Reclaiming the energy of a schedule: models and algorithms

Transcription

1 CONCURRENCY AN COMPUTATION: PRACTICE AN EXPERIENCE Concurrency Computat.: Pract. Exper. 0; 5:505 5 Publshed onlne 5 July 0 n Wley Onlne Lbrary (wleyonlnelbrary.com)..889 Reclamng the energy of a schedule: models and algorthms Gullaume Aupy, Anne Benot,, *,, Fanny ufossé and Yves Robert, ENS Lyon, Unversté de Lyon, LIP laboratory, UMR 5668, ENS Lyon CNRS INRIA UCBL, Lyon, France Insttut Unverstare de France, Pars, France SUMMARY We consder a task graph to be executed on a set of processors. We assume that the mappng s gven, say by an ordered lst of tasks to execute on each processor, and we am at optmzng the energy consumpton whle enforcng a prescrbed bound on the executon tme. Although t s not possble to change the allocaton of a task, t s possble to change ts speed. Rather than usng a local approach such as backfllng, we consder the problem as a whole and study the mpact of several speed varaton models on ts complexty. For contnuous speeds, we gve a closed-form formula for trees and seres parallel graphs, and we cast the problem nto a geometrc programmng problem for general drected acyclc graphs. We show that the classcal dynamc voltage and frequency scalng (VFS) model wth dscrete modes leads to an NP-complete problem, even f the modes are regularly dstrbuted (an mportant partcular case n practce, whch we analyze as the ncremental model). On the contrary, the Vdd-hoppng model that allows to swtch between dfferent supply voltages (V ) whle executng a task leads to a polynomal soluton. Fnally, we provde an approxmaton algorthm for the ncremental model, whch we extend for the general VFS model. Copyrght 0 John Wley & Sons, Ltd. Receved 8 Aprl 0; Revsed Aprl 0; Accepted June 0 KEY WORS: energy models; complexty; b-crtera optmzaton; algorthms; schedulng. INTROUCTION Theenergy consumpton of computatonal platforms has recently become a crtcal problem, both for economc and envronmental reasons []. As an example, the Earth Smulator requres about MW (Mega Watts) of peak power, and PetaFlop systems may requre 00 MW of power, nearly the output of a small power plant (00 MW). At $00 per MW.Hour, peak operaton of a PetaFlop machne may thus cost $0,000 per hour []. Current estmates state that coolng costs $ to $ per watt of heat dsspated []. Ths s just one of the many economcal reasons why energy-aware schedulng has proved to be an mportant ssue n the past decade, even wthout consderng battery-powered systems such as laptops and embedded systems. As an example, the Green500 lst ( provdes rankngs of the most energy-effcent supercomputers n the world, therefore rasng even more awareness about power consumpton. To help reduce energy dsspaton, processors can run at dfferent speeds. Ther power consumpton s the sum of a statc part (the cost for a processor to be turned on) and a dynamc part, whch s a strctly convex functon of the processor speed, so that the executon of a gven amount of work costs more power f a processor runs n a hgher mode [4]. More precsely, a processor runnng at speed s dsspates s watts [5 9] per tme-unt; hence, consumes s d joules when operated durng d unts of tme. Faster speeds allow for a faster executon, but they also lead to a much hgher (supra-lnear) power consumpton. *Correspondence to: Anne Benot, LIP, ENS Lyon, 46 allée d Itale, 6964 Lyon Cedex 07, France. E-mal: Anne.Benot@ens-lyon.fr A two-page extended abstract of ths work appears as a short presentaton n SPAA 0. Copyrght 0 John Wley & Sons, Ltd.

2 506 G. AUPY ET AL. Energy-aware schedulng ams at mnmzng the energy consumed durng the executon of the target applcaton. Obvously, t makes sense only f t s coupled wth some performance bound to acheve, otherwse, the optmal soluton always s to run each processor at the slowest possble speed. In ths paper, we nvestgate energy-aware schedulng strateges for executng a task graph on a set of processors. The man orgnalty s that we assume that the mappng of the task graph s gven, say by an ordered lst of tasks to execute on each processor. There are many stuatons n whch ths problem s mportant, such as optmzng for legacy applcatons, or accountng for affntes between tasks and resources, or even when tasks are pre-allocated [0], for example, for securty reasons. In such stuatons, assume that a lst-schedule has been computed for the task graph, and that ts executon tme should not exceed a deadlne. We do not have the freedom to change the assgnment of a gven task, but we can change ts speed to reduce energy consumpton, provded that the deadlne s not exceeded after the speed change. Rather than usng a local approach such as backfllng [, ], whch only reclams gaps n the schedule, we consder the problem as a whole, and we assess the mpact of several speed varaton models on ts complexty. More precsely, we nvestgate the followng models: Contnuous model. screte model. Processors can have arbtrary speeds, and can vary them contnuously: ths model s unrealstc (any possble value of the speed, say p e, cannot be obtaned), but t s theoretcally appealng []. A maxmum speed, s max, cannot be exceeded. Processors have a dscrete number of predefned speeds (or frequences), whch correspond to dfferent voltages that the processor can be subjected to [4]. Swtchng frequences s not allowed durng the executon of a gven task, but two dfferent tasks scheduled on a same processor can be executed at dfferent frequences. Vdd-Hoppng model. Ths model s smlar to the ISCRETE one, except that swtchng modes durng the executon of a gven task s allowed: any ratonal speed can be smulated, by smply swtchng, at the approprate tme durng the executon of a task, between two consecutve modes [5]. Note that V usually represents the supply voltage, hence the name V-HOPPING. Incremental model. In ths varant of the ISCRETE model, we ntroduce a value ı that corresponds the mnmum permssble speed ncrement, nduced by the mnmum voltage ncrement that can be acheved when controllng the processor CPU. Ths new model ams at capturng a realstc verson of the IS- CRETE model, where the dfferent modes are spread regularly nstead of arbtrarly chosen. Our man contrbutons are the followng. For the CONTINUOUS model, we gve a closed-form formula for trees and seres parallel graphs, and we cast the problem nto a geometrc programmng problem [6] for general drected acyclc graphs (AGs). For the V-HOPPING model, we show that the optmal soluton for general AGs can be computed n polynomal tme, usng a (ratonal) lnear program. Fnally, for the ISCRETE and INCREMENTAL models, we show that the problem s NP-complete. Furthermore, we provde approxmaton algorthms that rely on the polynomal algorthm for the V-HOPPING model, and we compare ther soluton wth the optmal CONTINUOUS soluton. The paper s organzed as follows. We start wth a survey of related lterature n Secton. We then provde the formal descrpton of the framework and of the energy models n Secton, together wth a smple example to llustrate the dfferent models. The next two sectons consttute the heart of the paper: n Secton 4, we provde analytcal formulas for contnuous speeds and the formulaton nto the convex optmzaton problem. In Secton 5, we assess the complexty of the problem wth all the dscrete models: ISCRETE, V-HOPPING and INCREMENTAL, and we dscuss approxmaton algorthms. Fnally, we conclude n Secton 6.

3 RECLAIMING THE ENERGY OF A SCHEULE 507. RELATE WORK Reducng the energy consumpton of computatonal platforms s an mportant research topc, and many technques at the process, crcut desgn, and mcro-archtectural levels have been proposed [7 9]. The dynamc voltage and frequency scalng (VFS) technque has been extensvely studed, because t may lead to effcent energy/performance trade-offs [,,, 0 ]. Current mcroprocessors (for nstance, from AM [4] and Intel [5]) allow the speed to be set dynamcally. Indeed, by lowerng supply voltage, hence processor clock frequency, t s possble to acheve mportant reductons n power consumpton, wthout necessarly ncreasng the executon tme. We frst dscuss dfferent optmzaton problems that arse n ths context. Then we revew energy models... VFS and optmzaton problems When dealng wth energy consumpton, the most usual optmzaton functon conssts n mnmzng the energy consumpton, whle ensurng a deadlne on the executon tme (.e., a real-tme constrant), as dscussed n the followng papers. In [4], Okuma et al. demonstrate that voltage scalng s far more effectve than the shutdown approach, whch smply stops the power supply when the system s nactve. Ther target processor employs just a few dscretely varable voltages. e Langen and Juurlnk [6] dscuss leakage-aware schedulng heurstcs that nvestgate both dynamc voltage scalng (VS) and processor shutdown, because statc power consumpton due to leakage current s expected to ncrease sgnfcantly. Chen et al. [7] consder parallel sparse applcatons, and they show that when schedulng applcatons modeled by a drected acyclc graph wth a well-dentfed crtcal path, t s possble to lower the voltage durng non-crtcal executon of tasks, wth no mpact on the executon tme. Smlarly, Wang et al. [] study the slack tme for non-crtcal jobs, they extend ther executon tme and thus reduce the energy consumpton wthout ncreasng the total executon tme. Km et al. [] provde power-aware schedulng algorthms for bag-of-tasks applcatons wth deadlne constrants, based on dynamc voltage scalng. Ther goal s to mnmze power consumpton as well as to meet the deadlnes specfed by applcaton users. For real-tme embedded systems, slack reclamaton technques are used. Lee and Sakura [7] show how to explot slack tme arsng from workload varaton, thanks to a software feedback control of supply voltage. Prathpat [] dscusses technques to take advantage of run-tme varatons n the executon tme of tasks; t determnes the mnmum voltage under whch each task can be executed, whle guaranteeng the deadlnes of each task. Then, experments are conducted on the Intel StrongArm SA-00 processor, whch has eleven dfferent frequences, and the Intel PXA50 XScale embedded processor wth four frequences. In [8], the goal of Xu et al. s to schedule a set of ndependent tasks, gven a worst case executon cycle (WCEC) for each task, and a global deadlne, whle accountng for tme and energy penaltes when the processor frequency s changng. The frequency of the processor can be lowered when some slack s obtaned dynamcally, typcally when a task runs faster than ts WCEC. Yang and Ln [] dscuss algorthms wth preempton, usng VS technques; substantal energy can be saved usng these algorthms, whch succeed to clam the statc and dynamc slack tme, wth lttle overhead. Because an ncreasng number of systems are powered by batteres, maxmzng battery lfe also s an mportant optmzaton problem. Battery-effcent systems can be obtaned wth smlar technques of dynamc voltage and frequency scalng, as descrbed by Lahr et al. n [8]. Another optmzaton crteron s the energy-delay product, because t accounts for a trade-off between performance and energy consumpton, as for nstance dscussed by Gonzalez and Horowtz n [9]. We do not dscuss further these latter optmzaton problems, because our goal s to mnmze the energy consumpton, wth a fxed deadlne. In ths paper, the applcaton s a task graph (drected acyclc graph), and we assume that the mappng, that s, an ordered lst of tasks to execute on each processor, s gven. Hence, our problem s closely related to slack reclamaton technques, but nstead of focusng on non-crtcal tasks as for nstance n [], we consder the problem as a whole. Our contrbuton s to perform an exhaustve complexty study for dfferent energy models. In the next paragraph, we dscuss related work on each energy model.

4 508 G. AUPY ET AL... Energy models Several energy models are consdered n the lterature, and they can all be categorzed n one of the four models nvestgated n ths paper, that s, CONTINUOUS, ISCRETE, V-HOPPING, or INCREMENTAL. The CONTINUOUS model s used manly for theoretcal studes. For nstance, Yao et al. [0], followed by Bansal et al. [], am at schedulng a collecton of tasks (wth release tme, deadlne and amount of work), and the soluton s the tme at whch each task s scheduled, and also the speed at whch the task s executed. In these papers, the speed can take any value, hence followng the CONTINUOUS model. We beleve that the most wdely used model s the ISCRETE one. Indeed, processors have currently only a few dscrete number of possble frequences [, 4, 4, 5]. Therefore, most of the papers dscussed earler follow ths model. Some studes explot the contnuous model to determne the smallest frequency requred to run a task, and then choose the closest upper dscrete value, as for nstance [] and []. Recently, a new local dynamc voltage scalng archtecture has been developed, based on the V-HOPPING model [5,,]. It was shown n [7] that sgnfcant power can be saved by usng two dstnct voltages, and archtectures usng ths prncple have been developed (see for nstance [4]). Compared wth tradtonal power converters, a new desgn wth no needs for large passves or costly technologcal optons has been valdated n a STMcroelectroncs CMOS 65 nm low-power technology [5]. To the best of our knowledge, ths paper ntroduces the INCREMENTAL model for the frst tme. The man ratonale s that future technologes may well have an ncreased number of possble frequences, and these wll follow a regular pattern. For nstance, note that the SA-00 processor, consdered n [], has eleven frequences that are equdstant, that s, they follow the INCRE- MENTAL model. Lee and Sakura [7] explot dscrete levels of clock frequency as f, f=, f=,..., where f s the master (.e., the hgher) system clock frequency. Ths model s closer to the ISCRETE model, although t exhbts a regular pattern smlar to the INCREMENTAL model. Our work s the frst attempt to compare these dfferent models: on the one hand, we assess the mpact of the model on the problem complexty (polynomal vs NP-hard), and on the other hand, we provde approxmaton algorthms buldng upon these results. The closest work to ours s the paper by Zhang et al. [], n whch the authors also consder the mappng of drected acyclc graphs, and compare the ISCRETE and the CONTINUOUS models. We go beyond ther work n ths paper, wth an exhaustve complexty study, closed-form formulas for the contnuous model, and the comparson wth the V-HOPPING and INCREMENTAL models.. FRAMEWORK Frst, we detal the optmzaton problem n Secton.. Then, we descrbe the four energy models n Secton.. Fnally, we llustrate the models and motvate the problem wth an example n Secton.... Optmzaton problem Consder an applcaton task graph G.V, E/, wth n jvjtasks denoted as V ft, T, :::, T n g, and where the set E denotes the precedence edges between tasks. Task T has a cost w for 6 6 n. We assume that the tasks n G have been allocated onto a parallel platform made up of dentcal processors. We defne the executon graph generated by ths allocaton as the graph G.V, E/, wth the followng augmented set of edges: E E: f an edge exsts n the precedence graph, t also exsts n the executon graph; f T and T are executed successvely, n ths order, on the same processor, then.t, T / E. The goal s to mnmze the energy consumed durng the executon whle enforcng a deadlne on the executon tme. We formalze the optmzaton problem n the smpler case, where each task s executed at constant speed. Ths strategy s optmal for the CONTINUOUS model (by a

5 RECLAIMING THE ENERGY OF A SCHEULE 509 convexty argument) and for the ISCRETE and INCREMENTAL models (by defnton). For the V-HOPPING model, we reformulate the problem n Secton 5.. For each task T V, b s the startng tme of ts executon, d s the duraton of ts executon, and s s the speed at whch t s executed. We obtan the followng formulaton of the MINENERGY.G, / problem, gven an executon graph G.V, E/ and a deadlne ; thes values are varables, whose values are constraned by the energy model (Secton.). Mnmze P n s d subject to () w s d for each task T V () b C d 6 b j for each edge.t, T j / E () b C d 6 for each task T V (v) b > 0 for each task T V () Constrant () states that the whole task can be executed n tme d usng speed s. Constrant () accounts for all dependences, and constrant () ensures that the executon tme does not exceed the deadlne. Fnally, constrant (v) enforces that startng tmes are non-negatve. The energy consumed throughout the executon s the objectve functon. It s the sum, for each task, of the energy consumed by ths task, as we detal n the next secton. Note that d w =s, and therefore the objectve functon can also be expressed as P n s w. Note that, whatever the energy model, there s a maxmum speed that cannot be exceeded, denoted s max. We pont out that there s a soluton to the mnmzaton problem f and only f there s a soluton wth s s max for all 6 6 n. Such a soluton would correspond to executng each task as early as possble (accordng to constrants () and (v)) and as fast as possble. The optmal soluton then slows down tasks to save as much energy as possble, whle enforcng the deadlne constrant. There s no guarantee on the unqueness of the soluton, because t may be possble to modfy the begnnng tme of a task wthout affectng the energy consumpton, f some of the constrants () are not tght... Energy models In all models, when a processor operates at speed s durng d tme-unts, the correspondng consumed energy s s d, whch s the dynamc part of the energy consumpton, followng the classcal models of the lterature [5 9]. Note that we do not take statc energy nto account, because all processors are up and alve durng the whole executon. We now detal the possble speed values n each energy model, whch should be added as a constrant n Equaton (). In the CONTINUOUS model, processors can have arbtrary speeds, from 0 to a maxmum value s max, and a processor can change ts speed at any tme durng executon. In the ISCRETE model, processors have a set of possble speed values, or modes, denoted as s, :::, s m. There s no assumpton on the range and dstrbuton of these modes. The speed of a processor cannot change durng the computaton of a task, but t can change from task to task. In the V-HOPPING model, a processor can run at dfferent speeds s, :::, s m, as n the prevous model, but t can also change ts speed durng a computaton. The energy consumed durng the executon of one task s the sum, on each tme nterval wth constant speed s, of the energy consumed durng ths nterval at speed s. In the INCREMENTAL model, we ntroduce a value ı that corresponds to the mnmum permssble speed (.e., voltage) ncrement. That means that possble speed values are obtaned as s C ı, where s an nteger such that s max. Admssble speeds ı le n the nterval Œ, s max. Ths new model ams at capturng a realstc verson of the ISCRETE model, where the dfferent modes are spread regularly between s and s m s max, nstead of beng arbtrarly chosen. It s ntended as the modern counterpart of a potentometer knob!

6 50 G. AUPY ET AL. Fgure. Executon graph for the example... Example Consder an applcaton wth four tasks of costs w, w, w and w 4, and one precedence constrant T! T. We assume that T and T are allocated, n ths order, onto processor P, whle T and T 4 are allocated, n ths order, on processor P. The resultng executon graph G s gven n Fgure, wth two precedence constrants added to the ntal task graph. The deadlne on the executon tme s.5. We set the maxmum speed to s max 6 for the CONTINUOUS model. For the ISCRETE and V-HOPPING models, we use the set of speeds s.d/, s.d/ 5 and s.d/ 6. Fnally, for the INCREMENTAL model, we set ı, and s max 6, so that possble speeds are s./, s./ 4,ands./ 6. We am at fndng the optmal executon speed s for each task T ( 6 6 4), that s, the values of s that mnmze the energy consumpton. Wth the CONTINUOUS model, the optmal speeds are non-ratonal values, and we obtan s. C 5= / ' 4.8I s s 5 '.56I s = s 4 s '.8. 5= Note that all speeds are lower than the maxmum s max. These values are obtaned thanks to the formulas derved n Secton 4. The energy consumpton s then E.c/ opt P 4 w s.s C.s C.s ' The executon tme s w s C max w s, w Cw 4 s, and wth ths soluton, t s equal to the deadlne (actually, both processors reach the deadlne, otherwse we could slow down the executon of one task). For the ISCRETE model, f we execute all tasks at speed s.d/ 5, we obtan an energy E A better soluton s obtaned wth s s.d/ 6, s s s.d/ and s 4 s.d/ 5, whch turns out to be optmal: E.d/ opt 6 C. C / 4 C Note that E.d/ opt >E opt,.c/ that s, the optmal energy consumpton wth the ISCRETE model s much hgher than the one acheved wth the CONTINUOUS model. Indeed, n ths case, even though the frst processor executes durng =6C= tme unts, the second processor remans dle because =6 C = C =5.4 <. The problem turns out to be NP-hard (Secton 5.), and the soluton has been found by performng an exhaustve search. Wth the V-HOPPING model, we set s s.d/ 5; for the other tasks, we run part of the tme at speed s.d/ 5, and part of the tme at speed s.d/ n order to use the dle tme and lower the energy consumpton. T s executed at speed s.d/ durng tme 5 and at speed s.d/ 6 durng tme (.e., the frst processor executes durng tme =5 C 5=6 C =0.5, and 0 all the work for T s carred out: 5=6 C 5 =0 w ). T s executed at speed s.d/ (durng tme =5), and fnally T 4 s executed at speed s.d/ durng tme 0.5 and at speed s.d/ durng tme =5 (.e., the second processor executes durng tme =5 C =5 C 0.5 C =5.5, and all the work for T 4 s carred out: 0.5 C 5 =5 w 4 ). Ths set of speeds turns out to be optmal (.e., t s the optmal soluton of the lnear program ntroduced n Secton 5.), wth an energy consumpton E.v/ opt.=5 C =0 C =5 C =5/ 5 C.5=6 C 0.5/ 44. As

7 RECLAIMING THE ENERGY OF A SCHEULE 5 expected, E.c/ opt 6 E.v/ opt 6 E opt,.d/ thats,thev-hopping soluton stands between the optmal CONTINUOUS soluton, and the more constraned ISCRETE soluton. For the INCREMENTAL model, the reasonng s smlar to the ISCRETE case, and the optmal soluton s obtaned by an exhaustve search: all tasks should be executed at speed s./ 4, wth an energy consumpton E./ opt > E opt..c/ It turns out to be better than ISCRETE and V-HOPPING, because t has dfferent dscrete values of energy that are more approprate for ths example. We conclude the study of ths smple example wth a short dscusson on the energy savngs that can be acheved. All three models have a maxmum speed s max 6. Executng the four tasks at maxmum speed leads to consumng an energy E max Such an executon completes wthn a delay. We clearly see the trade-off between executon tme and energy consumpton here, because we gan more than half the energy by slowng down the executon from to.5. Note that wth, we can stll slow down task T to speed 4, and stll gan a lttle over the brute force soluton. Hence, even such a toy example allows us to llustrate the benefts of energy-aware schedules. Obvously, wth larger examples, the energy savngs wll be even more dramatc, dependng upon the range of avalable speeds and the tghtness of the executon deadlne. In fact, the maxmal energy gan that can be acheved s not bounded: when executng each task as slow as possble (nstead of as fast as possble), we gan smax Wtotal,whereW total s the sum of all task weghts, and ths quantty can be arbtrarly large. One of the man contrbutons of ths paper s to provde optmal energy-aware algorthms for each model (or guaranteed polynomal approxmatons for NP-complete nstances). 4. THE CONTINUOUS MOEL Wth the CONTINUOUS model, processor speeds can take any value between 0 and s max. Frst, we prove that, wth ths model, the processors do not change ther speed durng the executon of a task (Secton 4.). Then, we derve n Secton 4. the optmal speed values for specal executon graph structures, expressed as closed form algebrac formulas, and we show that these values may be rratonal (as already llustrated n the example n Secton.). Fnally, we formulate the problem for general AGs as a convex optmzaton program n Secton Prelmnary lemma Lemma (constant speed per task) In all optmal soluton wth the CONTINUOUS model, each task s executed at constant speed, that s, a processor does not change ts speed durng the executon of a task. Suppose that n the optmal soluton, there s a task whose speed changes durng the executon. Consder the frst tme-step at whch the change occurs: the computaton begns at speed s from tme t to tme t 0, and then contnues at speed s 0 untl tme t 00. The total energy consumpton for ths task n the tme nterval ŒtI t 00 s E.t 0 t/s C.t 00 t 0 /.s 0 /. Moreover, the amount of work carred out for ths task s W.t 0 t/ s C.t 00 t 0 / s 0. If we run the task durng the whole nterval ŒtI t 00 at constant speed W=.t 00 t/, the same amount of work s carred out wthn the same tme. However, the energy consumpton durng ths nterval of tme s now E 0.t 00 t/.w=.t 00 t//. By convexty of the functon x 7! x, we obtan E 0 <Ebecause t<t 0 <t 00. Ths contradcts the hypothess of optmalty of the frst soluton, whch concludes the proof. 4.. Specal executon graphs 4... Independent tasks. Consder the problem of mnmzng the energy of n ndependent tasks (.e., each task s mapped onto a dstnct processor, and there are no precedence constrants n the executon graph), whle enforcng a deadlne.

8 5 G. AUPY ET AL. Proposton (ndependent tasks) When G s composed of ndependent tasks ft, :::, T n g, the optmal soluton to MINENER- GY.G, / s obtaned when each task T ( 6 6 n) s computed at speed s w.iftheres ataskt such that s >s max, then the problem has no soluton. For task T, the speed s corresponds to the slowest speed at whch the processor can execute the task, so that the deadlne s not exceeded. If s >s max, the correspondng processor wll never be able to complete ts executon before the deadlne; therefore, there s no soluton. To conclude the proof, we note that any other soluton would meet the deadlne constrant; and therefore, the s s should be such that w s 6, whch means that s > w. These values would all be hgher than the s s of the optmal soluton, and hence would lead to a hgher energy consumpton. Therefore, ths soluton s optmal Lnear chan of tasks. Ths case corresponds for nstance to n ndependent tasks ft, :::, T n g executed onto a sngle processor. The executon graph s then a lnear chan (order of executon of the tasks), wth T! T C,for 6 <n. Proposton (lnear chan) When G s a lnear chan of tasks, the optmal soluton to MINENERGY.G, / s obtaned when each task s executed at speed s W, wth W P n w. If s>s max, then there s no soluton. Suppose that n the optmal soluton, tasks T and T j are such that s < s j. The total energy consumpton s E opt.wedefnes such that the executon of both tasks runnng at speed s takes the same amount of tme than n the optmal soluton, that s,.w C w j /=s w =s C w j =s j : s.w Cw j / w s j Cw j s s s j. Note that s <s<s j (t s the barycenter of two ponts wth postve mass). We consder a soluton such that the speed of task T k,for 6 k 6 n, wth k and k j, s the same as n the optmal soluton, and the speed of tasks T and T j s s. By defnton of s, the executon tme has not been modfed. The energy consumpton of ths soluton s E, where E op t E w s C w j sj.w C w j /s, that s, the dfference of energy wth the optmal soluton s only mpacted by tasks T and T j, for whch the speed has been modfed. By convexty of the functon x 7! x, we obtan E opt >E, whch contradcts ts optmalty. Therefore, n the optmal soluton, all tasks have the same executon speed. Moreover, the energy consumpton mzed when the speed s as low as possble, whle the deadlne s not exceeded. Therefore, the executon speed of all tasks s s W=. Corollary A lnear chan wth n tasks s equvalent to a sngle task of cost W P n w. Indeed, n the optmal soluton, the n tasks are executed at the same speed, and they can be replaced by a sngle task of cost W, whch s executed at the same speed and consumes the same amount of energy Fork and jon graphs. Let V ft, :::, T n g. We consder ether a fork graph G.V [ft 0 g, E/, wth E f.t 0, T /, T V g, or a jon graph G.V [ft 0 g, E/, wth E f.t, T 0 /, T V g. T 0 s ether the source of the fork or the snk of the jon. Theorem (fork and jon graphs) When G s a fork (resp. jon) executon graph wth n C tasks T 0, T, :::, T n, the optmal soluton to MINENERGY.G, / s the followng:

9 RECLAIMING THE ENERGY OF A SCHEULE 5 the executon speed of the source (resp. snk) T 0 s s 0 for the other tasks T, 6 6 n, wehaves s 0 P n w w P n w C w 0 f s 0 6 s max. Otherwse, T 0 should be executed at speed s 0 s max, and the other speeds are s w, wth 0 0 w 0 s max, f they do not exceed s max (Proposton for ndependent tasks). Otherwse, there s no soluton. If no speed exceeds s max, the correspondng energy consumpton s Pn C w 0 w mne.g, /. Let t 0 w 0 s 0. Then, the source or the snk requres a tme t 0 for executon. For 6 6 n, taskt must be executed wthn a tme t 0 so that the deadlne s respected. Gven t 0, we can compute the speed s for task T usng Theorem, because the tasks are ndependent: s w s t 0 w 0 s 0 w 0. The objectve s therefore to mnmze P n 0 w s, whch s a functon of s 0 nx nx w s w 0 s0 C 0 w s0.s 0 w 0 / s 0 w 0 C P n w f.s.s 0 w 0 / 0 /. Let W P n w.inordertofndthevalueofs 0 that mnmzes ths functon, we study the functon f.x/,forx>0. f 0.x/ x w 0 C W x W, and therefore f 0.x/ 0.x w 0 /.x w 0 / for x.w C w 0 /=. We conclude that the optmal speed for task T 0 s s 0.P n w / Cw 0 f s 0 6 s max. Otherwse, T 0 should be executed at the maxmum speed s 0 s max, because t s the s bottleneck task. In any case, for 6 6 n, the optmal speed for task T s s w 0 s 0 w 0. Fnally, we compute the exact expresson of mne.g, / f.s 0 /,whens 0 6 s max 0 f.s 0 / s0 W w 0 W! W C w 0 A W C w 0 C w.s 0 w 0 / 0, whch concludes the proof. W = ;, Corollary (equvalent tasks for speed) Consder a fork or jon graph wth tasks T, n, and a deadlne, and assume that the speeds n the optmal soluton to MINENERGY.G, / do not exceed s max. Then, these speeds are the same as n the optmal soluton for nc ndependent tasks T0 0, T 0, :::, T n 0,wherew0 0 P n w Cw 0, and, for 6 6 n, w 0 w0 0 w. P. n w / Corollary (equvalent task for energy) Consder a fork or jon graph G and a deadlne, and assume that the speeds n the optmal soluton to MINENERGY.G, / do not exceed s max. We say that the graph G s equvalent to the graph G.eq/, consstng of a sngle task T.eq/ 0 of weght w.eq/ 0 P n w C w 0, because the mnmum energy consumpton of both graphs are dentcal: mne.g, /=mne.g.eq/, / Trees. We extend the results on a fork graph for a tree G.V, E/ wth jv jn C tasks. Let T 0 be the root of the tree; t has k chldren tasks, whch are each themselves the root of a tree. A tree can therefore be seen as a fork graph, where the tasks of the fork are trees.

10 54 G. AUPY ET AL. The prevous results for fork graphs naturally lead to an algorthm that peels off branches of the tree, startng wth the leaves, and replaces each fork subgraph n the tree, composed of a root T 0 and k chldren, by one task (as n Corollary ) that becomes the unque chld of T 0 s parent n the tree. We say that ths task s equvalent to the fork graph, because the optmal energy consumpton wll be the same. The computaton of the equvalent cost of ths task s carred out thanks to a call to the eq procedure, whle the tree procedure computes the soluton to MINENERGY.G, / (Algorthm ). Note that the algorthm computes the mnmum energy for a tree, but t does not return the speeds at whch each task must be executed. However, the algorthm returns the speed of the root task, and t s then straghtforward to compute the speed of each chldren of the root task, and so on. Theorem (tree graphs) When G s a tree rooted n T 0 (T 0 V,whereV s the set of tasks), the optmal soluton to MINENERGY.G, / can be computed n polynomal tme O.jV j /. Let G be a tree graph rooted n T 0. The optmal soluton to MINENERGY.G, / s obtaned wth a call to tree.g, T 0, /, and we prove ts optmalty recursvely on the depth of the tree. Smlar to the case of the fork graphs, we reduce the tree to an equvalent task that, f executed alone wthn a deadlne, consumes exactly the same amount of energy. The procedure eq s the procedure that reduces a tree to ts equvalent task (Algorthm ). If the tree has depth 0, then t s a sngle task, eq.g, T 0 / returns the equvalent cost w 0,and the optmal executon speed s w 0 (Proposton ). There s a soluton f and only f ths speed s not greater than s max, and then the correspondng energy consumpton s w 0, as returned by the algorthm.

11 RECLAIMING THE ENERGY OF A SCHEULE 55 Assume now that for any tree of depth <p, eq computes ts equvalent cost, and tree returns ts optmal energy consumpton. We consder a tree G of depth p rooted n T 0 : G T 0 [fg g,where each subgraph G s a tree, rooted n T, of maxmum depth p. As n the case of forks, we know that each subtree G has a deadlne x, wherex w 0 s 0,ands 0 s the speed at whch task T 0 s executed. By nducton hypothess, we suppose that each graph G s equvalent to a sngle task, T 0,ofcostw0 (as computed by the procedure eq). We can then use the results obtaned on forks to compute w.eq/ 0 (see proof of Theorem ) w.eq/ 0 X.w 0 /! C w 0. Fnally, the tree s equvalent to one task of cost w.eq/ 0,andf w.eq/ 0 6 s max, the energy w.eq/ 0 w 0 s max consumpton s, and no speed exceeds s max. Note that the speed of a task s always greater than the speed of ts successors. Therefore, f w.eq/ 0 >s max, we execute the root of the tree at speed s max and then process each subtree G ndependently. Of course, there s no soluton f >; and otherwse, we perform the recursve calls to tree to process each subtree ndependently. Ther deadlne s then w 0 s max. To study the tme complexty of ths algorthm, frst note that when callng tree.g, T 0, /,there mght be at most jv j recursve calls to tree, once at each node of the tree. Wthout accountng for the recursve calls, the tree procedure performs one call to the eq procedure, whch computes the cost of the equvalent task. Ths eq procedure takes a tme O.jV j/, because we have to consder the jv j tasks, and we add the costs one by one. Therefore, the overall complexty s n O.jV j / Seres-parallel graphs. We can further generalze our results to seres-parallel graphs (SPGs), whch are bult from a sequence of compostons (parallel or seres) of smaller-sze SPGs. The smallest SPG conssts of two nodes connected by an edge (such a graph s called an elementary SPG). The frst node s the source, whereas the second one s the snk of the SPG. When composng two SGPs n seres, we merge the snk of the frst SPG wth the source of the second one. For a parallel composton, the two sources are merged, as well as the two snks, as llustrated n Fgure. We can extend the results for tree graphs to SPGs, by replacng step by step the SPGs by an equvalent task (procedure cost n Algorthm ): we can compute the equvalent cost for a seres or parallel composton. However, because t s no longer true that the speed of a task s always larger than the speed of ts successor (as was the case n a tree), we have not been able to fnd a recursve property on the tasks (a) Two SPGs before composton. (b) Parallel composton. (c) Seres composton. Fgure. Composton of seres-parallel graphs (SPGs).

12 56 G. AUPY ET AL. that should be set to s max, when one of the speeds obtaned wth the prevous method exceeds s max. The problem of computng a closed form for a SPG wth a fnte value of s max remans open. Stll, we have the followng result when s max C: Theorem (seres-parallel graphs) When G s an SPG, t s possble to compute recursvely a closed form expresson of the optmal soluton of MINENERGY.G, /, assumng s max C, n polynomal tme O.jV j/, wherev s the set of tasks. Let G be a seres-parallel graph. The optmal soluton to MINENERGY.G, / s obtaned wth a call to SPG.G, /, and we prove ts optmalty recursvely. Smlar to trees, the man dea s to peel the graph off, and to transform t untl there remans only a sngle equvalent task that, f executed alone wthn a deadlne, would consume exactly the same amount of energy. The procedure cost s the procedure that reduces a tree to ts equvalent task (Algorthm ). The proof s carred out by nducton on the number of compostons requred to buld the graph G, p. Ifp 0, G s an elementary SPG consstng n two tasks, the source T 0 and the snk T. It s therefore a lnear chan, and therefore equvalent to a sngle task whose cost s the sum of both costs, w 0 C w (see Corollary for lnear chans). The procedure cost returns therefore the correct equvalent cost, and SPG returns the mnmum energy consumpton. Let us assume that the procedures return the correct equvalent cost and mnmum energy consumpton for any SPG consstng of <pcompostons. We consder an SPG G wth p compostons. By defnton, G s a composton of two smaller-sze SPGs, G and G, and both of these SPGs have strctly fewer than p compostons. We consder G 0 and G0, whch are dentcal to G

13 RECLAIMING THE ENERGY OF A SCHEULE 57 and G, except that the cost of ther source and snk tasks are set to 0 (these costs are handled separately), and we can reduce both of these SPGs to an equvalent task, of respectve costs w 0 and w0, by nducton hypothess. There are two cases: If G s a seres composton, then after the reducton of G 0 and G0, we have a lnear chan n whch we consder the source T 0 of G, the snk T of G (whch s also the source of G ), and the snk T of G. The equvalent cost s therefore w 0 C w 0 C w C w 0 C w, thanks to Corollary for lnear chans. If G s a parallel composton, the resultng graph s a fork-jon graph, and we can use Corollares and to compute the cost of the equvalent task, accountng for the source T 0 and the snk T : w 0 C.w 0 / C.w 0 / C w. Once the cost of the equvalent task of the SPG has been computed wth the call to cost.g/, the optmal energy consumpton s.cost.g//. Contrarly to the case of tree graphs, because we never need to call the SPG procedure agan because there s no constrant on s max, the tme complexty of the algorthm s the complexty of the cost procedure. There s exactly one call to cost for each composton, and the number of compostons n the SPG s n O.jV j/. All operatons n cost can be carred out n O./, hence a complexty n O.jV j/. 4.. General AGs For arbtrary executon graphs, we can rewrte the MINENERGY.G, / problem as follows: P n u Mnmze w subject to () b C w u 6 b j for each edge.t, T j / E () b C w u 6 for each task T V () u > s max for each task T V (v) b > 0 for each task T V Here, u =s s the nverse of the speed to execute task T. We now have a convex optmzaton problem to solve, wth lnear constrants n the non-negatve varables u and b.infact,theobjectve functon s a posynomal, so we have a geometrc programmng problem [6, Secton 4.5] for whch effcent numercal schemes exst. In addton, such an optmzaton problem wth a smooth convex objectve functon s known to be well-condtoned [5]. However, as llustrated on smple fork graphs, the optmal speeds are not expected to be ratonal numbers but nstead arbtrarly complex expressons (we have the cubc root of the sum of cubes for forks, and nested expressons of ths form for trees). From a computatonal complexty pont of vew, we do not know how to encode such numbers n polynomal sze of the nput (the ratonal task weghts and the executon deadlne). Stll, we can always solve the problem numercally and get fxed-sze numbers that are good approxmatons of the optmal values. In the followng, we show that the total power consumpton of any optmal schedule s constant throughout executon. Whle ths mportant property does not help to desgn an optmal soluton, t shows that a schedule wth large varatons n ts power consumpton s lkely to waste a lot of energy. We need a few notatons before statng the result. Consder a schedule for a graph G.V, E/ wth n tasks. Task T s executed at constant speed s (Lemma ) and durng nterval Œb, c : T begns ts executon at tme b and completes t at tme c. The total power consumpton P.t/of the schedule at tme t s defned as the sum of the power consumed by all tasks executng at tme t X P.t/ s. 66n, tœb,c Theorem 4 Consder an nstance of CONTINUOUS, and an optmal schedule for ths nstance, such that no speed s equal to s max. Then the total power consumpton of the schedule throughout executon s constant. ()

14 58 G. AUPY ET AL. We prove ths theorem by nducton on the number of tasks of the graph. Frst, we prove a prelmnary result: Lemma Consder a graph G.V, E/ wth n > tasks, and any optmal schedule of deadlne. Lett be the earlest completon tme of a task n the schedule. Smlarly, let t be the latest startng tme of a task n the schedule. Then, ether G s composed of ndependent tasks, or 0<t 6 t <. Task T s executed at speed s and durng nterval Œb, c.wehavet mn 66n c and t max 66n b. Clearly, 0 6 t, t 6 by defnton of the schedule. Suppose that t <t. Let T be a task that ends at tme t,andt one that starts at tme t. Then: ÀT V,.T, T/ E (otherwse, T would start after t ), therefore, t ; ÀT V,.T, T / E (otherwse, T would fnsh before t ); therefore t 0. Ths also means that all tasks start at tme 0 and end at tme. Therefore, G s only composed of ndependent tasks. Back to the proof of the theorem, we consder frst the case of a graph wth only one task. In an optmal schedule, the task s executed n tme, and at constant speed (Lemma ), hence wth constant power consumpton. Suppose now that the property s true for all AGs wth at most n tasks. Let G be a AG wth n tasks. If G s exactly composed of n ndependent tasks, then we know that the power consumpton of G s constant (because all task speeds are constant). Otherwse, let t be the earlest completon tme, and t the latest startng tme of a task n the optmal schedule. Thanks to Lemma, we have 0<t 6 t <. Suppose frst that t t t 0. There are three knds of tasks: those begnnng at tme 0 and endng at tme t 0 (set S ), those begnnng at tme t 0 and endng at tme (set S ), and fnally those begnnng at tme 0 and endng at tme (set S ). Tasks n S execute durng the whole schedule duraton, at constant speed, hence ther contrbuton to the total power consumpton P.t/s the same at each tme-step t. Therefore, we can suppress them from the schedule wthout loss of generalty.. The energy Next, we determne the value of t 0.LetA P T S w,anda P T S w consumpton between 0 and t 0 s A, and between t t0 0 and, ts A. t 0 /. The optmal energy con- sumpton s obtaned wth t 0 A. Then, the total power consumpton of the optmal schedule A CA! s the same n both ntervals, hence at each tme-step: we derve that P.t/,whch A CA s constant. Suppose now that t <t. For each task T,letw 0 be the number of operatons executed before t, and w 00 the number of operatons executed after t (wth w 0 C w00 w ). Let G 0 be the AG G wth executon costs w 0,andG00 be the AG G wth executon costs w 00. The tasks wth a cost equal to 0 are removed from the AGs. Then, both G 0 and G 00 have strctly fewer than n tasks. We can therefore apply the nducton hypothess. We derve that the power consumpton n both AGs s constant. Because we dd not change the speeds of the tasks, the total power consumpton P.t/ n G s the same as n G 0 f t<t, hence a constant. Smlarly, the total power consumpton P.t/n G s the same as n G 00 f t>t, hence a constant. Consderng the same parttonng wth t nstead of t, we show that the total power consumpton P.t/ s a constant before t, and also a constant after t.butt <t, and the ntervals Œ0, t and Œt, overlap. Altogether, the total power consumpton s the same constant throughout Œ0,, whch concludes the proof.

15 RECLAIMING THE ENERGY OF A SCHEULE ISCRETE MOELS In ths secton, we present complexty results on the three energy models wth a fnte number of possble speeds. The only polynomal nstance s for the V-HOPPING model, for whch we wrte a lnear program n Secton 5.. Then, we gve NP-completeness results n Secton 5., and approxmaton results n Secton 5., for the ISCRETE and INCREMENTAL models. 5.. The Vdd-hoppng model Theorem 5 Wth the V-HOPPING model, MINENERGY.G, / can be solved n polynomal tme. Let G be the executon graph of an applcaton wth n tasks, and a deadlne. Let s, :::, s m be the set of possble processor speeds. We use the followng ratonal varables: for 6 6 n and 6 j 6 m, b s the startng tme of the executon of task T,and.,j/ s the tme spent at speed s j for executng task T.TherearenCnmn.m C / such varables. Note that the total executon tme of task T s P m j.,j/. The constrants are as follows: n, b > 0: startng tmes of all tasks are non-negatve numbers; n, b C P m j.,j/ 6 : the deadlne s not exceeded by any task; 8 6, 0 6 n such that T! T 0, b C P m j.,j/ 6 b 0: a task cannot start before ts predecessor has completed ts executon; n, P m j.,j/ s j > w :taskt s completely executed. Pn P The objectve functon s then mn m j.,j/sj. The sze of ths lnear program s clearly polynomal n the sze of the nstance, all n.m C / varables are ratonal, and therefore t can be solved n polynomal tme [6]. 5.. NP-completeness results Theorem 6 Wth the INCREMENTAL model (and hence the ISCRETE model), MINENERGY.G, / s NP-complete. We consder the assocated decson problem: gven an executon graph, a deadlne, and a bound on the energy consumpton, can we fnd an executon speed for each task such that the deadlne and the bound on energy are respected? The problem s clearly n NP: gven the executon speed of each task, computng the executon tme and the energy consumpton can be carred out n polynomal tme. To establsh the completeness, we use a reducton from -Partton [7]. We consder an nstance I of -Partton: gven n strctly postve ntegers a, :::, a n, does there exst a subset I of f, :::, ng such that P I a P I a?lett P n a. We buld the followng nstance I of our problem: the executon graph s a lnear chan wth n tasks, where task T has sze w a ; the processor can run at m dfferent speeds; s and s, (.e.,, s max, ı ); L T =; E 5T. Clearly, the sze of I s polynomal n the sze of I. Suppose frst that nstance I has a soluton I. For all I, T s executed at speed, otherwse t s executed at speed. The executon tme s then P I a C P I a = T, and the

16 50 G. AUPY ET AL. energy consumpton s E P I a C P I a 5T E. Both bounds are respected, and therefore the executon speeds are a soluton to I. Suppose now that I has a soluton. Because we consder the ISCRETE and INCREMENTAL models, each task run ether at speed or at speed. LetI f j T s executed at speed g. Note that we have P I a T P I a. The executon tme s 0 P I a C P I a = T C. P I a /=. Because the deadlne s not exceeded, 0 6 T =, and therefore P I a 6 T. For the energy consumpton of the soluton of I,wehaveE 0 P I a C P I a T C P I a. Because E 0 6 E 5T, we obtan P I a 6 T, and hence P I a 6 T. Because P I a C P I a T, we conclude that P I a P I a T, and therefore I has a soluton. Ths concludes the proof. 5.. Approxmaton results Here, we explan for the INCREMENTAL and ISCRETE models, how the soluton to the NP-hard problem can be approxmated. Note that, gven an executon graph and a deadlne, the optmal energy consumpton wth the CONTINUOUS model s always lower than that wth the other models, whch are more constraned. Theorem 7 Wth the INCREMENTAL model, for any nteger K>0,theMINENERGY.G, / problem can be approxmated wthn a factor. C ı /. C K /, n a tme polynomal n the sze of the nstance and n K. Consder an nstance I nc of the problem wth the INCREMENTAL model. The executon graph G has n tasks, s the deadlne, ı s the mnmum permssble speed ncrement, and, s max are the speed bounds. Moreover, let K>0be an nteger, and let E nc be the optmal value of the energy consumpton for ths nstance I nc. We construct the followng nstance I vdd wth the V-HOPPING model: the executon graph and the deadlne are the same as n nstance I nc, and the speeds can take the values ( C ), K 066N where N s such that s max s not exceeded: N.ln.s max / ln. //= ln C. K AsN s asymptotcally of order O.K ln.s max //, the number of possble speeds n I vdd, and hence the sze of I vdd, s polynomal n the sze of I nc and K. Next, we solve I vdd n polynomal tme, thanks to Theorem 5. For each task T,lets.vdd/ be the average speed of T n ths soluton: f the executon tme of the task n the soluton s d, then s.vdd/ w n =d ; E vdd s the optmal energy consumpton o obtaned wth these speeds. Let mn u C u ı j C u ı > s.vdd/ be the smallest speed n I nc that s larger s.algo/ than s.vdd/. There exsts such a speed snce, because of the values chosen for I vdd, s.vdd/ 6 s max. The values s.algo/ can be computed n tme polynomal n the sze of I nc and K. LetE algo be the energy consumpton obtaned wth these values. In order to prove that ths algorthm s an approxmaton of the optmal soluton, we need to prove that E algo 6. C ı /. C K / E nc. For each task T, s.algo/ ı 6 s.vdd/ 6 s.algo/. Because 6 s.vdd/,wedervethats.algo/ E algo X w s.algo/ 6 X 6 s.vdd/. C ı /. Summng over all tasks, we obtan w s.vdd/ C ı 6 E vdd C ı.

17 RECLAIMING THE ENERGY OF A SCHEULE 5 Next, we bound E vdd thanks to the optmal soluton wth the CONTINUOUS model, E con.leti con be the nstance where the executon graph G, the deadlne, the speeds and s max are the same as n nstance I nc, but now admssble speeds take any value between and s max.lets.con/ be the optmal contnuous speed for task T, and let 0 6 u 6 N be the value such that C u 6 s.con/ 6 C uc s K K. In order to bound the energy consumpton for I vdd, we assume that T runs at speed s, nstead of s.vdd/. The soluton wth these speeds s a soluton to I vdd, and ts energy consumpton s E > E vdd. From the prevous nequaltes, we deduce that s 6 s.con/ C K, and by summng over all tasks, E vdd 6 E X w s X 6 w s.con/ 6 E con C 6 E nc C. K K C K Proposton For any nteger ı>0, any nstance of MINENERGY.G, / wth the CONTINUOUS model can be approxmated wthn a factor. C ı / n the INCREMENTAL model wth speed ncrement ı. For any nteger K>0, any nstance of MINENERGY.G, / wth the ISCRETE model can be approxmated wthn a factor C s C, K wth max6<m fs C s g, n a tme polynomal n the sze of the nstance and n K. For the frst part, let s.con/ be the optmal contnuous speed for task T n nstance I con ; E con s the optmal energy consumpton. For any task T,lets be the speed of I nc such that s ı<s con 6 s..let Then, s.con/ 6 s C ı.lete be the energy wth speeds s. E con 6 E C ı. E nc be the optmal energy of I nc. Then, E con 6 E nc C ı For the second part, we use the same algorthm as n Theorem 7. The same proof leads to the approxmaton rato wth nstead of ı. 6. CONCLUSION In ths paper, we have assessed the tractablty of a classcal schedulng problem, wth task preallocaton, under varous energy models. We have gven several results related to CONTINUOUS speeds. However, whle these are of conceptual mportance, they cannot be acheved wth physcal devces, and we have analyzed several models enforcng a bounded number of achevable speeds, a.k.a. modes. In the classcal ISCRETE model that arses from VFS technques, admssble speeds can be rregularly dstrbuted, whch motvates the V-HOPPING approach that mxes two consecutve modes optmally. Although computng optmal speeds s NP-hard wth dscrete modes, t has polynomal complexty when mxng speeds. Intutvely, the V-HOPPING approach allows for smoothng out the dscrete nature of the modes. An alternate (and smpler n practce) soluton to V-HOPPING s the INCREMENTAL model, where one stcks wth unque speeds durng task executon as n the ISCRETE model, but where consecutve modes are regularly spaced. Such a model can be made arbtrarly effcent, accordng to our approxmaton results. Altogether, ths paper has lad the theoretcal foundatons for a comparatve study of energy models. In the recent years, we have observed an ncreased concern for green computng, and a rapdly