dual varables, we have D D. Alternatve power functon: We wll consder the total cost of our algorthm w.r.t. the power functon ˆfs) = C Γ f f s C ). We let Ê denote the energy cost of the algorthm w.r.t. ths power functon, and let âlg = F + Ê denote the total cost of the algorthm w.r.t. ths power functon, notng that the flow tme s unaffected. By Lemma 2.9, we have Lemma 2.. For any nstance, we have Ê E and âlg alg. By Lemma 2. and Lemma 2., t suffces to bound the rato between the ncrease n the total cost of the algorthm w.r.t. power functon ˆf when a new ob arrves and the ncrease n the alternatve dual obectve, denoted as âlg and D respectvely. Next, suppose a new ob arrves at tme t and s assgned to machne. We wll account for the ncremental costs of the algorthm and the dual by relatng them to the ncremental cost n an magnary nstance n whch we are usng the conservatve greedy algorthm. We wll call the current nstance I and the magnary nstance I. More precsely, let there be a sngle machne n I that s dentcal to machne n I. For each ncomplete ob n I at tme t before the arrval of ob ), we put an dentcal ob wth the same remanng volume n I at tme. Then, we wll consder also releasng ob n I at tme. By our constructon of I and the fact that the C-aggressve greedy algorthm s runnng exactly C-tmes faster than the conservatve one, there s a one-to-one mappng between the tmelne after t n I and the tmelne n I as specfed n the next lemma, whose proof s straghtforward and omtted. Lemma 2.2. For any tme t t, the remanng weght of each ob n I at tme t s the same as that n I at tme Ct t), both before and after the release of ob. Let F and E denote the ncrease n weghted fractonal flow tme and energy respectvely n I cond-

11 toned on runnng the conservatve greedy both before and after releasng ob. Recall that n the conservatve greedy algorthm, we have F = Γ f ) E. We let F and Ê denote the ncrease n weghted fractonal flow tme and energy w.r.t. ˆf) n I due to the arrval of ob. The next lemma accounts for the the ncrease n flow tme due to the arrval of ob as a fracton of E. The proof s rather straghtforward from Lemma 2.2 so we omt t. Lemma 2.3. F = C F = C Γ f ) E. The next lemma bound the ncrease n energy due to the arrval of ob by a fracton of E. Lemma 2.4. Ê = CΓ f E. Proof. For any t t, suppose the C-aggressve greedy algorthm runs at speed s at tme t. Then, the energy cost w.r.t. ˆf) n I from tme t to t + dt s ˆfs)dt = C Γ f f s C )dt. Further, by Lemma 2.2 and that the C-aggressve greedy algorthm s runnng C- tmes faster than the conservatve one, the conservatve greedy algorthm runs at speed s C at tme Ct t) n I. So the energy cost of from tme Ct t) to Ct t) + Cdt n I s f s C )Cdt, a C Γ f + fracton of the correspondng energy cost n I. Integratng for all t t, we get that the energy cost w.r.t. ˆf) n I after tme t s exactly C Γ f tmes the total energy cost n I. As ths relaton holds both before and after the release of ob, the lemma follows. The next lemma establsh the relaton between the ncremental cost of the alternatve dual due to β t s and E. Lemma 2.5. C f Cβ t )dt = C 2 Γ f )E. Proof. [Sketch] Recall that β t = C f ) ŵ t ). By the convexty of f, we have C f Cβ t ) = C ŵt. So we have that C f Cβ t )dt equals C tmes the weghted fractonal flow tme of nstance I. Hence, we have C f Cβ t )dt = C F = C 2 F. The lemma then follows from F = Γ f ) E. The next lemma follows from the fact that the C- aggressve greedy s runnng at exactly C tme the speed of conservatve greedy and our choce of β t. Lemma 2.6. α C F + E ) = C Γ f E. Proof. [Sketch] Consder the tme t at whch the new ob would be nserted accordng to HDF w.r.t. the orgnal nstance wthout ob ). Let t denote the tme at whch would be nserted n the magnary nstance. By our choce of speed scalng, we have t t) = C t. Moreover, β t = C f ) ŵ t ). So by our choce of α and an analyss smlar to Lemma 2.7, we get that α s at least C tme the total ncrease n weghted fractonal flow tme plus energy n I,.e., α C E + F ). Fnally, we are ready to derve the compettve rato of the C-aggressve greedy algorthm. Proof. [Theorem 2.3] Puttng together Lemma 2.3 to Lemma 2.6, we have that the ncremental costs of the algorthm w.r.t. ˆf) s ) âlg = F + Ê = C Γ f ) + C Γ f E and the ncremental costs of the alternatve dual s D = α f Cβ t )dt C C Γ f ) C 2 Γ f ) E. Smplfyng the rato between âlg and D from the above equatons proves Theorem 2.3. Optmal choce of aggressveness By optmzng our choce of C for < Γ f 2, Γ f = 3, and asymptotcally optmzng t for general Γ f, we get the followng corollary. We also show ts asymptotc) optmalty n Secton 5 wth an almost matchng lower bound. Corollary 2.. The fractonal C-aggressve greedy algorthm s: 2Γ f -compettve for < Γ f 2, wth C = ; 5.58-compettve for Γ f = 3, wth C.68; 8Γ f log 2 Γ f -compettve for general Γ f, wth C = + ln Γ f Γ f. Proof. The compettve ratos for < Γ f 2 and Γ f = 3 are easy to verfy. So we omt the tedous calculaton here. Next, consder the asymptotcal bound for general Γ f. Wth our choce of C = + ln Γ f Γ f, the denomnator of Γ f,c s Γ f C Γ f + = ln Γ f = log 2 Γ f / log 2 e).

12 On the other hand, note that and C = + ln Γ f Γ f C Γ f = = + ln + Γ f )) Γ f + Γ f 2 Γ f + ln Γ ) Γf f < e ln Γ f = Γ f e Γ f. So puttng together, we get that the numerator of Γ f,c s CC Γ f + Γ f ) < + Γ ) ) f 2 Γ f e Γ f + Γ f = 2 2 ) Γ f e + ) Γ f Γ f ) 2 e + 2 Γ f. Fnally, by the above dscusson and that 2 e + 2) log 2 e) < 4 and Theorem 2.3 we prove the clamed asymptotc compettve rato for arbtrary Γ f. In partcular, for polynomal power functons fs) = s α, we have the followng: Corollary 2.2. The fractonal C-aggressve greedy algorthm s: 2α-compettve for < α 2, wth C = ; 5.58-compettve for α = 3, wth C.68; 8α log 2 α ln α α. -compettve for general α, wth C = + We remark that wth fs) = s α the speed scalng of PERW also falls nto the framework of C-aggressve greedy algorthm. Further, ts choce of C α = α ) s also asymptotcally optmal when α 2. We omt ths calculaton n ths paper.) So our result can be vewed as a formal ustfcaton of the optmalty of PERW. Smlar remark apples to the ntegral case. We also note that for mnmzng fractonal flow tme plus energy, we can drop the thrd term n the prmal obectve and drop a factor of 2 n the compettve rato e.g., Lemma 2.6). As a result, we can get an α-compettve algorthm for < α 2, and a compettve algorthm for α = 3, hence the clamed ratos n Table 2. We omt the detals n ths paper. 3 Weghted ntegral flow-tme plus energy In ths secton, we wll dscuss the problem of onlne schedulng for mnmzng weghted ntegral) flow-tme plus energy. The problem for weghted ntegral flow tme has the same nput, output, and constrants as the fractonal flow tme verson. The only dfference s the obectve. Next let us formally defne the weghted ntegral flow tme of an nstance gven a schedule. Suppose ob s completed at tme c, then the weghted ntegral flow tme s w c r ). : An equvalent formula for the same s as follows. Let A t denote the set of obs such that have been released before or at tme t but have not been completed accordng to the schedule tll tme t,.e., r t and s t dt < v. t [r, ]: t)= The weghted ntegral flow tme s equal to A t: w dt. So the man dfference s that when a ob s partally completed, the entre weght of the ob wll contrbute to the weghted ntegral flow tme, whle only the ncomplete fracton wll contrbute to the weghted fractonal flow tme. Convex programmng relaxaton and the dual Smlar to the fractonal, we wll use the prmal-dual analyss va followng convex program for the problem of mnmzng ntegral flow tme plus energy and consder ts dual program: P nt ) mnmze r ρ t r )s t dt + fs t )dt + r f ) w )s t dt 3.5), t : :r s t t = s t s 3.6) : t v 3.7) r D nt ) maxmze,, t r : α v α f β t )dt ρ t r ) + β t + f ) w ) Here we use the same notaton as n the fractonal case so we wll omt the detaled explanatons of the

13 Integral conservatve greedy algorthm Speed scalng: Choose speed s t s.t. f f s t )) equals the ntegral remanng weght on machne. Set duals β t = f s t ) s.t. f β t ) equals the ntegral remanng weght on machne, also for future tmes based on the planned schedule currently. Job selecton and ob assgnment: Upon arrval of a new ob, assgn t to a machne and nsert t nto the processng queue of such that we mnmze ρ t r ) + β t + w w f ) w)dw, where t would be the completon tme of the predecessor of ob n the queue. Set α so that the correspondng constrant s tght. Update the β t s for machne. Fgure 4: The conservatve greedy onlne schedulng algorthm for mnmzng weghted ntegral flow tme plus energy wth arbtrary power functons convex programs. The only change s the thrd term n the prmal program and the correspondng part n the dual). Ths s because condtoned on beng allocated to machne, the optmal cost for ob n a sngleob nstance w.r.t. ntegral flow tme plus energy s v f ) w ). Hence, the share of the optmal sngleob cost for the st v dt fracton of ob s processed on machne from t to t + dt s f ) w )s t dt. Algorthms Smlar to the fractonal case, we wll consder the conservatve greedy algorthm, whch wll use the optmal speed scalng assumng there are no future obs, and a more general famly of C-aggressve greedy algorthms. The man dfference comparng to the fractonal case s the ob selecton rule on a sngle machne s no longer HDF. Instead, the algorthms wll combne the ob assgnment rule ob selecton rule by mantanng a processng queue for each machne. The machnes wll process the obs n ther queues n order. When a new ob arrves, the algorthm wll nsert the new ob to an poston n one of the processng queue accordng to the dual varables. The formal descrptons of the algorthms are presented n Fgure 4 and Fgure 5. The ntegral conservatve/aggressve greedy algorthms obtan the followng compettve rato for general power functons and polynomal power functons respectvely. Theorem 3.. The ntegral conservatve greedy algorthm s 2Γ f -compettve for mnmzng weghted ntegral flow tme plus energy, where recall that Γ f = max s f f s)) fs) +. Corollary 3.. The ntegral conservatve greedy algorthm s 2α-compettve for power functon fs) = s α for mnmzng weghted ntegral flow tme plus energy. Theorem 3.2. The ntegral C-aggressve greedy algorthm s 2Γ f,c -compettve for mnmzng weghted ntegral flow tme plus energy wth power functon f, where Γ f,c = CCΓ f + Γ f ) Γ f C Γ f + By asymptotcally optmzng our choce of C, we get the followng corollares, whose optmalty wll be presented n Secton 5. The proofs are dentcal to the ntegral case and hence omtted. Corollary 3.2. The ntegral C-aggressve greedy algorthm s 2Γ f -compettve for < Γ f 2, wth C = ; 5.58-compettve for Γ f = 3, wth C.68; 8Γ f log 2 Γ f -compettve for general Γ f, wth C = + ln Γ f Γ f. Corollary 3.3. For mnmzng weghted ntegral flow tme plus energy w.r.t. power functon fs) = s α, the fractonal C-aggressve greedy algorthm s 2α-compettve for < α 2, wth C = ; 5.58-compettve for α = 3, wth C.68; 8α log 2 α ln α α. -compettve for general α, wth C = + Compettve rato The prmal dual analyss of the ntegral case s almost dentcal to the fractonal case. So here we wll only sketch the analyss of the conservatve algorthm and explan the man dfferences between the ntegral case and the fractonal case. Extendng the.

14 Integral C-aggresve greedy algorthm Speed scalng: Choose speed s t s.t. f f st C )) equals the ntegral remanng weght on machne. Set duals β t = C f st C ) s.t. f Cβ t ) equals the ntegral remanng weght on machne, also for future tmes based on the planned schedule currently. Job selecton and ob assgnment: Upon arrval of a new ob, assgn t to a machne and nsert t nto the processng queue of such that we mnmze ρ t r ) + β t + w w f ) w)dw, where t would be the completon tme of the predecessor of ob n the queue. Set α so that the correspondng constrant s tght. Update the β t s for machne. Fgure 5: The C-aggressve greedy onlne schedulng algorthm for mnmzng weghted ntegral flow tme plus energy wth arbtrary power functon analyss from conservatve to aggressve algorthms s farly straghtforward usng technques we ntroduce n Secton 2.5. So the detals wll be omtted. Next, we wll show that α s an upper bound on the ncrease n flow tme plus energy due the ob. Ths s the man techncal component of the analyss of the compettve rato. The clam follows easly from the next lemma, whose proof s smlar to Lemma 2.7 and wll be sketched below. Lemma 3.. Suppose s a ob n the queue of machne whose scheduled completon tme s t before the arrval of ob. Then, the ncrease n total costs f we nsert ob to the queue of machne rght after s at most ) v ρ t r ) + β t + f ) w ) Proof. [Sketch] Note that the algorthm uses the optmal speed scalng, assumng no future obs. To bound the ncrease n flow tme plus energy when we nsert ob to the queue of machne rght after, t suffces to bound the ncrease n flow tme plus energy when we use a sub-optmal speed scalng. In partcular, we wll let the obs scheduled before use the same speed as before the arrval of, and use the optmal speed scalng after that. In ths case, the ncrease n flow tme plus energy wll be the flow tme due to ob watng untl tme t,.e., w t r ), plus the ncrease n flow tme of ob and obs scheduled after ) plus energy due to processng ob,.e., v f ) W t ) + w ) where W t ) s the ntegral remanng weght of obs on machne at tme t. By the concavty of f ), ncrease n flow tme s at most v f ) W t )) + f ) w ) ). The lemma then follows by f β t ) = W t ). Note that β t remans the same whle machne s processng the same ob. So the mnmal value of v ρ t r ) + β t + f ) w ) ) must be acheved n the completon tme t of one of the obs on. As a smple corollary of Lemma 3., we have: Corollary 3.4. The value of α s at least the ncrease n flow tme plus energy due to ob. Now we are ready to analyze the compettve rato of the ntegral conservatve greedy algorthm. Proof. [Theorem 3.] Frst, note that t follows from our defnton of the prmal program P nt that ts optmal obectve s at most twce that of the optmal flow tme plus energy by any offlne) algorthm. Further, by our choce of β t s, the contrbuton of f β t )dt s the weghted ntegral flow tme of the algorthm. So combnng wth Corollary 3.4 we get that upon the arrval of new obs, the ncrease n the dual obectve s at least the ncrease n energy of the algorthm. Fnally, va the same argument as n the fractonal case we have that the weghted ntegral flow tme of the algorthm s at most Γ f tmes the energy. So the theorem follows. 4 Resource augmentaton Further, as a smple applcaton of our prmal dual approach, we manage to gve smpler proofs for ether matchng or mproved compettve ratos for several onlne schedulng problems wth resource augmentaton. In the resource augmentaton settng, we wll compare to a weaker offlne benchmark n the sense that gven the same energy, the offlne algorthm can only run at + ɛ) fracton of the speed of the onlne algorthm. Theorem 4.. The fractonal/ntegral conservatve greedy algorthm s + ɛ)-speed and 2 ɛ + ) -