Procrastination Scheduling for Fixed-Priority Tasks with Preemption Thresholds

Transcription

1 Procrastnaton Schedulng for Fxed-Prorty Tasks wth Preempton Thresholds XaoChuan He, Yan Ja Insttute of Network Technology and Informaton Securty School of Computer Scence Natonal Unversty of Defense Technology Changsha, Chna Abstract. Dynamc Voltage Scalng (DVS), whch adjusts the clock speed and supply voltage dynamcally, s an effectve technque n reducng the energy consumpton of embedded real-tme systems. However, the longer a job executes, the more energy n the leakage current the devce/processor consumes for the job. Procrastnaton schedulng, where task executon can be delayed to maxmze the duraton of dle ntervals by keepng the processor n a sleep/shutdown state even f there are pendng tasks wthn the tmng constrants mposed by performance requrements, has been proposed to mnmze leakage energy dran. Ths paper targets energy-effcent fxed-prorty wth preempton threshold schedulng for perodc real-tme tasks on a unprocessor DVS system wth non-neglgble leakage power consumpton. We propose a two-phase algorthm. In the frst phase, the executon speed,.e., the supply voltage of each task are determned by applyng off-lne algorthms, and n the second phase, the procrastnaton length of each task s derved by applyng on-lne smulated work-demand tme analyss, and thus the tme moment to turn on/off the system s determned on the fly. A seres of smulaton experments was evaluated for the performance of our algorthms. The results show that our proposed algorthms can derve energy-effcent schedules. 1 Introducton Low power utlzaton has been an mportant ssue for hardware manufacturng for next-generaton portable, scalable, and sophstcated embedded systems. To reduce the power consumpton wthout the sacrfce of performance, archtectural technques have been proposed to dynamcally trade the performance and power consumpton. Dynamc Voltage Scalng (DVS), whch adjusts the supply voltage and ts correspondng clock frequency dynamcally, s one of the most effectve low-power desgn technque for embedded real-tme systems. Snce the energy consumpton of CMOS crcuts has a quadratc dependency on the supply voltage, lowerng the supply voltage s one of the most effectve ways of reducng the energy consumpton. In many real-tme applcatons, average or worst-case task response tme s an mportant non-functonal desgn requrement of the system. For example, to mantan the system stablty, many embedded real-tme systems must complete the tasks before

2 ther deadlnes. For real-tme systems targetng commercal varable voltage mcroprocessors, snce lowerng the supply voltage also decreases the maxmum achevable clock speed [1], energy-effcent task schedulng s to reduce supply voltage dynamcally to the lowest possble level whle satsfyng the tasks tmng constrants. In the past decade, energy-effcent task schedulng wth varous deadlne constrants receved extensve attenton, especally for the mnmzaton of the energy consumpton of the dynamc voltage scalng part n a unprocessor envronment [2]. Recently, researchers have started explorng energy-effcent schedulng wth the consderatons of leakage current snce the power consumpton resultng from leakage current s comparable to the dynamc power dsspaton [3]. To reduce the energy consumpton resultng from leakage current, a system mght be turned off (to enter a dormant mode). For perodc real-tme tasks, Jejurkar et al. [4] and Lee et al. [5] proposed energy-effcent schedulng on a unprocessor by procrastnaton schedulng to decde when to turn off the system. Jejurkar and Gupta [3] then further consdered real-tme tasks that mght complete earler than ts worst-case estmaton by extendng the algorthms presented n [4]. Fxed-prorty preemptve (FPP) schedulng algorthms and fxed-prorty non-preemptve (FPNP) schedulng algorthms are two mportant classes of real-tme schedulng algorthms. To obtan the benefts of both FPP and FPNP algorthms, there are several other algorthms tryng to fll the gap between them. The fxed-prorty wth preempton threshold (FPPT) schedulng algorthm [6] s one of them. Under FPPT, each task has a par of prortes: regular prorty and preempton threshold, where the preempton threshold of a task s hgher than or equal to ts regular prorty. The preempton threshold represents the tasks runnng-tme preempton prorty level. It prevents the preempton of the task from other tasks, unless the preemptng tasks prorty s hgher than the preempton threshold of the current runnng task. Saksena and Wang have shown that task sets scheduled wth FPPT can have sgnfcant schedulablty mprovements over task set usng fxed prortes [6]. Ths paper consders energy-effcent FPPT schedulng of perodc real-tme tasks on a unprocessor whose dynamc voltage scalng porton mght be turned off for further energy savng. We further combne procrastnaton schedulng wth dynamc voltage scalng to mnmze the total statc and dynamc energy consumpton of the system. An on-lne algorthm was developed to calculate the respectve procrastnaton nterval for each task. A seres of smulaton experments was also evaluated for the performance of our algorthms. The results show that our proposed algorthms can derve energyeffcent schedules. The rest of ths paper s organzed as follows: Secton 2 defnes the leakage-aware energy-effcent FPPT schedulng problem n a unprocessor system. Prelmnary results are shown n Secton 2. The proposed algorthms are n Secton 3. Expermental results for the performance evaluaton of the proposed algorthms are presented n Secton 4. Secton 5 s the concluson.

3 2 System Model 2.1 Task Model Ths study deals wth the fxed prorty preemptve schedulng of tasks n a real-tme systems wth hard constrants,.e., systems n whch the respect of tme constrants s mandatory. The actvtes of the system are modeled by perodc tasks. The model of the system s defned by a task set T of cardnalty n, T = {τ 1, τ 2,..., τ n }. The j th job of task τ s denoted as J,j. The ndex, j, for jobs of a task s started from zero. A perodc task τ s characterzed by a 3-tuple (C, T, D ) where each request of τ, called nstance, has an executon CPU cycles (denoted as C ), and a relatve deadlne (denoted as D ). T tme unts separate two consecutve nstances of τ (hence T s the perod of the task). Gven a set T of n tasks, the hyper-perod of T, denoted by L, s defned so that L/T s an nteger for any task τ n T. The number of jobs n the hyper-perod of task τ s L/T. For example, L s the least common multple (LCM) of the perods of tasks n T when the perods of tasks are all nteger numbers. We focus on the case that all of the tasks arrve at tme 0. We also assocate wth each task τ a unque prorty {1, 2,..., n} such that contenton for resources s resolved n favor of the job wth the hghest prorty that s ready to run. The analyss presented n secton 3 uses the concept of busy and dle perods [7]. These are defned as follows: A level- busy perod s a contnuous tme nterval durng whch the notonal run-queue contans one or more tasks of actve prorty level or hgher. Smlarly, a level- dle perod s a tme nterval durng whch the run-queue s free of level or hgher prorty tasks. We note that the run queue may become momentarly free of level- tasks, when one tasks completes and another s released. Ths appears n our formulaton as an dle perod of zero length. 2.2 Power Consumpton and Executon Models We explore energy-effcent schedulng on a dynamc voltage scalng (DVS) processor. The power consumpton s contrbuted by the dynamc power consumpton resultng from the chargng and dschargng of gates on the CMOS crcuts and the statc power consumpton resultng from leakage current. The dynamc power consumpton P d of the dynamc voltage scalng part of the processor s a functon of the adopted processor speed f: P d = C eff V 2 DD f (1) f = αk (V DD V TH ) α V DD (2) where k s a devce related parameter, V TH s the threshold voltage, C eff s the effectve swtchng capactance per cycle and α ranges from 2 to 1.2 dependng on the devce technology. Snce power vares lnearly wth the clock speed and the square

4 of the voltage, adjustng both can produce cubc power reductons, at least n theory. The statc power consumpton P s of the system comes from the leakage current of the processor, system I/O devces, and RAM. It mght be modeled as a nonnegatve constant, as n [8], or a lnear functon of the supply voltage (a sub-lnear functon of the executon speed) [9], [3], [4], [10]. The power consumpton of processor s denoted by P, whch s the sum of the dynamc and statc power consumpton. We consder systems n whch P(f) s a convex and ncreasng functon, and P(f)/f s a convex functon, smlarly to [11], [4]. Recent processors support multple varable voltage and frequency levels for energy effcent operaton of the system. Let the avalable frequences be {FLK 1, FLK 2,..., FLK s } n ncreasng order of frequency and the correspondng voltage levels be {v 1, v 2,..., v s }. We assume that the CPU speed f of task τ can be changed between a mnmum speed FLK 1 (mnmum supply voltage necessary to keep the system functonal) and a maxmum speed FLK s. In our framework, the voltage/speed changes take place only at context swtch tme and whle state savng nstructons execute. If not neglgble, the voltage change overhead can be ncorporated nto the worst-case workload of each task. The system could enter the dormant mode (or be turned off) whenever needed. The power consumpton of the system s treated as 0 when t s n the dormant mode [8] by scalng the statc power consumpton. We consder systems that could be turned on/off at nstant. When needed, turnng the system off mght further reduce the energy consumpton. The energy consumpton to turn off the system s assumed to be neglgble, but t mght requre addtonal energy to turn on the system [12]. We denote E sw as the energy of the swtchng overhead from the dormant mode to the actve mode. For the rest of ths paper, we say the system s dle at tme nstant t, f the processor does not execute any task at tme nstant t. When the system s actve and dle, the processor executes NOP nstructons and must be at processor speed FLK 1 to mnmze the energy consumpton. Let P I be the power consumpton when the system s dle and actve, where P I = P(FLK 1 ). 2.3 Crtcal Speed The crtcal speed ˆf s defned as the avalable speed of the processor to execute a cycle wth the mnmum energy consumpton. Because of the convexty of P(f), executng at a common speed for a CPU cycle mnmzes the energy consumpton. Hence, the energy consumpton to execute a CPU cycle at speed f s P(f)/f. Snce the power consumpton functon P(f) s a convex and ncreasng functon, where P(f)/f s merely a convex functon. P(f)/f s mnmzed when f s equal to f, wth d(p(f )/f ) df = 0. As a result, to mnmze the executon energy consumpton of T, we do not have to consder schedules that execute jobs at any lower speed than f snce we could execute jobs at speed f wth lower energy consumpton and less executon tme. If f s between FLK 1 and FLK s, we know that ˆf s f. If f s less than FLK 1, ˆf s set to FLK 1 to satsfy the hardware constrant. Smlarly, f f s greater than FLK s, ˆf s set to FLK s, and jobs are executed at FLK s to mnmze the energy consumpton. As a result, ˆf s mn{max{f, FLK 1 }, FLK s }. Executng a job of task τ at any speed less than ˆf

5 would ether consume more energy than that at ˆf wth more executon tme or volate the speed constrant. We assume that f could be obtaned effcently or pre-determned as a specfed parameter n the nput. 2.4 Problem defnton The problem consdered n ths paper s as follows: Defnton 1. (Leakage-Aware Energy-Effcent Schedulng for FPPT, LAEES-FPPT) Consder a set T of n ndependent tasks ready at tme 0. Each perodc task τ T s assocated wth a computaton requrement equal to C CPU-cycles and ts perod T, where the relatve deadlne of τ s equal to D. And each task τ T s assgned wth a unque prorty {1, 2,..., n} and a preempton threshold γ {1, 2,..., n} (γ ), where s used to compete for processor and gamma s used to protect τ from unnecessary task preemptons after τ starts. The power consumpton functon P(f) s a convex and ncreasng functon, whle P(f)/f s merely a convex functon. The processor s wth a dscrete spectrum of the avalable speeds n [FLK 1, FLK s ]. The energy of the swtchng overhead from the dormant mode to the actve mode of a system s E sw, and the power consumpton when the system s actve and dle s P I. The problem s to mnmze the energy consumpton n the hyper-perod L of tasks n T n the schedulng of fxed-prorty tasks wth preempton thresholds n T wthout mssng the tmng constrants. A schedule of a task set T s an assgnment of the avalable processor speeds for each correspondng task executon, where the job arrvals of each task τ T satsfy ts tmng constrant D. A schedule s feasble f no job msses ts deadlne. A schedule s optmal for the LAEES-FPPT problem, f t s feasble, and ts energy consumpton s the mnmum among all feasble schedules. For the rest of ths paper, let S be an optmal schedule for T. For a schedule, an dle nterval s a maxmal nterval when the system s dle, whle an executon nterval s a maxmal nterval when the processor executes some jobs. The system mght be turned off or be at the actve mode n an dle nterval, whle the system s actve n an executon nterval. For any set X, let X be the cardnalty of the set. For example, I S s the number of dle ntervals n schedule S n (0, L]. If the dle nterval s greater than E sw /P I, turnng off the system s worthwhle. Let t θ be the threshold dle nterval E sw /P I. If the dle nterval s greater than t θ, the longer the dle nterval s, the more the energy saved by turnng off the system. The energy consumpton of a schedule S, denoted as E(S), conssts of two parts: the executon energy consumpton φ(s) and the dle energy consumpton ε(s). The executon energy consumpton s the sum of the energy consumpton of the executons of jobs n S n the tme nterval (0, L]. The dle energy consumpton s the sum of the energy consumpton n the ntervals n (0, L] n whch the system does not execute any job. Let υ(t, S) be the speed at tme nstant t n schedule S. The executon energy consumpton φ(s) n E(S) s L P(υ(t, S))dt. The dle energy consumpton ε(s) n 0 E(S) s the summaton of E sw tmes the number of nstances that the system s turned from the dormant mode to the actve mode and P I tmes the total nterval length that the system s dle and actve n (0, L].

6 3 Proposed Algorthms Ths secton presents a two-phase algorthm for perodc real-tme tasks. The algorthms determne, n the frst phase, the executon speed,.e., the supply voltage, of each task, and n the second phase the moment to turn on/off the system on the fly. 3.1 An On-lne Procrastnaton Algorthm to Mnmze the Energy Leakage: LA-FPPT Let S e be the resultng FPPT schedule by applyng some off-lne dvs algorthms / [13]. For brevty, let C be the executon tme of a job of task τ n S e, C = C f opt. The frst phase of the proposed algorthm [13] decdes the executon speed of tasks n T to meet the tmng constrants and mnmze the executon energy consumpton. The second phase s to reduce the dle energy consumpton by turnng the system off on the fly. The dea behnd schedulng on the fly s to lengthen and aggregate the dle ntervals so that the resultng dle tme s long enough to turn off the system. The determnaton of dle ntervals can be done by procrastnatng the arrval tme of the next job to the system, as n [4],[14] for EDF schedulng, and n [9],[11] for fxed-prorty schedulng. In [9],[4] procrastnaton s done by computng the maxmum procrastnaton ntervals of all of the tasks n T based on the system utlzaton, whle the dle ntervals n [14],[11] are determned by procrastnatng the remanng jobs as late as possble. In ths secton, we proposes an on-lne smulated work-demand analyss approach to the determnaton of dle ntervals. If a job completes at tme nstant t, and the ready queue s empty, we have to decde whether the system should be turned off or dle. Let r (t) be the arrval tme of the next job of τ for any τ n T arrved after tme nstant t, t.e., r (t) = T. Let d (t) be the next deadlne on an nvocaton of task τ after T tme nstant t,.e., d (t) = r (t) + D. Our formulaton stems from consderng the schedulablty of each fxed-prorty task wth preempton thresholds at tme nstant t. We focus on fndng the maxmum (t), whch may be stolen at prorty level, durng the nterval [t, t+d (t)), whlst guaranteeng that task τ meets ts deadlne. (Note, Sπ max (t) amount of dle nterval, S max may not actually be avalable for dle due to the constrants on hard deadlne tasks wth prortes lower than. We return to ths pont later). To guarantee that task τ wll meet ts deadlne, we need to analyze the worst case scenaro from tme t onwards. We therefore assume that all tasks τ j are re-nvoked at ther earlest possble next release r j (t) and subsequently wth a perod of T j. (t), t s nstructve to vew the nterval [t, t + d (t)) as comprsng a number of level- busy and dle perods. Any level- dle tme between the completon of task τ and ts deadlne could be swapped for task τ s procrastnaton nterval Z wthout causng the deadlne to be mssed. Hence the maxmum procrastnaton nterval Z whch may be stolen s In attemptng to determne the maxmum guaranteed dle tme, S max equal to the total level- dle tme n the nterval. We use ths result to calculate S max (t). We frst derve equaton 3 usng technques gven n [15]. Although the ready queue s empty at tme nstant t, two components stll determne the extent of the busy perod under the nfluence of procrastnaton schedulng:

7 1. For the task τ k wth prorty π k < < γ k, τ k s released workload just before the start of busy perod 2. For the task τ j wth prorty π j > γ, τ j s released workload durng the busy perod The second component mples a recursve defnton. As the processng released ncreases monotoncally wth the length of the busy perod, a recurrence relaton can be used to fnd w (t): w m+1 (t) = S π (t) + max k,π k <<γ k C k + j,π j> ( w m π (t) x j (t) T j C j ) The term S π (t) represents the begnnng of level- dle tme from tme t. The recurrence relaton begns wth wπ 0 (t) = 0 and ends when wπ m+1 (t) = wπ m (t) or wπ m+1 (t) > d (t). Proof of convergence follows from analyss of smlar recurrence relatons by Audsley et al [15]. The fnal value of w π (t) defnes the length of the busy perod. Alternatvely, we may vew t+w π (t) as defnng the start of a level- dle tme. Gven the start of a level- dle tme, wthn the nterval [t, t + d (t)), the end of the dle tme, whch may be converted to procrastnaton nterval of task τ, occurs ether at the next release of a task τ j wth prorty π j > or at the end of the nterval. Equaton 4 gves the length, l (t, w π (t)), of the level- dle tme. l (t, w π (t)) = mn [ ] d (t) w( (t), ) wπ (t) r mn j(t) j,π T j γ j T j + r j (t) w π (t) where the term d (t) w π (t) means that the end of level- dle tme come about wπ (t) r at the end of [t, t + d (t)), the term j(t) T j + r j (t) descrbe the workload T j contrbuted by task τ j n the level- busy perod, whose length s denoted by w π (t). Combnng equatons 3 and 4, our method for determnng the maxmum dle tme, (t), proceeds as follows: S max 1. The dle tme whch may be derved, S π (t), s ntally set to zero 2. Equaton 3 s used to compute the end of a busy perod n the nterval [t, t + d (t)) 3. The end of the busy perod s used as the start of an dle perod by equaton 4 whch returns the length of contguous dle tme. 4. The dle tme, S π (t) s ncremented by the amount of dle tme found n step If the deadlne on task τ has been reached, then the maxmum dle tme whch can be derved s gven by S π (t). Otherwse, we repeat steps 2 to 5. The pseudo-codes of dynamc procrastnaton algorthm at tme nstant t when the ready queue s empty, and a job completes are shown n Algorthm 1. 4 Case Studes and Smulatons Secton 3 showed that our two-phase algorthm (EE-FPPT [13] + LA-PFFT) wll always render the controlled leakage current n CMOS crcuts and reduced energy consumptons that wll mantan the schedulablty of the workload. we use randomly-generated workloads to examne broad trends across a range of desgn ponts. (3) (4)

8 Algorthm 1 On-lne Algorthm to Mnmze Energy Leakage 1: procedure DYNAMIC PROCRASTINATION(t) a job completes at t and the ready queue s empty 2: sort T by ascendng prorty order 3: for ( = 1; n; n) do 4: r (t) t T T 5: d (t) r (t) + D 6: S π (t) 0 7: w m+1 (t) 0 (t) d (t)) do (t) 8: whle (w m+1 9: w m (t) w m+1 10: w m+1 (t) = S (t) + max k,π k < <γ k C k + 11: f (wπ m (t) = wπ m+1 (t)) then S π (t) S π (t) + l (t, wπ m (t)) wπ m+1 (t) wπ m+1 (t) + l (t,wπ m (t)) 12: end f j,π j ( ) w m π (t) r j(t) C j 13: end whle 14: Sπ max (t) S (t) 15: revse the arrval tme r (t) of job J,t by settng r (t) r (t) + Sπ max 16: end for 17: f ( mn r (t) t > t θ ) then τ T 18: turn the system off at tme t and turn on at mn r (t) τ T 19: else 20: reman on the actve mode 21: end f 22: end procedure T j (t) We nvestgate workload characterstcs that affect the energy savng capablty attanable through LA-FPPT. We now smulate and analyze randomly generated systems of tasks to better understand our approaches. The power consumpton functon of the system speed f was set as P(f) = f The normalzed total energy was adopted as the performance metrcs. The normalzed total energy of an algorthm for an nput nstance s the energy consumpton of the derved soluton n (0, L] dvded by the energy consumpton by applyng the orgnal FPPT schedulng wthout processor slowdown, procrastnaton and by turnng off the system when the dle nterval s long enough. We tred two dfferent expermental settngs. The frst experment nvestgate separately the effect of the swtchng overhead E sw, the system utlzaton on the lmted energy consumpton acheved by our methods. To cover a wde range of desgn ponts, 20,000 real-tme task sets wth 10 tasks each were randomly generated. These were created so 1000 have a utlzaton of 50%, 1000 have 52% utlzaton, and so on up to 90%. For each group of task sets who hold the same utlzaton, those were created so 20 have a E sw of 0.03, 20 have 0.04, and so on up to The second one focused on

9 the mpact of the number of tasks and E sw (0.17), another 20,000 real-tme task sets wth system utlzaton 67% each were randomly generated too. Those were created so 1000 nclude 5 ndependent tasks, 1000 nclude 6 ndependent tasks, and so on up to 25 tasks. Task perods s assgned randomly n the range [1, 100] wth a unform probablty dstrbuton functon. Moreover, task deadlnes were set equal to ther respectve perods (for smplcty, though not necessary). Tasks WCETs were set to ncur the requred overall system utlzaton. All 40,000 real-tme task sets generated were schedulable wth a fully preemptve polcy. Fg. 1. Experment I results for power savng of our approaches Average Normalzed Energy Average Normalzed Energy EE-FPPT EE-FPPT + LA-FPPT 0.4 EE-FPPT EE-FPPT + LA-FPPT E sw Utlzaton (a) Energy consumpton produced for LA- (b) Energy consumpton rses wth system FPPT rses wth large E sw, but keep almost constant for EE-FPPT, wth system systems. utlzaton, but soars up for hgh-utlzaton utlzaton = 0.67 Fg. 2. Experment II results for power savng of our approaches EE-FPPT EE-FPPT + LA-FPPT Utlzaton = 30% Utlzaton = 40% Utlzaton = 50% Utlzaton = 60% Average Normalzed Energy Percent of Systems Number of Tasks Normalzed Energy Consumpton (a) Energy consumpton declnes wth the (b) EE-FPPT + LA-FPPT dramatcally accomplshes the energy savngs, even for ncrement of the number of tasks, on the condton that system utlzaton = 0.67 hgh-utlzaton systems. and E sw = 0.17

10 Usng the MPTA, the total energy produced by each system was computed and normalzed to the energy requred by the orgnal verson of the system. The average normalzed energy were then plotted as a functon of E sw, the system utlzaton and the number of tasks n turn. The results are shown n fgures 2(a), 2(b) and 3(a) respectvely. In Fgure 2(a), the more the swtchng overhead E sw was, the more the normalzed energy consumpton was for schedules derved from Algorthms LA-FPPT. When E sw s relatvely small (E sw 0.18 ), the energy consumpton from leakage current n CMOS crcuts stll have ltter nfluence on the total energy consumpton, thus the more E sw, the less normalzed energy consumpton. In Fgure 2(b), our algorthms (EE-FPPT + LA-FPPT) outperformed orgnal Algorthm FPPT when the system utlzaton was greater than When the system utlzaton was large enough, procrastnaton mght create two (or more) dle ntervals to turn the system off, but the orgnal FPPT schedule mght make the system dle for a short nterval and turn the system off for a longer nterval. As a result, the energy consumpton of procrastnaton schedules mght consume more energy than the orgnal FPPT schedule when the system utlzaton s large enough. Moreover, when the utlzaton for task executon s large, the mprovement on dle energy consumpton s margnal snce task executon domnates the total energy consumpton. In Fgure 3(a), for all the smulated algorthms, the normalzed energy consumpton decreased for small number of tasks wth n 12, and was steady for n > 12. Ths s because the resultng utlzaton of a task was large when n was small n the expermental setup, and, hence, there was only lttle room for procrastnaton to save energy. For task sets wth n > 12, the maxmum procrastnaton nterval was domnated by tasks wth small perods, and, hence, the mprovement became margnal. Another nterestng property s the dstrbuton of the 20,000 systems of Experment I among the dfferent normalzed power consumpton levels. Fgure 3(b) show ths dstrbuton for the overall system utlzaton levels of 30%, 40%, 50%, and 60%, respectvely. As can be seen, the workloads scheduled wth the fxed-prorty schemes depend on the system utlzaton level to some extent. 5 Conclusons In ths paper we dscuss the energy-effcent schedulng problem of perodc realtme tasks by applyng FPPT polcy on a unprocessor dynamc voltage scalng system that can go nto the dormant mode for energy effcency. We propose a two-phase schedulng algorthm. In the frst phase, the executon speed,.e., the supply voltage, of each task s determned by applyng off-lne algorthms. In the second phase, the tme moment to turn on/off the system s determned on the fly. Theoretcal analyss shows that our 1 proposed algorthms could derve schedulng solutons wth at most max{, 2} (U bd ) 2 tmes of the energy consumpton of optmal solutons, where the term U bd represents the breakdown utlzaton [16] of a task set. A seres of smulaton experments was evaluated to demonstrate the performance of the proposed algorthms. Our expermental results show that our approaches can accomplsh dramatc energy savngs as the same tme keep the schedulablty of task set.

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