A Quasi-Static Approach to Minimizing Energy Consumption in Real-Time Systems under Reward Constraints

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1 n Proc. Intl. Confrnc on Ral-Tm and mbddd Computng Systms and Applcatons, 2006, pp A Quas-Statc Approach to Mnmzng nrgy Consumpton n Ral-Tm Systms undr Rward Constrants Lus Aljandro Cortés 1 Ptru ls 2 Zbo Png 2 aljandro.corts@volvo.com ptl@da.lu.s zbp@da.lu.s 1 Volvo Truck Corporaton 2 Lnköpng Unvrsty Gothnburg, Swdn Lnköpng, Swdn Abstract In som ral-tm applcatons, t s dsrabl to trad off prcson for tmlnss. For such systms, consdrd typcally undr th Imprcs Computaton modl, a functon assgns rward to th applcaton dpndng on th amount of computaton allottd to t. Also, many such applcatons run on battry-powrd dvcs whr th nrgy consumpton s of utmost mportanc. W addrss n ths papr th problm of nrgy mnmzaton for Imprcs-Computaton systms that hav rward and tm constrants. W propos a Quas-Statc (QS) approach that xplots, wth low on-ln ovrhad, th dynamc slack that arss from varatons n th actual numbr of xcuton cycls: frst, at dsgn-tm, a st of solutons ar computd and stord (off-ln phas); scond, th slcton among th prcomputd assgnmnts s lft for run-tm, basd on actual valus of tm and rward (on-ln phas). 1 Introducton Powr and nrgy consumpton hav bcom qut mportant dsgn consdratons. Dynamc Voltag Scalng (DVS) tchnqus [9] ar a wll-known approach for rducng th nrgy consumpton n ral-tm systms. By lowrng th supply voltag quadratc savngs n nrgy consumpton can b achvd whl prformanc s dgradd n approxmatly lnar fashon. At th sam tm, for crtan ral-tm applcatons approxmat but tmly rsults ar accptabl, for xampl, fuzzy mags n tm ar oftn prfrabl to prfct mags too lat. Imprcs Computaton (IC) tchnqus [5] hav bn usd for studyng such systms. Tasks ar composd of mandatory and optonal parts, both of whch must b fnshd by th dadln, although th optonal part can b lft ncomplt at th xpns of th qualty of rsults (a functon assgns rward dpndng on th amount of computaton allottd to th optonal part). On th on hand, DVS tchnqus, whch allow th trad-off btwn nrgy consumpton and prformanc, hav manly bn appld to hard ral-tm systms (no rward aspct consdrd). On th othr hand, IC approachs, whch mak t possbl to trad off prcson for tmlnss, hav untl now dsrgardd th nrgy aspcts. Rusu t al. [10] proposd an approach n whch rward, nrgy, and dadlns ar consdrd n th sam framwork. Th problm s to mz th total rward wthout xcdng th nrgy budgt or th dadlns. Ths approach solvs statcally th optmzaton problm and consquntly consdrs only worst cass. A smlar problm was dscussd by Cortés t al. [4] but, as opposd to [10], th dynamc slack, causd by tasks compltng arlr than n th worst cas, s xplotd by usng a QS approach. In ths papr w also dal wth ral-tm systms for whch t s possbl to trad off prcson for tmlnss as wll as nrgy consumpton for prformanc. Th problm addrssd n ths papr (somhow a mrror problm to th on n [4]) s to mnmz th nrgy consumpton subjct to a mnmum total-rward constrant and dadlns. W am at fndng th voltag lvls at whch ach task runs and ts numbr of optonal cycls such that th objctv functon s optmzd and th constrants satsfd. A statc soluton mpls fndng on Voltag/Optonalcycls (V/O) assgnmnt; t s pssmstc bcaus actual xcuton tms ar typcally far off from worst-cas valus. A dynamc soluton mpls rcomputng, vry tm a task complts, a V/O assgnmnt; although t can xplot th dynamc slack, th on-ln ovrhad s too hgh to mak th dynamc soluton applcabl n practc. W propos a quasstatc approach composd of two stps: frst, at dsgn tm, w comput a st of V/O assgnmnts (off-ln phas); scond, at run tm, on of th prcomputd V/O assgnmnts s slctd basd on actual valus of tm and accumulatd rward (on-ln phas). To our knowldg ths s th frst papr that consdrs th problm of nrgy mnmzaton n th fram of IC systms. A chf mrt of our approach s ts ablty to ffctvly xplot th dynamc slack at vry low on-ln ovrhad. 2 Prlmnars 2.1 Task and Archtctural Modls Th systm s capturd by a drctd acyclc graph G = (T, ) whr th nods T = {T 1, T 2,..., T n } corrspond to th computatonal tasks and th dgs ndcat th data dpndncs btwn tasks. For th sak of convnnc n th notaton, w assum that tasks ar namd accordng to a partcular xcuton ordr (as xpland latr n ths Subscton) that rspcts th data dpndncs. That s, task T +1 xcuts mmdatly aftr T, 1 < n. ach task T s composd of a mandatory part and an optonal part, charactrzd n trms of th numbr of CPU cycls M and O rspctvly. Th actual numbr of mandatory cycls M of a task T at a crtan actvaton of th systm s unknown bforhand but ls n th ntrval boundd by th bst-cas numbr of cycls M bc and th worst-cas numbr of cycls M, that s, M bc M M. Th xpctd numbr of mandatory cycls of a task T s dnotd M. Th optonal part of a task xcuts mmdatly aftr ts corrspondng mandatory part complts. For ach task T, thr s a dadln d by whch both mandatory and optonal parts of T must b compltd. For ach task T, thr s a rward functon R (O ) that taks as argumnt th numbr of optonal cycls O assgnd to T ; w assum that R (0) = 0. W consdr non-dcrasng concav 1 rward functons as thy captur th partcularts of most ral-lf applcatons [10]. Also, as dtald n Scton 4, th concavty of rward functons s xplotd for obtanng solutons to partcular optmzaton problms n polynomal tm. W assum also that thr s a valu O, for ach T, aftr whch no xtra rward s achvd, that s, R (O ) = R f O O. Th total rward s th sum P of ndvdual rward contrbutons and s dnotd R = T T R(O). Th rward producd P up to th complton of task T s dnotd RP (RP = j=1 R j(o j )). W 1 A functon f(x) s concav ff f (x) 0, that s, th scond drvatv s ngatv.

2 consdr a rward constrant, dnotd R mn, that gvs th lowr bound of th total rward that must b producd. W consdr that tasks ar non-prmptabl and hav qual rlas tm (r = 0, 1 n). All tasks ar mappd onto a sngl procssor and xcutd n a fxd ordr, dtrmnd off-ln, that rspcts th data dpndncs and accordng to an DF (arlst Dadln Frst) polcy. For non-prmptabl tasks wth qual rlas tm and runnng on a sngl procssor, DF gvs th optmal xcuton ordr [3] 2. T dnots th -th task n ths squnc. Th targt procssor supports voltag scalng and w assum that th voltag lvls can b vard n a contnuous way n th ntrval [V mn, V ]. If only a dscrt st of voltags ar supportd by th procssor, our approach can b adaptd by usng wll-known tchnqus for dtrmnng th dscrt voltag lvls that rplac th calculatd contnuous on [9]. In our QS approach w comput a numbr of V/O (Voltag/Optonal-cycls) assgnmnts. Th st of prcomputd V/O assgnmnts s stord n a ddcatd mmory as lookup tabls, on tabl LUT for ach task T. Th mum numbr of V/O assgnmnts that can b stord n mmory s fxd by th dsgnr and s dnotd N. 2.2 nrgy and Dlay Modls Th powr consumpton n CMOS crcuts s th sum of dynamc, statc (lakag), and short-crcut powr. Th shortcrcut componnt s nglgbl. Th dynamc powr s at th momnt th domnatng componnt. Howvr th lakag powr s bcomng an mportant factor n th ovrall powr dsspaton. For th sak of smplcty and clarty n th prsntaton of our das, w consdr only th dynamc nrgy consumpton. Nonthlss, th lakag nrgy and Adaptv Body Basng (ABB) tchnqus [1] can asly b ncorporatd nto th formulaton wthout changng our gnral approach. Th amount of dynamc nrgy consumd by task T s gvn by th followng xprsson [6]: = C V 2 (M + O ) (1) whr C s th ffctv swtchd capactanc, V s th supply voltag, and M + O s th total numbr of cycls xcutd by th task. Th nrgy ovrhad causd by swtchng from V to V j s as follows [6]:,j V = C r(v V j) 2 (2) whr C r s th capactanc of th powr ral. W also consdr, for th QS soluton, th nrgy ovrhad sl orgnatd from th nd to look up and slct on of th prcomputd V/O assgnmnts. Th way w stor th prcomputd assgnmnts maks th lookup and slcton procss tak O(1) tm. Thrfor sl s a constant valu. Also, ths valu s th sam for all tasks ( sl = sl, for 1 n). For consstncy rasons w kp th ndx n th notaton of th slcton ovrhad sl. Th nrgy ovrhad causd by onln opratons s dnotd dyn. In a QS soluton th on-ln ovrhad s just th slcton ovrhad ( dyn = sl ) [4]. Th xcuton tm of a task T xcutng M + O cycls at supply voltag V s [6]: V τ = k + O) (3) (V V th ) α(m whr k s a constant dpndnt on th procss tchnology, α s th saturaton vlocty ndx (also tchnology dpndnt, typcally 1.4 α 2), and V th s th thrshold voltag. Th 2 By optmal n ths contxt w man th task xcuton ordr that, among all fasbl ordrs, admts th V/O assgnmnt for whch th lowst total nrgy can b achvd. W hav dmonstratd n [3] that an DF xcuton ordr s th on that last constrants th spac of V/O solutons and hncforth optmal n th abov sns. tm ovrhad, whn swtchng from V to V j, s gvn by th followng xprsson [1]: δ,j V = p V V j (4) whr p s a constant. Th tm ovrhad for lookng up and slctng on V/O assgnmnt n th QS approach s dnotd δ sl and, as xpland abov, s constant and s th sam valu for all tasks. Th startng and complton tms of a task T ar dnotd s and t rspctvly, wth s + δ + τ = t whr δ capturs th total tm ovrhads. δ = δ 1, V + δ dyn th on-ln ovrhad. whr δ dyn s Not that n a QS soluton ths onln ovrhad s just th lookup and slcton tm, that s, = δ sl δ dyn. 3 Motvatonal xampl Bfor gong nto th prcs formulaton and th dtals of th soluton, w consdr n ths scton th xampl shown n Fg. 1. W assum non-dcrasng rward functons of th form R (O ) = K O, O O as wll as a rward constrant R mn = 8. As xpland n Subscton 2.1, tasks run accordng to th schdul T 1T 2T 3, fxd off-ln n conformty to an DF polcy. W consdr a procssor that prmts contnuous voltag scalng n th rang V. For th sak of clarty, n ths xampl w assum that transton ovrhads ar zro. T 1 T 3 T 2 R R K O O z } { Task M bc M M C [nf] d [µs] K R T T T Fg. 1. Motvatonal xampl Th optmal statc V/O assgnmnt for ths xampl s gvn by Tabl 1. Th assgnmnt gvs, for ach task T, th voltag V at whch T must run and th numbr of optonal cycls O that t must xcut n ordr to mnmz th nrgy consumpton, whl guarantng that dadlns ar mt and th rward constrant s satsfd. Task V [V] O T T T Tabl 1. Optmal statc V/O assgnmnt Th V/O assgnmnt gvn by Tabl 1 s optmal n th statc sns. It s th bst possbl that can b obtand offln wthout knowng th actual numbr of cycls xcutd by ach task. Howvr, th actual numbr of cycls, whch ar not known n advanc, ar typcally far off from th worstcas valus usd to comput such a statc assgnmnt. Ths pont s llustratd by th followng stuaton. Th frst task starts xcutng at V 1 = V, as rqurd by th statc assgnmnt. Assum that T 1 xcuts M 1 = (nstad of M1 = 80000) mandatory cycls and thn ts assgnd O 1 = 14 optonal cycls. At ths pont, knowng that T 1 has

3 compltd at t 1 = τ 1 = µs and that th rward producd s RP 1 = R 1 = , a nw V/O assgnmnt can accordngly b computd for th rmanng tasks amng at obtanng th mnmum total nrgy for th nw condtons. W consdr, for th momnt, th dal cas n whch such an on-ln computaton taks zro tm and nrgy. Obsrv that, for computng th nw assgnmnts, th worst cas for tasks not yt compltd has to b assumd as thr actual numbr of xcutd cycls s not known n advanc. Th nw assgnmnt gvs V 2 = V and O 2 = Thn T 2 runs at V 2 = V and lt us assum that t xcuts M 2 = (nstad of M2 = 50000) mandatory cycls and thn ts nwly assgnd O 2 = optonal cycls. At ths pont, th complton tm s t 2 = τ 1 + τ 2 = µs and th rward so far producd s RP 2 = R 1 +R 2 = Agan, a nw assgnmnt can b computd takng nto account th nformaton about complton tm and producd rward. Ths nw assgnmnt gvs V 3 = V and O 3 = For such a stuaton, n whch M 1 = 40000, M 2 = 30000, and M 3 = 80000, th V/O assgnmnt computd dynamcally (consdrng δ dyn = 0) s summarzd n Tabl 1(a). Accordng to ths assgnmnt and consdrng M 1 = 40000, M 2 = 30000, M 3 = 80000, th total nrgy (assumng also dyn = 0) s dyndal = µj. In ralty, howvr, th onln tm and mmory ovrhads causd by computng nw assgnmnts ar not nglgbl. Whn consdrng, for xampl, th ovrhad δ dyn = 40 µs th V/O assgnmnt computd dynamcally s vdntly dffrnt, as gvn by Tabl 1(b). Ths assgnmnt maks th total nrgy consumd (assumng that th on-ln computaton of V/O assgnmnts taks dyn = 35 µj) b dynral = µj. Th valus of δ dyn and dyn ar n practc svral ordrs of magntud hghr than th ons usd n ths hypothtcal xampl. For nstanc, for a systm wth 50 tasks, computng on such V/O assgnmnt usng a commrcal solvr taks a fw sconds. vn on-ln hurstcs, whch produc approxmat rsults, hav long xcuton tms. Ths mans that a dynamc V/O schdulr mght produc solutons that ar actually nfror to th statc on (n trms of total nrgy consumd) or, vn wors, a dynamc V/O schdulr mght not b abl to fulfll th gvn tm and rward constrants. (a) δ dyn = 0 Task V [V] O T T T (b) δ dyn = 40 µs Task V [V] O T T T Tabl 2. Dynamc V/O assgnmnts (for M 1 = 40000, M 2 = 30000, M 3 = 80000) Obsrv that for th stuaton of numbr of mandatory cycls consdrd abov (M 1 = 40000, M 2 = 30000, M 3 = 80000), th total nrgy consumd whn usng th statc assgnmnt of Tabl 1 s st = µj whl th dal dynamc assgnmnt of Tabl 1(a) gvs dyndal = µj, that s, nrgy savngs of 22%. Ths shows that mportant nrgy savngs mght b achvd by xplotng th dynamc slack. At th sam tm, th dynamc assgnmnt of Tabl 1(b), whch taks nto account th on-ln ovrhads, gvs a total nrgy dynral = µj, or 10% mor than th statc assgnmnt; ths nfror rsult s causd by th tm and nrgy on-ln ovrhads. Th abov fgurs llustrat that th dynamc slack can ffcntly b xplotd only f mthods wth low on-ln ovrhads ar usd. In our QS approach w comput at dsgn-tm a numbr of V/O assgnmnts, whch ar slctd at run-tm by th so-calld QS V/O schdulr (at vry low ovrhad). W can dfn, for nstanc, a QS st of assgnmnts for th xampl dscussd n ths subscton, as gvn by Tabl 3. Although ths st was obtand by usng th partcular soluton w propos n Scton 5 (n whch th numbr of optonal cycls s frozn as xpland latr), t llustrats wll th ssnc of th QS approach. Ths assgnmnts wr computd consdrng th slcton ovrhads δ sl = 0.3 µs and sl = 0.3 µj. At run-tm, upon complton of ach task, V and O ar slctd from th prcomputd st accordng to th gvn condton. Task Condton V [V] O T T 2 f t 1 84 µs ls f t µs ls T 3 f t µs ls f t µs ls Tabl 3. Prcomputd st of V/O assgnmnts For th stuaton M 1 = 40000, M 2 = 30000, M 3 = and th st gvn by Tabl 3, th QS V/O schdulr would do as follows. Task T 1 s run at V 1 = V and s allottd O 1 = 14 optonal cycls. Snc, whn compltng T 1, t 1 = τ 1 = µs, V 2 = 1.285/O 2 = s slctd by th QS V/O schdulr. Task T 2 runs undr ths assgnmnt so that, whn t fnshs, t 2 = τ 1 + δ2 sl + τ 2 = µs. Thn V 3 = 1.321/O 3 = s slctd and task T 3 s xcutd accordngly. Tabl 4 summarzs th slctd assgnmnt. Th nrgy consumd, whn usng ths V/O assgnmnt s qs = µj (compar to dyndal = µj, dynral = µj, and st = µj). Two mportant facts can b notd from th xampl: frst, th QS soluton qs outprforms clarly th dynamc on dyn ral bcaus of th larg ovrhads of th lattr; scond, th QS soluton qs s not far from th dal cas of a dynamc V/O schdulr dyn dal wth zro ovrhads. Task V [V] O T T T Tabl 4. QS V/O assgnmnt (for M 1 = 40000, M 2 = 30000, M 3 = 80000) slctd from th st of Tabl 3 4 Problm Formulaton In ths papr undr th framwork of th Imprcs Computaton modl w dscuss th problm of mnmzng th nrgy consumpton consdrng that thr s a mnmum total rward that must b dlvrd by th systm as wll as tm constrants n th form of dadlns that must b mt. In what follows w prsnt th prcs formulaton of rlatd problms as wll as th partcular problm addrssd n ths papr. Rcall that th task xcuton ordr s prdtrmnd, wth T bng th -th task n ths squnc.

4 Statc V/O Assgnmnt: Fnd, for ach task T, 1 n, th voltag V and th numbr of optonal cycls O that mnmz C r (V 1 V ) 2 + C =1 V 2 (M + O ) (5) 1, V subjct to V mn V V (6) V s +1 = t = s +p V 1 V + k (V V th ) α(m +O ) d (7) δ 1, V τ R (O ) R mn (8) =1 Th abov formulaton can b xpland as follows. Th objctv functon to b mnmzd s th total nrgy, whch s th sum of th voltag-swtchng nrgs 1, V and th nrgy consumd by ach task (q. (5)). Th voltag V for ach task T must b n th rang [V mn, V ] (q. (6)). Th complton tm t s th sum of th start tm s, th voltag-swtchng tm δ 1,, V and th xcuton τ, and tasks must complt bfor thr dadlns d (q. (7)); not that th worst-cas numbr of mandatory cycls has to b assumd n ordr to guarant th dadlns. Th total rward has to b at last R mn (q. (8)). Whn solvng th abov problm, for tractablty rasons, w consdr O as a contnuous varabl and thn round th rsult down. By ths, wthout gnratng th optmal soluton, w obtan a soluton that s vry nar to th optmal on bcaus on clock cycl s a vry fn-grand unt (tasks xcut typcally hundrds of thousands of clock cycls) [3]. It can also b notd that n th abov problm th objctv as wll as th constrant functons ar convx 3. Thrfor w hav a convx non-lnar programmng (NLP) formulaton [11] and hnc th problm can b solvd usng polynomaltm mthods [8]. Dynamc V/O Schdulr: Th followng s th problm that a dynamc V/O schdulr must solv vry tm a task T c complts. It s consdrd that tasks T 1,..., T c hav alrady compltd (th rward producd up to th complton of T c s RP c and th complton tm of T c s t c ). Fnd V and O, for c + 1 n, that mnmz dyn + V 1, + C V 2 (M + O ) =c+1 subjct to V mn V V (10) s +1 = t = s +δ dyn +δ V R (O ) R mn RP c =c+1 1, +k V (V V th ) α(m +O ) d τ (9) (11) (12) whr δ dyn and dyn ar th tm and nrgy ovrhad of computng dynamcally V and O for task T. Th problm solvd by th abov dynamc V/O schdulr corrsponds to an nstanc of th statc V/O assgnmnt problm (for c + 1 n and takng nto account t c and RP c ), but consdrng δ dyn and dyn. Howvr, a spculatv vrson of th dynamc V/O schdulr can b formulatd as follows. Such a dynamc spculatv V/O schdulr producs bttr rsults than ts non-spculatv countrpart, as shown by th xprmntal rsults of Scton 6. 3 Obsrv that th functon abs cannot b usd drctly n mathmatcal programmng bcaus t s not dffrntabl n 0. Howvr, thr xst tchnqus for obtanng quvalnt formulatons [1]. Dynamc Spculatv V/O Schdulr: Th followng s th problm that a dynamc spculatv V/O schdulr must solv vry tm a task T c complts. It s consdrd that tasks T 1,..., T c hav alrady compltd (th rward producd up to th complton of T c s RP c and th complton tm of T c s t c). Fnd V and O, for c + 1 n, that mnmz dyn + V 1, + C V 2 (M + O ) =c+1 (13) subjct to V mn V V (14) s +1 = t = s +δ dyn +δ V R (O ) R mn RP c =c+1 s +1 = t = s + δdyn 8 > < τ = >: 1, +k V (V V th ) α(m +O ) d (15) τ (16) + δ V 1, + τ d (17) V k (V V th ) α(m + O ) f = c + 1 V k (V V th ) α(m + O ) f > c + 1 (18) whr δ dyn and dyn ar, rspctvly, th tm and nrgy ovrhad of computng dynamcally V and O for task T. qs. (13)-(16) ar bascally th sam as qs. (9)-(12) xcpt that th xpctd numbr of mandatory cycls M s usd nstad of th worst-cas numbr of mandatory cycls M n th constrant corrspondng to th dadlns (s qs. (11) and (15)). Th constrant gvn by q. (15) dos not guarant by tslf th satsfacton of dadlns bcaus f th actual numbr of mandatory cycls s largr than M, dadln volatons mght ars. Thrfor an addtonal constrant, as gvn by qs. (17) and (18), s ntroducd. It xprsss that: th nxt task T c+1, runnng at V c+1, must mt ts dadln (T c+1 wll run at th computd V c+1 ); th othr tasks T, c + 1 < n, runnng at V, must also mt th dadlns (th othr tasks T mght run at a voltag dffrnt from th valu V computd n th currnt traton, bcaus solutons obtand upon complton of futur tasks mght produc dffrnt valus). Guarantng th dadlns n ths way s possbl bcaus nw assgnmnts ar smlarly rcomputd vry tm a task fnshs. Th dynamc spculatv V/O schdulr prsntd abov solvs th V/O assgnmnt problm spculatng that tasks wll xcut thr xpctd numbr of mandatory cycls but lavng nough room for ncrasng th voltag so that futur tasks, f ndd, run fastr and thus mt th dadlns. W consdr that th nrgy dal consumd by a systm, whn th V/O assgnmnts ar computd by such a dynamc spculatv V/O schdulr n th dal cas δ dyn = 0 and dyn = 0, s th lowr bound on th total nrgy that can practcally b achvd wthout knowng n advanc how many mandatory cycls tasks wll xcut and wthout accptng rsks rgardng dadln of rward volatons. Although th dynamc V/O assgnmnt problm can b solvd n polynomal-tm, th tm and nrgy for solvng t ar n practc vry larg and thrfor unaccptabl at runtm for practcal applcatons. In our approach w prpar off-ln a numbr of V/O assgnmnts, on of whch s to b slctd by th QS V/O schdulr. Upon fnshng a task T c, th QS V/O schdulr chcks th complton tm t c and th rward RP c producd up to complton of T c, and looks up an assgnmnt n LUT c. From th lookup tabl LUT c th QS V/O schdulr gts th pont (t c, RP c), whch s th closst to (t c, RP c) such that

5 t c t c and RP c RP c, and slcts V /O corrspondng to (t c, RP c). Th goal s to mak th systm consum as lttl nrgy as possbl, whn usng th assgnmnts slctd by th QS V/O schdulr. Th problm w dscuss n th rst of th papr s: St of V/O Assgnmnts: Fnd a st of N assgnmnts such that: N N ; th V/O assgnmnt slctd by th QS V/O schdulr guarants th dadlns (s P +δ sl +δ 1,+ V n τ = t d ) and th rward constrant ( =1 R(O) R mn ), and so that th total nrgy qs s mnmal. As dscussd n Scton 5, for a task T, potntally thr xst nfntly many possbl valus for t and RP. Thrfor, n ordr to approach th thortcal lmt dal, t would b ndd to comput an nfnt numbr of V/O assgnmnts, on for ach (t, RP ). Th problm s thus how to slct at most N ponts n ths nfnt spac such that th nrgy consumd, whn usng th rspctv V/O assgnmnts, s as clos as possbl to dal. 5 Computng th St of V/O Assgnmnts For ach task T, thr s a tm-rward spac of possbl valus of complton tm t and rward RP producd up to complton of T, as dpctd n Fg. 2. ach pont n ths spac dfns a V/O assgnmnt for th nxt task T +1: f T fnshd at t a and th producd rward was RP a, th assgnmnt for th nxt task would b V +1 = V a /O +1 = O a (that s, T +1 would run at V a and xcut O a optonal cycls). Th valus V a and O a ar thos that an dal dynamc spculatv V/O schdulr would gv n th cas t = t a, RP = RP a (rcall that w am at matchng dal whch s th nrgy consumd whn th V/O assgnmnts producd by such an dal dynamc spculatv V/O schdulr ar usd). Dffrnt ponts (t, RP ) dfn dffrnt V/O assgnmnts as shown n Fg. 2. Not also that for a gvn valu t thr mght b dffrnt vald valus of RP, and ths s du to th fact that dffrnt prvous V/O assgnmnts can lad to th sam t but stll dffrnt RP. ach task T j xcuts Mj bc cycls at V. Th ntrvals [t mn, t ] and [0, RP ] bound th tm-rward spac as shown n Fg. 3. RP RP t mn t Fg. 3. Boundars of th tm-rward spac A gnrc charactrzaton of th tm-rward spac s not possbl bcaus rward functons vary from task to task as wll as from systm to systm: w cannot drv a gnral xprsson that rlats th rward R wth th xcuton tm τ and hnc a charactrzaton of th t -RP spac s not possbl. On altrnatv for slctng ponts n th tm-rward spac would b to consdr a msh-lk confguraton, n whch th spac s dvdd n rctangular aras and ach ara s covrd by on pont (th lowr-rght cornr covrs th rctangl) as dpctd n Fg. 4. Th drawback of ths approach s twofold: frst, th boundars n Fg. 3 dfn a tm-rward spac that nclud ponts that cannot happn, for xampl, th pont (t mn, RP ) s not fasbl bcaus t mn occurs whn no optonal cycls ar xcutd whras RP rqurs all tasks T j xcutng Oj optonal cycls; scond, th numbr of rqurd ponts for covrng th spac s a quadratc functon of th granularty of th msh, whch mans that too many ponts mght b ncssary for achvng an accptabl granularty. RP RP t RP V O a +1= V a +1= O a a ( t, RP ) V O b t +1= V b +1= O b b (, RP ) t mn Fg. 4. Ponts n a msh confguraton t t Fg. 2. Tm-rward spac In ordr to slct N ponts n th t -RP spac and accordngly comput th st of N assgnmnts, t s frst ndd to dtrmn th boundars of ths spac (for ach task T ). N s th numbr of assgnmnts to b stord n th lookup tabl LUT, aftr dstrbutng th mum numbr N of assgnmnts among tasks. Th boundars of th t -RP spac can b obtand by computng th xtrm valus of t (arlst and latst complton tms) and of RP (mnmum and mum rward producd up to task T ), consdrng V mn, V, M bc j, Mj, and Oj, P 1 j. Th - = P j=1 Rj(O j ) and th = j=1 Rj(0) = 0. Th occurs whn ach task T j x- mum producd rward s RP mnmum rward s smply RP mn mum complton tm t cuts M j + O j cycls at V mn, whl t mn t happns whn W hav optd for a soluton whr w frz th assgnd optonal cycls, that s, for ach task T w fx O to a valu O computd off-ln. Thus, n th soluton proposd n ths papr, for any actvaton of th systm, T wll nvarably xcut O optonal cycls. In ths way, w transform th orgnal problm nto a classcal voltag-scalng problm wth dadlns snc th only varabls now ar V. Ths mans that w rduc th bdmnsonal tm-rward spac nto a on-dmnson spac (tm s now th only dmnson). Ths approach gvs vry good rsults as shown by th xprmntal valuaton prsntd n Scton 6. By frzng th optonal cycls O, although th spac of solutons s constrand, good rsults can b achvd bcaus th only varabl that affcts th rward dmnson s O and, for ths problm, th rward s a constrant. Thus, f th optonal cycls ar frozn n such a way that th rward constrant s satsfd, thr s stll nough room for xplotng th dynamc slack causd by tasks xcutng lss mandatory

6 cycls than n th worst cas: thr s no gan by runnng mor optonal cycls and accordngly producng mor rward than rqurd by th rward constrant. Th way w obtan th fxd valus O s th followng. W consdr th nstanc of th problm as formulatd by qs. (13)-(18) that th dynamc spculatv V/O schdulr solvs at th vry bgnnng, bfor any task s xcutd (c = 0). Th soluton gvs partcular valus of V and O, 1 n. For ach task, th numbr of optonal cycls gvn by ths soluton s takn as th fxd valu O n our approach. Onc th numbr of optonal cycls has bn fxd to O, th only varabls ar V and th problm bcoms that of voltag scalng for nrgy mnmzaton wth tm constrants. For th sak of compltnss, w nclud blow ts formulaton. Th rward constrant dsappars from th formulaton bcaus, by fxng th optonal cycls as xpland abov, t s guarantd that th total rward wll b at last R mn. Dynamc Voltag Schdulr: Th followng s th problm that a dynamc voltag schdulr must solv vry tm a task T c complts. It s consdrd that tasks T 1,..., T c hav alrady compltd (th complton tm of T c s t c ). Fnd V, for c + 1 n, that mnmz dyn + V 1, + C V 2 (M + O ) =c+1 (19) subjct to V mn V V (20) s +1 = t = s +δ dyn +δ V s +1 = t = s + δdyn 8 > < τ = >: 1, +k V (V V th ) α(m +O ) d (21) τ + δ V 1, + τ d (22) V k (V V th ) α(m + O ) f = c + 1 V k (V V th ) α(m + O ) f > c + 1 whr δ dyn and dyn computng dynamcally V for task T. (23) ar th tm and nrgy ovrhad of In our QS approach, onc th numbr of assgnd optonal cycls has bn frozn to O, w tak N ponts t j along th ntrval [t mn, t ]; th arlst complton tm t mn occurs whn ach of th prvous tasks T j (nclusv T ) xcut thr mnmum numbr of cycls Mj bc and O j optonal cycls at mum voltag V, whl t occurs whn ach task T j xcuts Mj + O j cycls at V mn. Thn w comput and stor th rspctv voltag sttngs V j +1 that mnmz th total nrgy whn T complts at t j, accordng to th formulaton gvn by qs. (19)-(23). It should b notd that for th computaton of th voltag V j +1, th tm and nrgy ovrhads δ dyn = δ sl and dyn = sl (ndd for slctng voltags at run-tm) ar takn nto account. ach on of th ponts, togthr wth ts corrspondng assgnmnt, covrs a rgon as ndcatd n Fg. 5. Th QS schdulr slcts on of th stord assgnmnts basd on th actual complton tm. If, for xampl, task T complts at t, t a < t t b, th QS V/O schdulr wll slct th prcomputd assgnmnt V b /O. Not that w hav ncludd n Fg. 5 th optonal cycls O for th sak of makng clarr th natur of our approach. Howvr, n practc, thr s no nd to stor th numbr of optonal cycls n th lookup tabls LUT snc, onc ths ar frozn, task T wll nvarably xcut O optonal cycls. Th psudocod corrspondng to th computatons prformd off-ln for obtanng th st of assgnmnts s gvn by Algorthm 1. Frst, th mum numbr N of as- Condton V +1 O +1 f t t a V a O ls f t t b V b O ls f t t c V c O ls V d O RP RP = Σ k=1 R k ( O k ) t mn Fg. 5. Illustraton of a lookup tabl t a t b t c sgnmnts that ar to b stord s dstrbutd among tasks (ln 1). A straghtforward approach s to dstrbut thm unformly among th dffrnt tasks, so that ach lookup tabl contans th sam numbr of assgnmnts. Howvr, t s mor ffcnt to dstrbut th assgnmnts accordng to th sz of th ntrval [t mn, t ], n such a way that th lookup tabls of tasks wth largr ntrvals gt mor ponts. Thn w comput th soluton of th problm formulatd by qs. (13)-(18) for c = 0 and w frz th numbr of optonal cycls accordng to ths soluton (ln 2). Snc th assgnmnt V 1 s nvarably th sam (task T 1 runs always at th sam voltag lvl), ths s th only on stord for th frst task (ln 3). Th valu V 1 s takn from th soluton obtand n ln 2. For vry task T, 1 n 1, w comput th ntrval [t mn, t ] (ln 5): t mn s th sum of xcuton tms τk mn gvn by q. (3) wth V, Mk bc, and O k and tm ovrhads δ k ; t s th sum of xcuton tms τk gvn by q. (3) wth V mn, Mk, and O k and tm ovrhads δ k. W tak thn N qually-spacd ponts t j along [tmn, t ] (ln 7) and, for ach such pont, w comput th rspctv assgnmnt V j +1 (w solv th dynamc voltag scalng problm as formulatd by qs. (19)-(23) assumng that th complton tm of T s t j ) and stor t accordngly n th j-th poston n th lookup tabl LUT (ln 8). nput: Th mum numbr N of assgnmnts output: Th st of assgnmnts t 1: dstrbut N among tasks (T gts N ponts) 2: solv nstanc c = 0 of th problm gvn by qs. (13)- (18); tak th soluton and mak O := O, 1 n 3: stor V 1 n LUT 1 [1] 4: for 1, 2,..., n 1 do 5: t mn δ k 6: for j 1, 2,..., N do 7: t j := [(N j)t mn P := k=1 τ k mn + δ k ; t := P k=1 τ k + + j t ]/N 8: comput V j +1 for tj and stor t n LUT [j] 9: nd for 10: nd for Algorthm 1: Off-ln phas Th st of assgnmnts, prpard off-ln, s usd on-ln by th QS schdulr as outlnd by Algorthm 2. Ths algorthm s calld vry tm a task complts: upon fnshng task T, th lookup tabl LUT s consultd. Th ndx j of th tabl ntry s calculatd vry asly (ln 1). Computng drctly th ndx j, nstad of sarchng through th tabl LUT, s possbl bcaus th ponts t j stord n LUT ar qually-spacd. Th voltag sttng stord n LUT [j] s rtrvd (ln 2) and fnally th voltag at whch task T +1 must run as wll as th numbr of optonal cycls t must xcut s rturnd as a V/O assgnmnt (ln 3). Notc that Algorthm 2 has a tm complxty O(1), whch mans that t

7 th on-ln opraton prformd by th QS schdulr taks constant tm and nrgy. Mor mportantly, du to th smplcty of th algorthm, ths lookup and slcton procss s svral ordrs of magntud chapr than th on-ln computaton by th dynamc spculatv V/O schdulr. nput: Actual complton tm t of T and lookup tabl LUT as wll as fxd optonal cycls O +1 output: Th assgnmnt V +1 /O +1 for th nxt task T +1 1: j := N (t t mn )/(t t mn ) 2: V +1 := assgnmnt stord n LUT [j]; O +1 := O +1 3: rturn V +1 /O +1 Algorthm 2: On-ln phas In summary, n our QS soluton to th problm of mnmzng nrgy subjct to tm and rward constrants, w frst fx off-ln th numbr of optonal cycls assgnd to ach task, by takng th valus O as gvn by th soluton to th problm formulatd by qs. (13)-(18) (nstanc c = 0). Thus th orgnal problm s rducd to QS voltag scalng for nrgy mnmzaton. Th voltag-scalng problm n a QS framwork had prvously bn dscussd by Andr t al. [2]. In th on-dmnson spac of possbl complton tms, w slct ponts and comput th corrspondng voltag assgnmnts as dscussd abov. For ach task, a numbr of voltag sttngs ar stord n ts rspctv lookup tabl. Not that ths tabls contan only voltag valus as th numbr of optonal cycls has alrady bn fxd off-ln. At run-tm, th voltag valus ar smply obtand by consultng th rspctv lookup tabl ach tm a task complts (rcall that th voltag sttng rad from th tabl dpnds on th complton tm); ths voltag valu togthr wth th numbr of optonal cycls fxd off-ln mak up th V/O assgnmnt for th nxt task. 6 xprmntal valuaton Th approach proposd n ths papr has bn valuatd through a larg a larg numbr of xprmnts usng numrous synthtc bnchmarks. Such synthtc xampls corrspond to randomly gnratd task graphs that contan btwn 10 and 100 tasks. vry pont n th plots of th xprmntal valuaton prsntd n ths Scton (Fgs. 6 through 9) corrsponds to th avrag of th rsults of 75 synthtc task graphs, rsultng ovrall n mor than 2500 prformd xprmnts. W adoptd th tchnology-dpndnt paramtrs from [6], whch corrspond to a procssor n a 0.18 µm CMOS fabrcaton procss. Th rward functons w usd along th xprmnts ar of th form R (O ) = α O + β O + γ 3 O, wth coffcnts α, β, and γ chosn randomly. Th frst st of xprmnts valdats th clam that th dynamc spculatv V/O schdulr (whch solvs th problm formulatd by qs. (13)-(18)) outprforms th nonspculatv on (whch solvs th problm formulatd by qs. (9)-(12)). Fg. 6 shows th avrag nrgy savngs (rlatv to a statc V/O assgnmnt) as a functon of th dadln slack (th rlatv dffrnc btwn th dadln and th complton tm whn worst-cas numbr of mandatory cycls ar xcutd at th mum voltag such that th rward constrant s guarantd). Th hghst savngs can b obtand for systms wth small dadln slack: th largr th dadln slack s, th lowr th voltags gvn by a statc assgnmnt can b (tasks can run slowr), and thrfor th dffrnc n nrgy consumd by a statc and a dynamc soluton s smallr. Th xprmnts whos rsults ar prsntd n Fg. 6 wr prformd consdrng th dal cas of zro tm and nrgy on-ln ovrhads and show clarly that a dynamc spculatv V/O schdulr prforms bttr (that s, producs hghr nrgy savngs) than ts non-spculatv countrpart. Avrag nrgy Savngs [%] Dadln Slack [%] Spculatv Non-spculatv Fg. 6. Comparson of th spculatv and non-spculatv dynamc V/O schdulrs In a scond st of xprmnts w valuatd th QS approach proposd n ths papr, n trms of th nrgy savngs achvd by t wth rspct to th optmal statc soluton. In ths st of xprmnts w dd tak nto consdraton th tm and nrgy ovrhads ndd for slctng th voltag sttngs among th prcomputd ons. In ths xprmnts w consdr that th tm and nrgy ovrhads ndd for slctng th assgnmnts by th QS schdulr ar δ sl = 450 ns and sl = 400 nj. Ths ar ralstc valus as slctng a prcomputd assgnmnt taks only tns of cycls and th accss tm and nrgy consumpton (pr accss) of, for xampl, a low-powr Statc RAM ar around 70 ns and 20 nj rspctvly [7]. Fg. 7(a) shows th nrgy savngs by our QS approach for thr cass: 2, 5, and 50 ponts (assgnmnts stord n th lookup tabls) pr task. Mor ponts pr task produc naturally hghr nrgy savngs but vn wth a coupl of ponts pr task, as shown by th plot, vry sgnfcant nrgy savngs can b achvd (clos to 20% for systms wth tght dadlns). Fg. 7(b) also shows th nrgy savngs achvd by th QS approach, but ths tm as a functon of th rato btwn th worst-cas numbr of cycls M and th bst-cas numbr of cycls M bc. In ths xprmnts w hav consdrd systms wth a dadln slack of 10%. As th rato M /M bc ncrass, th dynamc slack bcoms largr and thrfor thr s mor room for xplotng t n ordr to rduc th total nrgy consumd by th systm. In a thrd st of xprmnts w valuatd th qualty of th soluton gvn by th QS approach prsntd n ths scton wth rspct to th thortcal lmt that could b achvd wthout knowng n advanc th actual numbr of xcuton cycls (th nrgy consumd whn a dynamc spculatv V/O schdulr s usd, n th dal cas of zro ovrhads δ dyn = 0 and dyn = 0). In ordr to mak a far comparson, n ths st of xprmnts, w consdrd also zro ovrhads for th QS approach (δ sl = 0 and sl = 0). Fg. 8 shows th dvaton dv = ( qs dal )/ dal as a functon of th numbr of prcomputd voltags (ponts pr task), whr dal s th total nrgy consumd for th cas of an dal dynamc spculatv V/O schdulr and qs s th total nrgy consumd for th cas of a QS schdulr that slcts voltags from lookup tabls prpard as xpland n

8 Avrag nrgy Savngs [%] Avrag nrgy Savngs [%] Dadln Slack [%] QS (50 ponts/task) QS (5 ponts/task) QS (2 ponts/task) (a) Influnc of th dadln slack Rato M /M bc QS (50 ponts/task) QS (5 ponts/task) QS (2 ponts/task) (b) Influnc of th rato M /M bc Fg. 7. Comparson of th QS and statc solutons Scton 5. In ths st of xprmnts w hav consdrd systms wth dadln slack of 20%. It must b notd that qs corrsponds to th proposd QS approach n whch w fx th numbr of optonal cycls and th prcomputd assgnmnts ar only voltag sttngs, whras dal corrsponds to th dynamc V/O schdulr that rcomputs both voltag and numbr of optonal cycls vry tm a task complts. vn so, wth rlatvly fw ponts pr task t s possbl to gt vry clos to th thortcal lmt, for nstanc, for 20 ponts pr task th avrag dvaton s just 0.4%. Avrag Dvaton [%] Numbr of Ponts pr Task Fg. 8. Comparson of th QS and dal dynamc solutons Fnally, n a fourth st of xprmnts w took nto consdraton ralstc valus for th on-ln ovrhads δ dyn and dyn (ndd for rcomputng at run-tm voltag valus and numbr of optonal cycls) of th dynamc spculatv V/O schdulr as wll as th on-ln ovrhads δ sl and sl (ndd for lookng up th tabls and slctng on of th prcomputd assgnmnts) of th QS schdulr. Th ovrhad valus usd n ths xprmnts wr takn from [10], whr hurstc mthods wr usd for solvng a smlar problm. Fg. 9 shows th avrag nrgy savngs by th dynamc and QS approachs (takng as basln th nrgy consumd whn usng a statc approach). It shows that n practc th dynamc approach maks th nrgy consumpton hghr than n th statc soluton (ngatv savngs), a fact that s du to th hgh ovrhads ncurrd by computng on-ln assgnmnts by th dynamc V/O schdulr. Also bcaus of th hgh ovrhads, whn th systm has tght dadlns, th dynamc approach cannot vn guarant th tm constrants. On th contrary, th QS approach succds n xplotng th dynamc slack and thus rducng th nrgy consumpton bcaus of ts low on-ln ovrhads. Avrag nrgy Savngs [%] QS (5 ponts/task) Statc Dynamc Dadln Slack [%] Fg. 9. Comparson consdrng ralstc ovrhads 7 Conclusons W hav addrssd th problm of mnmzng th nrgy consumpton for ral-tm systms wth both rward and tm constrants n th fram of Imprcs Computaton systms. To th bst of our knowldg ths s th frst approach prsntd for ths partcular problm. Th proposd approach has as chf mrt th ablty to ffctvly xplot th dynamc slack, causd by tasks xcutng lss clock cycls than n th worst cas. Such a QS approach succds n xplotng th dynamc slack, yt ncurrng a vry low on-ln ovrhad, bcaus th complx tm- and nrgyconsumng parts of th computatons ar prformd off-ln, at dsgn-tm, lavng for run-tm only smpl lookup and slcton opratons. Th valuaton of our soluton has bn prformd usng a larg numbr of synthtc bnchmarks. Ths hav shown that sgnfcant rductons n th nrgy consumpton can b achvd wth our tchnqu, for nstanc, nrgy savngs of around 20% for systms wth tght dadlns. Rfrncs [1] A. Andr, M. Schmtz, P. ls, Z. Png, and B. Al- Hashm. Ovrhad-Conscous Voltag Slcton for Dynamc and Lakag nrgy Rducton of Tm-Constrand Systms. In Proc. DAT, pp , [2] A. Andr, M. Schmtz, P. ls, Z. Png, and B. Al-Hashm. Quas-Statc Voltag Scalng for nrgy Mnmzaton wth Tm Constrants. In Proc. DAT, pp , [3] L. A. Cortés. Vrfcaton and Schdulng Tchnqus for Ral-Tm mbddd Systms. PhD thss, Dpt. Computr and Informaton Scnc, Lnköpng Unvrsty, Mar [4] L. A. Cortés, P. ls, and Z. Png. Quas-Statc Assgnmnt of Voltags and Optonal Cycls for Maxmzng Rwards n Ral-Tm Systms wth nrgy Constrants. In Proc. DAC, pp , [5] J. W. S. Lu, W.-K. Shh, K.-J. Ln, and R. Bttat. Imprcs Computatons. Proc. I, 82(1):83 94, Jan [6] S. M. Martn, K. Flautnr, T. Mudg, and D. Blaauw. Combnd Dynamc Voltag Scalng and Adaptv Body Basng for Low Powr Mcroprocssors undr Dynamc Workloads. In Proc. ICCAD, pp , [7] NC Mmors. [8] Y. Nstrov and A. Nmrovsk. Intror-Pont Polynomal Algorthms n Convx Programmng. SIAM, Phladlpha, PA, [9] T. Okuma, H. Yasuura, and T. Ishhara. Softwar nrgy Rducton Tchnqus for Varabl-Voltag Procssors. I Dsgn & Tst of Computrs, 18(2):31 41, Mar [10] C. Rusu, R. Mlhm, and D. Mossé. Maxmzng Rwards for Ral-Tm Applcatons wth nrgy Constrants. ACM Trans. on mbddd Computng Systms, 2(4): , Nov [11] S. A. Vavass. Nonlnar Optmzaton: Complxty Issus. Oxford Unvrsty Prss, Nw York, NY, 1991.