Energy Aware Scheduling for Weighted Completion Time and Weighted Tardiness

Energy Aw are Sc heduling f or W eigh ted Completion Tim e and W eigh ted T ardiness Ro drigo A. C ar r a sco ∗ Garud Iy engar † Cliff Stein ‡ Ma y 2011, v.arXiv 4.6 Abstract The ever in c r easing adoption of mobile devices with limited ener gy storage capacity , on the o ne hand, and more aw arenes s of the environmental impact of ma ssive data centres and server po ols, o n the other hand, have both led to an incr e a sed interest in energy mana gement algorithms. The main con tr ibutio n o f this paper is to present several new constant factor approximation algorithms for ener gy aw are scheduling pro blems where the ob jective is to minimize weigh ted completion time plus the c o st of the energy consumed, in the o ne machine non- preemptive setting, while allowing r elease dates a nd deadlines.Unlik e previous known algorithms these new algorithms can handle general job-dep endent energy cost functions, extending the a pplication of thes e algo rithms to settings outside the typical CPU-energy one. These new settings include problems where in addition, or instead, of energy costs w e also hav e maintenance costs, wear a nd tear, repla c ement costs, etc., which in general depend on the sp eed at which the machin e runs but a lso dep end on the types of jobs pro cessed. O ur algo r ithms also extend to approximating weigh ted tar diness plus ener gy cost, an inherently mor e difficult pr oblem that has no t b een addressed in the literature. Keyw ords : energy aware scheduling, appr oximation algorithms, α -p oints, w eighted tardines s ∗ rac2159@co lumbia.edu . Department of Industrial Engineering & Op erations Research, Columbia U niversit y , Mudd 313, 500W 120th St reet, New Y ork, N Y 10027 . Researc h partially supp orted by NSF grants CCF-0728 733 and CCF-09156 81, and F u lbrigh t/Conicyt Chile S c h olarship. † garud@ieor .columbia.edu . Department of Ind ustrial Engineering & Op erations Research, Colum b ia Universit y , Mudd 314, 500W 120th St reet, New Y ork, NY 10027. Researc h partially sup p orted by NSF grant DMS-1016571, ONR gran t N000140 310514, and D OE gran t DE-F G02-08ER25856. ‡ cliff@ieor .columbia.edu . Department of Ind ustrial Engineering & Op erations Research, Colum b ia Universit y , Mudd 326,500W 120th Street, New Y ork, NY 10027. Researc h partially supp orted by NS F grants CCF-0728733 and CCF-09156 81. Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein 1 In tro du ction Managing energy consumption is a pr oblem of cr itical in terest throughout the w orld and throughout v arious indu stries. C omputing devices u se a large amoun t of energy , b oth in individual devices suc h as laptops and PD As and also in large industr ial u ses such as d atacen ters. F or example, Go ogle states that the serv ers in its d atacen ter, w h ic h are muc h more efficien t th an the a ve r age industry serv er, consume 1kJ p er query on a v erage [1]. In Jan uary 2011, just in the US, there were an av erage more than 400 million queries p er da y [2], and th us the tot al amount of energy consumed wa s 44 . 5 million kWh, equiv alen t to m ore than 4 , 000 a ve r age US households [3]. F urtherm ore, CPUs account for 50-60% of a t ypical computer’s energy consumption [4], making CP U en ergy managemen t v ery imp ortant . When sc hedu ling on su c h devices, it is imp ortan t not only to consider the relev an t qualit y of service (QoS) metrics suc h as mak espan or w eighted completion time, but also to tak e energy consu mption int o acco u n t. Most mo dern C PUs can b e run at m u ltiple sp eeds; the lo wer the sp eed, the less energy us ed , and the rela tionsh ip is device-dep endent, but t ypically sup erlinear. The tec hniqu e of scheduling while con trolling the sp eed of the p ro cessor is kn own as sp e e d sc aling. Starting with the work of Y ao, Demers, and Shenk er [25], there has by now b een tens of pap ers studying sc hedulin g p roblems in whic h energy consump tion is tak en in to accoun t. (See, for example, the surveys by Irani and Pruh s [19] and that by Alb ers [4]). There are thr ee main settings for energy a wa r e scheduling problem: optimizing a QoS metric with an energy budget [22, 23], minimizing energy sub ject to a QoS constrain t [7, 10, 11, 25], or optimizing some con ve x com bination of a sc hedu ling ob jectiv e and energy consumption [5, 6, 9, 12 ]. Und erlying the latter setting, whic h is in the one we will fo cus in this w ork, is an assump tion that b oth ener gy and time can b e (implicitly) con ve r ted in to a common un it, such as dollars. 1.1 Our r esults In this pap er w e consider t wo commonly stu died scheduling metrics, weighte d c ompletion time and weighte d tar diness , that ha ve n ot receiv ed atten tion in th e energy a ware s cheduling literature. Giv en a schedule in w hic h job i with weig ht w i , release time r i , and deadline d i is completed at time C i , the total weig hted completion time is P i w i C i . The tardin ess of a j ob is zero if it is completed b efore its deadline and otherwise equal to the amoun t b y which it misses, that is, T i = max { 0 , C i − d i } and total weig hted tardiness is P w i T i . F or b oth these metrics, w e consider the non-p reemptiv e, off-line p r oblem on one mac h in e, an d allo w arbitrary precedence constrain ts. F or th e we ighted completion time w e allo w arbitrary release dates as wel l. W e consider a metric that is a con vex combination of our sc hed u ling metric and en ergy cost. W e are not a ware of an y previous w ork on energy a ware sc hedu ling algorithms for these metrics. T here is a ric h literature on minimizing w eighte d completion time in th e absence of energy concerns (e.g. [20, 21, 24]), but w e are aw are of only one result ab out weig hted tardiness in the absence of energy concerns in the sp eed scaling/resource augmen tation literature [8], where a 2-mac hine, 24-sp eed 4-appro ximation algorithm is present ed. W eigh ted ta r diness, in p articular, is difficu lt to analyze b ecause, in contrast to most sc h ed uling ob j ectiv es, it is a non-linear fu nction of completion time. In our w ork we consider a more general mo del of energy cost than has previously b een used . The most common energy mo d el assumes that the rate at wh ic h pow er is consumed is a p olynomial function of sp eed of the form P ( s ) = s β for some constan t β ; t ypical v alues of β are 2 or 3. Some recen t w ork[6, 9] uses a m ore general p o wer function with minim u m regularit y conditions, lik e non- negativit y , b ut in all the cases the p o wer function d o es not d ep end on the job. F urthermore, most energy a ware algorithms assume cost functions that are closely r elated to energy consumption; ho wev er, in practice the actual energy cost is not simply a fu nction of energy consu m ption, it is a complicated fu nction of discounts, pricing, time of consumption, etc. W e consider a more 1 Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein general class of cost functions that are only restricted to b e non-negativ e and can b e different for differen t jobs. Beca u s e w e allo w job -d ep endent energy costs, our algorithms can b e used outside the CPU-energy setting, where energy cost generally are job indep end en t, and can b e applied to more general problems that ha v e additional sp eed-asso ciated costs. Examples of these costs are main tenance costs, w ear and tear of p arts, failure rates, etc. all of whic h not only dep end on the sp eed at wh ich the mac hin e r uns, but also the job b eing pro cessed. W e are not aw are of an y other work that allo w s suc h general cost s . F or the w eigh ted tardiness case we requir e an additional regularit y condition on the energy cost fun ctions that allo ws us to con trol its rate of gro wth . Our pap er con tains s ev eral results for d ifferen t sc h eduling problems, we state here the most general resu lts: Theorem 1.1. Given n jobs with pr e c e denc e c onstr aints and r ele ase dates and a gener al non- ne gative ener gy c ost function, ther e is an O (1) -appr oximation algorithm f or the pr oblem of non- pr e emptively minimizing a c onvex c ombination of weighte d c ompletion time and ener gy c ost. Theorem 1.2. Give n n jobs with pr e c e denc e c onstr aints and de ad lines and a ge ner al non-ne gative ener gy c ost function, ther e is an O (1) -appr oximation algorithm for the pr oblem of non-pr e emptively minimizing a c onvex c ombination of weighte d tar diness and ener gy c ost. The constants in th e O (1) are mo dest. Consider the case w h ere we are given a set of sp eeds S = { σ 1 , . . . , σ m } , at wh ic h the mac hine can run, with σ j ≤ (1 + δ ) σ j − 1 , and some ǫ > 0. T hen the algorithm for the w eigh ted completion time setting has a 4(1 + ǫ )(1 + δ )-appro ximation ratio when only precedence constraints exist, an d (3 + 2 √ 2)(1 + ǫ )(1 + δ )-appro ximation ratio w h en release dates are added. The algorithm for th e we ighted tard in ess setting has a 4 β (1 + ǫ ) β − 1 (1 + δ ) β − 1 - appro ximation ratio ev en with arbitrary pr ecedence constraints, wh ere β con trols the gro wth of the energy cost function. 1.2 Our Metho dology The p roblem of minimizing weig hted completion time in the com binatorial setting has b een we ll- studied. The work of P hillips, Stein, and W ein [20] and Hall, Sch ulz, Shm o ys, and W ein [17, 18] in tro d uced the idea of α -p oints , and these hav e b een used in muc h of the su bsequent w ork. The idea is that one fir st form ulates a time-indexed inte ger p rogram in w hic h decision v ariable x it is 1 if job i completes at time t , and then solv es its linear programming relaxation. F rom the solution to the relaxation, one computes the α -p oin t of eac h job, that is, the earliest time at whic h an α fraction of the job has completed in the relaxation. The exact in terpretation of when an α fraction completes dep en d s up on the particular pr oblem. On e u ses these α -p oin ts to infer an ord er on the jobs and then run s the jobs non-preemptive ly , resp ecting that order. T h ere are man y v ariants and extensions of these tec hniqu e includin g choosing α r andomly [13, 14] or choosing a d ifferent α for eac h j ob [15]. This tec hn ique has led to small constant factor ap p ro ximation algorithms for many w eight ed completion time scheduling problems [24]. The time-indexed int eger program (IP) formulatio n s for this pr ob lem are n ot typica lly of p oly- nomial size. Ho w ever, the interval-indexe d IP , introdu ced in [18], in whic h time is divid ed in to geometrica lly increasing interv als and jobs are assigned to interv als rather than in dividual time slots, is of p olynomial size. By usin g this linear pr ogram one obtains a p olynomial sized linear program from whic h it is still p ossib le to ap p ly the id eas of α -p oint s while suffering only a small additional d egradation of the app ro ximation ratio. In this pap er , we extend the interv al-indexed IP to handle sp eed s caling and then design new α -p oin t based r ou n ding algorithms to obtain the r esulting sc hedu les. In doing so we in tro d u ce the new concept of α -sp e e ds . W e assume, in Sections 2 , 3, and 4 , that w e ha ve a discrete set of m 2 Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein allo w able s p eeds S = { σ j } , and th at the rate of p o wer consumption is a p olynomial function of the sp eed. In Section 5.3 we describ e ho w to r emo v e these assump tions. Although the time-indexed IPs are easier to explain, du e to limited space, w e will describ e only the in terv al-indexed linear programs in this pap er. In our in terv al-indexed IP , a v ariable x ij t is 1 if job i run s at sp eed σ j and completes in in terv al t . W e can th en extend the standard in terv al-indexed in teger p rogramming form u lation to tak e the extra dimension of sp eed int o accoun t (see Section 2 for details). Once w e ha ve solv ed its linear program (LP) r elaxatio n , we need to no w determine b oth an α -p oint and α -sp e e d . The key insigh t is that by “summarizing” eac h dimension appr opriately , we are able to mak e the correct choic e for th e other dim en sion. A t a high level , we fi rst choose the α -p oint by “colla p sing” all pieces of a job that complete in the LP in in terv al t (these p ieces ha ve d ifferen t sp eeds), b eing esp ecially careful with the last inte r v al, where we m a y ha ve to c h o ose only some of the s p eeds. W e then u s e only the pieces of the job that complete b efore the α -p oin t to c ho ose th e sp eed, where the sp eed is c hosen by collapsing the time dimension and then in terpreting the result as a p robabilit y mass function (pmf ), where the probabilit y that the job is r un at sp eed σ j dep end s on the total amount of pro cessing done at that sp eed. W e then define the concept of α -sp eeds, whic h is related to the exp ected v alue und er this p mf, and run the job at this sp eed (see Section 3 for more details). W e com bine this new rounding metho d with extensions of the more traditional metho ds for dealing with p r ecedence constraints and r elease d ates to obtain our algorithms. F or w eighte d tardiness, w e emph asize again th at not m u c h is kno wn ab out approxima ting this problem, ev en in the absence of energy concerns. F or this problem, w e are able to use the same in terv al-indexed linear program, with the ob jectiv e function mo dified to tardiness. Because the linear p rogram is in terv al indexed, the non-linear ob jectiv e function is not a p roblem. After the solving the linear program, w e are able to sh o w that with only a constant fact or increase in energy (o v er the lo wer b oun d from the linear p rogram), w e obtain only a constant f actor (o v er the linear program) increase in tardin ess. Implicit in this analysis is the f act that jobs that receiv e 0 tardiness in the linear program will receiv e 0 tardin ess in o u r solution; in some sense the sp eed scaling mak es accomplishing this easier than in the com binatorial setting. W e n ote that our weigh ted tardiness algorithms d o es not wo r k in th e presence of release dates, as release d ates ma y stop us fr om b eing able to k eep jobs with 0 tardiness in the LP at 0 tardin ess in the s c hedule. Finally , in S ection 5, w e show how to extend our resu lts for the weig hted completion and w eight ed tardiness sc h eduling metrics to general energy cost f unctions. W e also show ho w to extend our r esults to the s etting where contin uous sp eeds are used and not ju st a discrete set S , while mainta in ing the same approxima tion ratio. 2 Problem F orm ulation 2.1 Problem Sett ing W e are give n n jobs, where job i h as a pr o cessing requirement of ρ i ∈ N + mac hine cycles, release time r i , and an asso ciated p ositiv e weig ht w i . Let s i denote the sp eed at w hic h job i r uns on the mac hine and C i denote its completion time. Let Π = { π (1) , . . . , π ( n ) } denote the order in wh ic h the job s are pro cessed, i.e. π ( k ) = i implies that job i is the k -th job to b e p r o cessed. Then C π ( i ) = max { r π ( i ) , C π ( i − 1) } + ρ π ( i ) s π ( i ) is the completion time of the i -th job to b e pro cessed, with C π (0) = 0. W e do not allo w pr eemption. Let S = { σ 1 , . . . , σ m } , b e the set of p ossible sp eeds at whic h the mac h ine can r un. W e will assume that σ j +1 ≤ (1 + δ ) σ j , for some δ > 0. This is a n atural assu mption b ecause actual sp eed scaling ac hieve d in CPUs is done via frequency m u ltipliers or d ivid ers. Although a discrete s et of sp eeds is probably the most common case for CPUs, in S ection 5.3 w e sho w that our algorithm has the s ame approxima tion r atio when a cont inuous set of sp eeds is used. 3 Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein Let E i ( s i ) denote the energy cost of runn ing job i at sp eed s i . F or sim p licit y w e initially consid er E i ( s i ) = v i ρ i s β − 1 i , w here β ≥ 2 and v i are kno wn constan ts. A s indicated earlier, an energy cost function of this form is the sta n dard model for these problems, although our mo del is m ore general b ecause the energy cost function is job-dep endent . In Section 5 we sho w that our algorithms also w ork for a m u c h larger class of job-dep endent energy cost functions. The ob jectiv e is to compute a feasible schedule (Π , C ), consisting of an order Π and completion times C , p ossibly sub ject to pr ecedence and /or release date constrain ts, and the v ector of job sp eeds s = { s 1 , . . . , s n } ∈ R n + that min imizes the total cost, f (Π , s ) = n X i =1 h v i ρ i s β − 1 i + w π ( i ) C π ( i ) i , (2.1) Since this fun ction is conv ex w e can assume, w.l.o.g., that eac h job r u ns at a constant sp eed. F or con ve n ience we w ill use an extended version of th e notation of Graham et al. [16] to r efer to the d ifferen t energy a wa r e sc h eduling problems, i.e. 1 | r i , pr ec | P E i ( s i ) + P w i C i , will refer to the problem setting with 1 mac hin e, with r i release dates, precedence constraint s , and the w eigh ted completion time as the sc h ed uling p erform an ce metric. S imilarly , the 1 | r i , pr ec | P E i ( s i ) + P w i T i will refer to the same setting, b ut with tard iness as the sc hedu ling p erformance metric. In all of them E i ( s i ) indicates that the energy cost is also added as a p erformance metric. 2.2 In terv al-Indexed F orm ulation W e no w mo dify and extend the inte r v al-indexed formulati on prop osed by Hall et al. [18] to accom- mo date s p eeds and energy cost. The int erv al-indexed formulatio n divides th e time horizon in to geometrical ly increasing inter- v als, and the co m pletion time of eac h job is assig n ed to one of th ese in terv als. S ince the completi on times are n ot asso ciated to a sp ecific time, th e completion times are not precisely kno w n b ut are lo w er b ounded. By controlli n g the growth of eac h in terv al one can obtain a sufficien tly tig ht b ound . The problem formulat ion is as f ollo w s. W e divide the time horizon into the follo wing geometri- cally increasing int erv als: [ κ, κ ], ( κ, (1 + ǫ ) κ ], ((1 + ǫ ) κ, (1 + ǫ ) 2 κ ], . . . , where ǫ > 0 is an arbitrary small constan t, and κ = ρ min σ max denotes the smallest in terv al size that will hold at least one whole job. W e defin e int erv al I t = ( τ t − 1 , τ t ], w ith τ 0 = κ and τ t = κ (1 + ǫ ) t − 1 . Th e interv al index ranges ov er { 1 , . . . , T } , w ith T = min {⌈ t ⌉ : κ (1 + ǫ ) t − 1 ≥ max n i =1 r i + P n i =1 ρ i σ 1 } ; and th us , w e h a v e a p olynomial n u m b er of indices t . Let x ij t =  1 , if job i r uns at a sp eed σ j and completes in the time in terv al I t = ( τ t − 1 , τ t ] 0 , otherwise . (2.2) By u sing the lo w er b ound s τ t − 1 of eac h time inte r v al I t , a lo w er b ound to (2.1) is wr itten as, min x n X i =1 m X j =1 T X t =1  v i ρ i σ β − 1 j + w i τ t − 1  x ij t . (2.3) The follo w in g are th e constrain ts r equired for the 1 | r i , pr ec | P E i ( s i ) + P w i C i problem: 1. Each job m ust fin ish in a unique time inte r v al and sp eed; ther efore for i = { 1 , . . . , n } : m X j =1 T X t =1 x ij t = 1 . (2.4) 4 Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein 2. S ince only one job can b e pro cessed at any give n time, the total pro cessing time of jobs up to time in terv al I t m us t b e at most τ t units. Thus, for t = { 1 , . . . , T } : n X i =1 m X j =1 t X u =1 ρ i σ j x ij u ≤ τ t . (2.5) 3. J ob i run ning at sp eed σ j requires ρ i σ j time units to b e pr o cessed, and considering that its release time is r i , then for i = { 1 , . . . , n } , j = { 1 , . . . , m } , and t = { 1 , . . . , T } : x ij t = 0 , if τ t < r i + ρ i σ j . (2.6) 4. F or i = { 1 , . . . , n } and t = { 1 , . . . , T } : x it ∈ { 0 , 1 } . (2.7) 5. T h e precedence constraint i 1 ≺ i 2 implies that job i 2 cannot finish in an interv al earlier th an i 1 . Th erefore for ev ery i 1 ≺ i 2 constrain t we ha v e th at for t = { 1 , . . . , T } : m X j =1 t X u =1 x i 1 j u ≥ m X j =1 t X u =1 x i 2 j u . (2.8) It is imp ortant to note that this in teger pr ogram only p r o vides a lo wer b ound for (2.1); in fact its optimal solution ma y not b e sc hedulable, since constraints (2 .5 ) do not imply that only one job can b e pro cessed at a single time, they only b ound the total amount of wo r k in ∪ t I t . 3 Appro ximation A lgorithm for W eigh ted Completion Time W e no w describ e the appr o ximation algorithm for the weigh ted completion time, called Schedule by α -inter v als and α -speeds (S AIAS) wh ich is displa yed in Figur e 3.1. Let ¯ x ij t denote the optimal s olution of the linear r elaxatio n of the in teger pr ogram (2.3)-(2.8), in wh ich we c hange constrain ts (2.7) for x ij t ≥ 0. In step 1 of th e algorithm we compute the optimal solution ¯ x and in step 2, giv en 0 ≤ α ≤ 1, w e compute the α -in terv al of job i , which is d efined as, τ α i = min    τ : m X j =1 τ X u =1 ¯ x ij u ≥ α    . (3.1) Schedul e by α -inter v a ls and α -speed s ( SAIAS) Inputs: set of jobs, α ∈ (0 , 1), ǫ > 0, set of sp eeds S = { σ 1 , . . . , σ m } . 1 Compute an optimal solution ¯ x to the linear relaxation (2.3)-(2.8). 2 Compute the α -in terv als τ α and the sets J t . 3 Compute an order Π α that h as the sets J t ordered in non-decreasing v alues of t and the jobs with in eac h set in a manner consistent with the precedence constrain ts. 4 Compute the α -sp eeds s α 5 Round down eac h s α i to the nearest sp eed in S and run job i at this roun d ed sp eed, ¯ s α i . 6 Set the i -th job to start at time m ax { r π ( i ) , ¯ C α π ( i − 1) } , wh ere ¯ C α π ( i − 1) is the completion time of the p revious job using the round ed α -sp eeds, and ¯ C α π (0) = 0. 7 return sp eeds ¯ s α and schedule (Π α , ¯ C α ). Figure 3.1: Schedule b y α -in terv als and α -sp eeds 5 Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein Since sev eral jobs ma y fin ish in th e same interv al, let J t denote the set of j obs that fi nish in in terv al I t , J t = { i : τ α i = t } , and we use these sets to determine the order Π α as d escrib ed in step 3. Next, in step 4, w e compute the α -sp eeds as follo ws. Since P m j =1 P τ α i u =1 ¯ x ij u ≥ α , we d efine auxiliary v ariable { ˜ x ij t } as: ˜ x ij t =      ¯ x ij t , t < τ α i max n min n ¯ x ij τ α i , α − P j − 1 l =1 ¯ x ilτ α i − β i o , 0 o , t = τ α i 0 , t > τ α i , (3.2) where β i = P m j =1 P τ α i − 1 u =1 ¯ x ij u < α . Note that with th is auxiliary v ariable P m j =1 P τ α i u =1 ˜ x ij u = α . T his is a k ey step that allo ws us to tru ncate the fractional solution so that for ev ery job i , the sum of ˜ x ij t up to time in terv al τ α i for eac h sp eed j can b e interpreted as a probabilit y mass function. W e define this pr obabilit y mass fu nction (pmf ) µ i = ( µ i 1 , . . . , µ im ) on the set of sp eeds S = { σ 1 , . . . , σ m } as µ ij = 1 α τ α i X u =1 ˜ x ij u . (3.3) Let ˆ s i define a random v ariable d istributed according to the pmf µ i , i.e. µ ij = P ( ˆ s i = σ j ). Then, th e α -sp eed of j ob i , s α i , is defined as follo ws: 1 s α i = E  1 ˆ s i  = m X j =1 µ ij σ j ⇒ s α i = 1 E h 1 ˆ s i i . (3.4) W e d efi ne the α -sp eeds using th e r ecipro cal of the s p eeds since the completion times are pro- p ortional to the recipro cals instead of the sp eeds, and we need to b ound completion times in the analysis of the algorithm. Next, in step 5, b ecause the α -sp eeds s α i do not necessarily b elong to the set of p ossible sp eeds S w e round them do wn to ¯ s α i , which is the nearest sp eed in the set suc h that ¯ s α i ≤ s α i . T h e follo w ing lemma b ounds the error in tro d uced b y this roundin g. Lemma 3.1. The c ost of the solution with the r ounde d down sp e e ds ¯ s α is at most (1 + δ ) times the c ost of the solution using the α -sp e e ds s α . Pr o of. T h e energy cost function E i ( s i ) is increasing so rounding do w n do es not in crease the energy cost, b ut the completion time is now larger. Let C α i b e the completion time of job i wh en the sp eeds s α are used and ¯ C α i when the rounded ones ¯ s α are used. Since the sp eeds are reduced at most by (1 + δ ), then (1 + δ ) ¯ s i α ≥ s α i , and we ha ve that, ¯ C α i = max { r i , ¯ C α i − 1 } + ρ ¯ s α i ≤ (1 + δ )  max { r i , C α i − 1 } + ρ s α i  = (1 + δ ) C α i , (3.5) whic h implies that P n i =1 w i ¯ C α i ≤ (1 + δ ) P n i =1 w i C α i and p ro ve s the lemma. Finally , in s teps 6 and 7 we compu te the completion times giv en the calculated sp eeds and return the set of sp eeds ¯ s α and the sc hedu le (Π α , ¯ C α ). W e no w analyse this algorithm’s p erformance for d ifferen t energy a ware sc h eduling problems. In th e follo wing sub sections w e will assume w .l.o.g. that τ α 1 ≤ τ α 2 ≤ . . . τ α n . 6 Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein 3.1 Single Mac hine Problem with Precedence Constrain ts W e firs t n eed to pro ve that the outpu t of the SAIAS algorithm is ind eed feasible. Lemma 3.2. If i 1 ≺ i 2 , then c onstr aint (2.8) implies that τ α i 1 ≤ τ α i 2 . Pr o of. Ev aluating the LP constrain t (2.8) corresp onding to i 1 ≺ i 2 , for t = τ α i 2 , we ha ve that, m X j =1 τ α i 2 X u =1 x i 1 j u ≥ m X j =1 τ α i 2 X u =1 x i 2 j u ≥ α, where the last inequalit y follo ws fr om the d efi nition of τ α i 2 . The c hain of in equ alities im p lies that P m j =1 P τ α i 2 u =1 x i 1 j u ≥ α , s o τ α i 1 ≤ τ α i 2 . Since the SAIAS algorithm sc hedu les jobs b y first ord ering the s ets J t in increasing order of t , and then orders the jobs within eac h set in a wa y that is consisten t with the precedence constraints, b y Lemma 3.2 it follo ws that the SAIAS algorithm p reserv es the p recedence constrain ts, and, therefore, the ou tp ut of the algorithm is feasible. Next, w e can pro ve the follo wing result. Theorem 3.1. The SA IAS algorithm with α = 1 2 is a 4(1 + ǫ )(1 + δ ) -appr oximation algorithm for the 1 | pr ec | P E i ( s i ) + P w i C i pr oblem, with E i ( s i ) = v i ρ i s β − 1 i . Pr o of. Let x ∗ ij t denote an optimal solution to the integ er problem (2.3)-(2.8), ¯ x ij t the f r actional solution of its linear relaxatio n , and ˜ x ij u the auxiliary v ariables calculated for the SAIAS algorithm. Since in (2.3) the completion time for jobs completed in in terv al I t is τ t − 1 , it follo ws that, n X i =1 m X j =1 T X t =1  v i ρ i σ β − 1 j + w i τ t − 1  ¯ x ij t ≤ n X i =1 m X j =1 T X t =1 v i ρ i σ β − 1 j x ∗ ij t + n X i =1 w i C ∗ i . (3.6) The energy terms of the algorithm’s solution are b ound ed as follo ws, v i ρ i ( s α i ) β − 1 = v i ρ i  1 s α i  − ( β − 1) = v i ρ i  E  1 ˆ s i  − ( β − 1) ≤ v i ρ i E "  1 ˆ s i  − ( β − 1) # = v i ρ i E h ˆ s β − 1 i i = v i ρ i m X j =1 µ ij σ β − 1 j , (3.7) where the inequalit y follo ws from J ensen’s I nequalit y app lied to the conv ex function 1 s β − 1 . Using the d efinition of µ ij in (3.3) and giv en that 0 ≤ α ≤ 1, ǫ > 0, and ˜ x ij t ≤ ¯ x ij t , it follo ws that, v i ρ i ( s α i ) β − 1 ≤ v i ρ i α m X j =1 τ α i X u =1 σ β − 1 j ˜ x ij u ≤ (1 + ǫ ) α (1 − α ) v i ρ i m X j =1 T X u =1 σ β − 1 j ¯ x ij u . (3.8) Since there are no release date constrain ts there is no id le time b etw een jobs, C α i = i X j =1 ρ j s α j = i X j =1 ρ j E  1 ˆ s j  = 1 α i X j =1 m X l =1 τ α j X u =1 ρ j σ l ˜ x j lu ≤ 1 α n X j =1 m X l =1 τ α i X u =1 ρ j σ l ¯ x j lu , (3.9) and fr om constraint (2.5) for t = τ α i w e get, C α i ≤ 1 α τ τ α i . 7 Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein Let ¯ C i = P m j =1 P T t =1 τ t − 1 ¯ x ij t denote the optimal fractional completion time give n b y the optimal solution of the r elaxed linear pr ogram (2.3)-(2.6). S in ce it is p ossible that P m j =1 P τ α i t =1 ¯ x ij t > α ; we define X (1) i = α − P m j =1 P τ α i − 1 t =1 ¯ x ij t and X (2) i = P m j =1 P τ α i t =1 ¯ x ij t − α , th us X (1) i + X (2) i = P m j =1 ¯ x ij τ α i , and we can r ewr ite ¯ C i = m X j =1 τ α i − 1 X t =1 τ t − 1 ¯ x ij t + τ τ α i − 1 X (1) i + τ τ α i − 1 X (2) i + m X j =1 T X t = τ α i +1 τ t − 1 ¯ x ij t , (3.10) and eliminating the lo w er terms of the p revious sum w e get that, ¯ C i ≥ τ τ α i − 1 X (2) i + m X j =1 T X t = τ α i +1 τ t − 1 ¯ x ij t ≥ τ τ α i − 1 X (2) i + m X j =1 T X t = τ α i +1 τ τ α i − 1 ¯ x ij t = τ τ α i − 1 (1 − α ) . (3.11) Because τ τ α i = (1 + ǫ ) τ τ α i − 1 , from (3.9) and (3.11) w e get that C α i ≤ (1+ ǫ ) α (1 − α ) ¯ C i ⇒ P n i =1 w i C α i ≤ (1+ ǫ ) α (1 − α ) P n i =1 w i ¯ C i . F rom this, (3.6) and (3.8) it follo ws that, n X i =1 v i ρ i ( s α i ) β − 1 + n X i =1 w i C α i ≤ (1 + ǫ ) α (1 − α )   n X i =1 m X j =1 T X t =1 v i ρ i σ β − 1 j x ∗ ij t + n X i =1 w i C ∗ i   , (3.12) and we set α = arg min 0 ≤ α ≤ 1 n 1 α (1 − α ) o = 1 2 , to minimize the b ou n d. By Lemma 3. 1 , w hic h b ounds the fi nal roun ding error, we get the d esired appr o ximation ratio. 3.2 Single Mac hine Problem with Precedence and Release Date Constraints W e n ow analyse the case with pr ecedence constraint s and release dates. Release dates mak es the problem s omewh at hard er since they can in tro d uce idle times b et ween jobs. Theorem 3.2. The SAIAS algorithm with α = √ 2 − 1 is a (3 + 2 √ 2)(1 + ǫ )(1 + δ ) -appr oximation algorithm for the 1 | r i , pr ec | P E i ( s i ) + P w i C i pr oblem, with E i ( s i ) = v i ρ i s β − 1 i . Pr o of. T h e b oun d for the energy terms computed in equation (3 .7 ) are still v alid w hen there is idle time b et wee n j obs, we ha ve that, v i ρ i ( s α i ) β − 1 ≤ (1 + ǫ ) α (1 − α ) v i ρ i m X j =1 T X u =1 σ β − 1 j ¯ x ij u ≤ (1 + ǫ )(1 + α ) α (1 − α ) v i ρ i m X j =1 T X u =1 σ β − 1 j ¯ x ij u . (3.13) When b ound ing th e completion time C α i , giv en the sorting done in step 3 of the S AIAS algo- rithm, n o w one has to consider all the jobs up to the ones in set J τ α i , and th us , C α i ≤ max j ∈{ J 1 ,...,J τ α i } r j + X j ∈{ J 1 ,...,J τ α i } ρ j s α j . (3.14) Since all jobs that ha ve b een at least partially pr o cessed up to time interv al I t need to b e released b efore τ t , it follo ws that m ax j ∈{ J 1 ,...,J τ α i } r j ≤ τ τ α i . On the other hand, w e also h a ve that, X j ∈{ J 1 ,...,J τ α i } ρ j s α j = 1 α X j ∈{ J 1 ,...,J τ α i } m X l =1 τ α j X u =1 ρ j σ l ˜ x j lu ≤ 1 α n X j =1 m X l =1 τ α i X u =1 ρ j σ l ¯ x j lu ≤ 1 α τ τ α i , (3.15) 8 Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein where the last inequalit y follo ws from constraint (2.5) with t = τ α i . Thus, C α i ≤ (1+ α ) α τ τ α i . Since ¯ C i = P m j =1 P T t =1 τ t − 1 ¯ x ij t , (3.11) is still v alid and b ecause τ τ α i = (1 + ǫ ) τ τ α i − 1 , we get, C α i ≤ (1 + ǫ )(1 + α ) α (1 − α ) ¯ C i ⇒ n X i =1 w i C α i ≤ (1 + ǫ )(1 + α ) α (1 − α ) n X i =1 w i ¯ C i . (3.16) Finally , from (3.13) and (3.16) it follo ws that, n X i =1 v i ρ i ( s α i ) β − 1 + n X i =1 w i C α i ≤ (1 + ǫ )(1 + α ) α (1 − α )   n X i =1 m X j =1 T X t =1 v i ρ i σ β − 1 j x ∗ ij t + n X i =1 w i C ∗ i   , (3.17) and by setting α = arg min 0 ≤ α ≤ 1 n (1+ α ) α (1 − α ) o = √ 2 − 1, and again using Lemma 3.1 to b ound th e sp eed-round ing error, we get th e requir ed appro xim ation r atio. If no precedence constrain ts and relea se dates exist, there are tw o v er s ions of this problem that can b e optimally solv ed in p olynomial time: when all w eights w i are equal, and when all jobs are of the same size (i.e. ρ i = ρ , ∀ i ) and all j ob s ha ve the same energy cost function. F or these cases w e ha ve the follo wing result: Theorem 3.3. If w i = w , ∀ i or ρ i v 1 β i = ξ , ∀ i then the or der Π is optimal if w π ( i ) ρ π ( i ) v 1 β π ( i ) ≥ w π ( i +1) ρ π ( i +1) v 1 β π ( i +1) , ∀ i ∈ { 1 , . . . , n − 1 } . Pr o of. F or simplicit y w e will d efi ne ξ i ≡ ρ i v 1 k i , q = k − 1 k , and K ≡ k ( k − 1) k − 1 k . First, du al form ulation of p roblem (2.1) w ith no precedence or release date constrain ts is giv en b y , min π F ( π ) = min π n X i =1 K ξ π ( i )   n X j = i w π ( j )   q . (3.18) W e now pro ve b oth cases by con tradiction usin g the d ual formulatio n . When w i = w , ∀ i , Th eorem 3.3 implies that in the optimal order ξ π ( i +1) ≥ ξ π ( i ) . By con tradic- tion, let π b e an optimal order suc h th at for some index k , ξ π ( k + 1) < ξ π ( k ) . F or this order the tota l cost is F ( π ) = n X i =1 K ξ π ( i )   n X j = i w   q = n X i =1 K ξ π ( i ) (( n − i + 1) w ) q , = K w q    ξ π ( k ) ( n − k + 1) q + ξ π ( k + 1) ( n − k ) q + n X i =1; i 6 = k ,k +1 ξ π ( i ) ( n − i + 1) q    . Let π k define the order where we switc h jobs k and k + 1 from order π , i.e. π k ( k ) = π ( k + 1) and π k ( k + 1) = π ( k ). Giv en this order w e ha ve that F ( π ) − F ( π k ) = K w q  ξ π ( k ) ( n − k + 1) q + ξ π ( k + 1) ( n − k ) q − ξ π ( k + 1) ( n − k + 1) q − ξ π ( k ) ( n − k ) q  , = K w q  ( n − k + 1) q ( ξ π ( k ) − ξ π ( k + 1) ) − ( n − k ) q ( ξ π ( k ) − ξ π ( k + 1) )  , = K w q  ξ π ( k ) − ξ π ( k + 1)  (( n − k + 1) q − ( n − k ) q )  . 9 Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein By our initial assumption the first term is p ositiv e (since ξ π ( k + 1) < ξ π ( k ) ) and the second one is alw a ys p ositiv e, hence F ( π ) − F ( π k ) > 0 wh ic h is a contradict ion, s ince that implies that π k has a smaller cost. F or the case wh en ξ i = ξ , ∀ i , Th eorem 3.3 implies that an order π is optimal then w π ( i ) ≥ w π ( i +1) . Let π b e an optimal order suc h that for some ind ex k , w π ( k ) < w π ( k + 1) . The total cost for this solution is F ( π ) = n X i =1 K ξ   n X j = i w π ( i )   q = K ξ    k X i =1   n X j = i w π ( i )   q +   n X j = k +1 w π ( i )   q + n X i = k +2   n X j = i w π ( i )   q    , = K ξ    k X i =1   n X j = i w π ( i )   q +   w π ( k + 1) + n X j = k +2 w π ( i )   q + n X i = k +2   n X j = i w π ( i )   q    . Let π k define the ord er where we switc h jobs k and k + 1 fr om order π . Give n this new order w e ha ve F ( π ) − F ( π k ) = K ξ      w π ( k + 1) + n X j = k +2 w π ( i )   q −   w π ( k ) + n X j = k +2 w π ( i )   q    > 0 , since w π ( k + 1) > w π ( k ) b y our initial assu mption, whic h is a con tradiction sin ce this r esult implies that ord er π k has a lo wer cost. 4 Extension to the W eigh ted T ardiness P roblem In this section we extend our results to the weigh ted tard in ess setting. W e still allo w f or arbitrary precedence constr aints b ut n o release dates. In th is case, eac h job i also h as a deadline d i . The tardiness T i of job i is defin ed as T i = max { 0 , C i − d i } , and the ob j ectiv e fu nction is no w given b y , g (Π , s ) = n X i =1 v i ρ i s β − 1 i + n X i =1 w π ( i )  C π ( i ) − d π ( i )  + . (4.1) W e no w formulate th e problem using a mo difi cation of the in terv al-and-sp eed-indexed formula- tion present ed in S ection 2. Because th e completion time can b e b oun ded by P m j =1 P T t =1 τ t − 1 x ij t , w e can b ound (4.1) from b elo w b y the follo wing optimization problem, min x n X i =1 m X j =1 T X t =1  v i ρ i σ β − 1 j + w i ( τ t − 1 − d i ) +  x ij t , (4.2) together with constr aints (2.4)-(2.8) f rom the inte r v al-indexed form ulation. Note that although the ob jectiv e (4.1) is non-linear, b ecause we ha ve a interv al-indexed formulatio n , (4.2 ) is linear. W e appro ximately solve (4.1) using the Sch edule by α -inter v als and α -speeds for T ardi- ness (SAIAS-T ) Algorithm displa yed in Figure 4.1. The main difference with the SAIAS algorithm, is th at in step 4 we scale up the α -sp eeds. This scaling mak es the completion time of th e relaxed LP comparable to the completion time of th e algorithm’s output, and th us job s that ha ve 0 tard in ess in the LP also hav e 0 tardiness in our algorithm. If w e roun ded sp eeds do wn, jobs with 0 tardin ess in the LP could, at a lo wer sp eed, miss their deadline, and thus the appro ximation ratio could b e arbitrary large. W e now analyse the algorithm assum ing w.l.o.g. that τ α 1 ≤ τ α 2 ≤ . . . ≤ τ α n . Sin ce Lemma 3.2 remains v alid, arguments identic al to those in Section 2 show th at the output of th e SAIAS- T algorithm is feasible; thus, w e ha ve the follo wing theorem: 10 Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein Schedul e by α -inter v a ls and α -speed s for T ar diness (SAIAS-T) Inputs: set of jobs, α ∈ (0 , 1), ǫ > 0, γ > 1, set of sp eeds S = { σ 1 , . . . , σ m } . 1 Compute an optimal solution ¯ x to the linear relaxation (4.2), (2.4)-(2 .8 ). 2 Compute the α -in terv als τ α and the sets J t as in the S AIAS algorithm. 3 Compute an order Π α that h as the sets J t ordered in non-decreasing v alues of t and the jobs with in eac h set in a manner consistent with the precedence constrain ts. 4 Compute the α -sp eeds s α and scale eac h s α i to ˜ s α i = γ s α i . 5 Round up eac h ˜ s α i to the next s p eed in S , ¯ s α i and r un eac h job i at this n ew sp eed. 6 Set the i -th job to start at time m ax { r π ( i ) , ¯ C α π ( i − 1) } , wh ere ¯ C α π ( i − 1) is the completion time of the p revious job using the round ed α -sp eeds, and ¯ C α π (0) = 0. 7 return sp eeds ¯ s α and schedule (Π α , ¯ C α ). Figure 4.1: Schedule b y α -in terv als and α -sp eeds for T ardiness Algorithm Theorem 4.1. The SA IAS-T algorithm with γ = (1+ ǫ ) α (1 − α ) and α = 1 2 is a 4 β (1 + ǫ ) β − 1 (1 + δ ) β − 1 - appr oximation algorithm f or the 1 | pr ec | P E i ( s i ) + P w i T i pr oblem, with E i ( s i ) = v i ρ i s β − 1 i . Pr o of. Let ¯ C i = P m j =1 P T t =1 τ t − 1 ¯ x ij t denote the optimal fractional completion time of the relaxed linear program. ( ¯ C i − d i ) + is a lo w er b oun d for the optimal tardiness ( C ∗ i − d i ) + , since P j t ( τ t − 1 − d i ) + ¯ x ij t ≥ ( ¯ C i − d i ) + . Thus, n X i =1 m X j =1 T X t =1 v i ρ i σ β − 1 j ¯ x ij t + n X i =1 w i  ¯ C i − d i  + ≤ n X i =1 m X j =1 T X t =1 v i ρ i σ β − 1 j x ∗ ij t + n X i =1 w i ( C ∗ i − d i ) + . (4.3) Let ˜ C α i denote the completion time of job i using sp eeds ˜ s α and C α i the one using sp eeds s α . Because th ere are no release date constr aints, th er e is no idle time in b etw een jobs ; therefore, ˜ C α i = i X j =1 ρ j ˜ s α j = 1 γ i X j =1 ρ j s α j = 1 γ C α i . (4.4) Since (3.9) remains v alid, it follo ws that C α i ≤ (1+ ǫ ) α (1 − α ) ¯ C i ⇒ ˜ C α i ≤ 1 γ (1+ ǫ ) α (1 − α ) ¯ C i . Th e key step is that by setting γ = (1+ ǫ ) α (1 − α ) , which mak es the t wo completion times comparable, we ha ve that, n X i =1 w i  ˜ C α i − d i  + ≤ n X i =1 w i  1 γ (1 + ǫ ) α (1 − α ) ¯ C i − d i  + = n X i =1 w i  ¯ C i − d i  + . (4.5) The energy term is b ound ed in a m an n er analogous to (3.8): v i ρ i ( ˜ s α i ) β − 1 = γ β − 1 v i ρ i ( s α i ) β − 1 ≤ (1 + ǫ ) β − 1 ( α (1 − α )) β v i ρ i m X j =1 T X t =1 σ β − 1 j ¯ x ij t , (4.6) where the last inequalit y follo ws fr om (3.8) th at remains v alid. F rom (4.3), (4.6 ), and (4.5) it follo w s that, n X i =1 v i ρ i ( ˜ s α i ) β − 1 + n X i =1 w i  ˜ C α i − d i  + ≤ (1 + ǫ ) β − 1 ( α (1 − α )) β   n X i =1 m X j =1 T X t =1 v i ρ i σ β − 1 j x ∗ ij t + n X i =1 w i ( C ∗ i − d i ) +   . 11 Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein Because sp eeds are rounded up, the completion ti m es, and th u s the tardiness can only impro v e, whereas the energy co st increases. S in ce at most we sp eed u p eac h job b y a factor (1 + δ ), w e hav e that, E i ( ¯ s α i ) ≤ E i ((1 + δ ) s α i ) = (1 + δ ) β − 1 E i ( s α i ) ⇒ n X i =1 E i ( ¯ s α i ) ≤ (1 + δ ) β − 1 n X i =1 E i ( s α i ) . (4.7) The app ro ximation ratio follo ws from setting α = arg min 0 ≤ α ≤ 1 n 1 ( α (1 − α )) β o = 1 2 . Clearly we could use (1+ ǫ ) β − 1 α β (1 − α ) β − 1 in (4.6) to compu te a tigh ter b oun d, bu t th e resulting expr ession is not as simple. W e are n ot able to extend this algorithm for the 1 | r i | P E i ( s i ) + P w i T i problem, sin ce it is based on sp eed scaling to mak e sur e that jobs are fi nished within a d esired time in terv al. When release d ates are presen t, we do not see h o w to arbitrarily redu ce the completion times. 5 Extension to General Energy Cost F unctions In this section we consider the extension to general energy c ost functions, as opp osed to simply energy c onsumption . W e b egin b y considering discrete sp eeds, as in th e previous sections, bu t in Section 5.3 we will relax this requ iremen t. Managers of d ata cen tres are clearly in terested in the energy cost metric, since they need to balance the p enalt y for violating the service lev el agreemen ts with the cost of energy . T he energy price cur ves for industrial consum er s are often qu ite complicated b ecause of energy contrac ts, discoun ts, real time pricing etc.; therefore it is very imp ortant to consid er general cost fu nctions in the sc hedu ling mo d el. Hence, in this section we use E i ( s i ) as the general en er gy cost f unction of run ning job i at sp eed s i . W e will require th at E i ( s i ) is n on-negativ e, just as in [6, 9], bu t n o other requiremen ts are needed for th e weigh ted completio n time setting. F or the wei ghted tardiness setting we will r equ ire an additional regularit y cond ition that b ounds the gro w th of the en er gy cost function. Since in p ractice the p ro cessor sp eed can b e d y n amically c hanged during the course of a job, one can replace the general cost fu nction by its lo wer conv ex en velo p e. Hence, without loss of generalit y , we can assume that E i ( s i ) is conv ex. F urthermore, since the mac h ine can only r un at the sp eeds in S , we can also consider that E i ( s ) is linear in b et ween these sp eeds. Hence, for eve r y s ∈ [ σ j , σ j +1 ] suc h that s = λσ j + (1 − λ ) σ j +1 , with λ ∈ [0 , 1], then E i ( s ) = λ E i ( σ j ) + (1 − λ ) E i ( σ j +1 ). Note that for b ound ing the energy cost terms in the w eigh ted completion time setting, we only used the fact that the energy consumption function E i ( s ) = v i ρ i s β − 1 is conv ex. Thus, the previous b ound s extend to our more general class of functions E i ( s ). I n the wei ghted tardiness case we required also a b ound on th e growth of the energy cost fun ction, which we will address in Section 5.2. 5.1 W eigh ted Completion Time Problem wit h General E nergy Cost The ob jectiv e function (2.3 ) is extended as follo ws, min x n X i =1 m X j =1 T X t =1 ( E i ( σ j ) + w i τ t − 1 ) x ij t , (5.1) where E i ( σ j ) are just co efficien ts. Give n that we only c hange the energy cost related terms, all th e completion time related b ound s computed previously are s till v alid. 12 Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein The only mo dification requ ir ed is in the rounding pro cedure at the end of the SAIAS algorithm, where it was done by r ounding d o wn the α -sp eeds. No w instead we will round them up or do wn suc h that E i ( ¯ s α i ) ≤ E i ( s α i ), whic h is alwa ys p ossible since E i ( s i ) is linear in b et ween the sp eeds in S . With this c hange Lemma 3.1 remains v alid and w e can extend the algo r ith m to our general energy cost fun ctions. Theorem 5.1. The SA IAS algorithm with α = 1 2 is a 4(1 + ǫ )(1 + δ ) -appr oximation algorithm for the 1 | pr ec | P E i ( s i ) + P w i C i pr oblem, for al l gener al non-ne gative ener gy c ost functions E i ( s ) . Pr o of. Because E i ( σ ), i = { 1 , . . . , n } are conv ex functions, (3.7) remains v alid since E i ( s α i ) = E i ( E [ ˆ s i ]) ≤ E [ E i ( ˆ s i )] = P m j =1 µ ij E i ( σ j ), and th u s, from the definition of µ ij , and from 0 ≤ α ≤ 1, ǫ > 0, and ˜ x ij t ≤ ¯ x ij t , n X i =1 E i ( s α i ) ≤ 1 α n X i =1 m X j =1 τ α i X t =1 E i ( σ j ) ˜ x ij t ≤ (1 + ǫ ) α (1 − α ) n X i =1 m X j =1 T X t =1 E i ( σ j ) ¯ x ij t . (5.2) The pr o of follo ws since the b oun ds for th e completion time in Th eorem 3.1 remain v alid, as w ell as Lemma 3.1. By th e same argum en t w e also ha ve that, Theorem 5.2. The SAIAS algorithm with α = √ 2 − 1 is a (3 + 2 √ 2)(1 + ǫ )(1 + δ ) -appr oximation algorithm for the 1 | r i , pr ec | P E i ( s i ) + P w i C i pr oblem, for al l gener al non-ne gative ener gy c ost functions E i ( s ) . 5.2 W eigh ted T ardiness Problem with General Energy Cost W e rep lace the energy term in (4.2 ) with the general en er gy cost term to obtain th e n ew ob jectiv e min x n X i =1 m X j =1 T X t =1  E i ( σ j ) + w i ( τ t − 1 − d i ) +  x ij t . (5.3) Since the SAIAS-T algorithm sp eeds u p th e jobs, we need to add th e follo wing regularit y condition f or the energy cost functions E i ( σ ) in order to obtain p erformance b ounds: Assumption 5.1. ∃ β ∈ N + , such that E i ( γ σ i ) ≤ γ β − 1 E i ( σ i ) , ∀ γ ≥ 1 . Theorem 5.3. The SAIAS-T algorithm with γ = (1+ ǫ ) α (1 − α ) and α = 1 2 , is a 4 β (1 + ǫ ) β − 1 (1 + δ ) β − 1 - appr oximation algorithm for the 1 | prec | P E i ( s i ) + P w i T i pr oblem, for al l non-ne gative ener gy c ost functions E i ( s ) that satisfy Assumption 5.1. Pr o of. As b efore, all the completio n time related b ound s (4.4) and (4.5) r emain v alid, so only a b ound analogous to (4.6) is needed. F rom Assumption 5.1 it follo ws that, E i ( ˜ s α i ) ≤ γ β − 1 E i ( s α i ) ≤ (1 + ǫ ) β − 1 α β (1 − α ) β m X j =1 T X t =1 E i ( σ j ) ¯ x ij t . (5.4) Th u s, from (4.5) it follo ws that, n X i =1 E i ( ˜ s α i ) + n X i =1 w i  ˜ C α i − d i  + ≤ (1 + ǫ ) β − 1 α β (1 − α ) β   n X i =1 m X j =1 T X t =1 E i ( σ j ) x ∗ ij t + n X i =1 w i ( C ∗ i − d i ) +   . Since we are rounding sp eeds up, equation (4.7) remains v alid and th us taking α = 1 2 completes the p ro of. 13 Energy Aware Sc h eduling v.arXiv 4.6 Carrasco, Iyengar, Stein 5.3 Con tinu ous Sp eeds As commen ted previously , our algorithms are also ap p licable for the case when a contin uous set of s p eeds is p ossible. In this case we mo d ify the SAIAS and SAIAS-T algorithms, eliminating the rounding step requir ed at the end of eac h algorithm. When the op erating r an ge of the mac h ine is giv en, i.e. the sp eed limits σ min and σ max , s in ce our IP requires a sp eed ind ex, we need to quantiz e the set [ σ min , σ max ] in m differen t sp eeds. W e can do this by setting σ 1 = σ min , and as b efore we define sp eed σ j = (1 + δ ) σ j − 1 , for some δ > 0, making su re that σ m ≥ σ max in order to co ver th e whole op erating range. Just by roundin g as describ ed in Section 5.1 for the w eigh ted completion time setting and roun d ing up f or th e w eigh ted tardiness setting we can pr o v e the follo wing lemma: Lemma 5.1. The opt i mal solution for the IP (2.3)- (2.8) is at most ( 1 + δ ) times t he optima l solution of the ener gy awar e pr oblem in the weighte d c ompletion time and c ontinuous sp e e d setting, and the optimal sol u tion for the IP (4.2), (2.4)- (2.8) is at most (1 + δ ) β − 1 times the optimal so lution of the ener gy awar e pr oblem in the weig hte d tar diness and c ontinuous sp e e d setting. The pro of is similar to Lemma 3.1 for the w eigh ted completion time and similar to equation (4.7) for the w eigh ted tardiness setting. Since there is n o additional round ing at the end of the algorithm, using Lemma 5.1 w e get th e same app ro ximation ratios as in Theorems 5.1, 5.2, and 5.3. When the op er ating range of the mac hine is not giv en, and we are in terested in determinin g a set S that co ve r s the optimal sp eeds from the con tinuous case, we n eed th e follo wing add itional regularit y condition on the energy cost functions: ∃ ξ < ∞ suc h that E j ( s i ) is incr easing ∀ s i ≥ ξ . It is easy to p r o ve that th is is a necessary and sufficien t conditions for the problem to b e we ll defined , and thus we can compute σ min and σ max suc h that the optimal sp eeds s ∗ i ∈ [ σ min , σ max ], for all i . Then w e can apply th e same pro cedu r e as b efore to quantize and build the set of sp eeds, and pro ceed to compute an appr o ximate solution. 6 Conclusion In this work w e describ ed new tec h niques for deve lopin g constant app ro ximation algorithms for energy aw are sc h eduling problems with v ery general job-dep endent energy cost functions, that w ork on b oth discr ete and cont inuous sp eed s ets. 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