Energy-Efficient Scheduling Fixed-Priority tasks with Preemption Thresholds on Variable Voltage Processors

Transcription

1 Energy-Effcent Schedulng Fxed-Prorty tasks wth Preempton Thresholds on Varable Voltage Processors XaoChuan He, Yan Ja Insttute of Network Technology and Informaton Securty School of Computer Scence Natonal Unversty of Defense Technology Changsha, Chna Abstract. Slowdown factors determne the extent of slowdown a computng system can experence based on functonal and performance requrements. 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. We address the problem of computng statc and dynamc slowdown factors n the FPPT algorthm. In ths paper, Suffcent constrants have been dentfed for the feasblty of the task set usng slowdown factors. We formulate ths problem of computng the statc slowdown factors for tasks as an nonlnear optmzaton problem to mnmze the total energy consumpton of the system. Our smulaton experments show on an average 17% 53% energy gans over FPPT schedulng polcy. 1 Introducton Dynamc Voltage Scalng (DVS), whch adjusts the supply voltage and ts correspondng clock frequency dynamcally, s an 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 technques for reducng the energy consumpton. There exst two prmary ways of reducng the power consumpton n embedded computng systems: processor shutdown and processor slowdown. Slowdown usng frequency and voltage scalng has been shown to be effectve n reducng the processor energy consumpton [1], [2], [3]. It s known that preemptablty s a necessary requrement to acheve hgher processor utlzaton and optmal slowdown [2],[4]. However, preemptve schedulng has ts addtonal costs as compared to non-preemptve schedulng. Though preemptve schedulng acheves hgher utlzaton, t s not always requred to preempt a lower prorty task. Fxed-Prorty schedulng wth preempton threshold (FPPT) [5],[6] allows a task to dsable preemptons from tasks up to a specfed preempton threshold prorty. Tasks wth a prorty greater than the preempton threshold prorty are

2 2 stll allowed to preempt. The preempton threshold schedulng model has been shown to reduce the run-tme costs by elmnatng unnecessary task preemptons. Furthermore, FPPT allows tasks to be parttoned nto non-preemptve groups to mnmze the number of threads [6] and the stack memory requrement [7], thereby leadng to scalable real-tme systems. Fxed-Prorty schedulng wth preempton threshold (FPPT) elmnates unnecessary context swtches, thereby savng energy. Slowdown usng frequency and voltage scalng s more effectve n reducng the energy consumpton [4], [2], [8], [1]. In ths paper, we ntegrate processor slowdown technques wth FPPT to enable scalable and energy effcency real-tme systems. Gven task wth predefned prortes and correspondng preempton thresholds, we propose an algorthm to compute the statc slowdown factors by formulatng the problem as a lnear optmzaton problem. Furthermore, we consder the energy consumpton of task set under dfferent preempton threshold assgnments. We gan as much as 17% 53% energy savngs over the usual FPPT schedulng algorthm. The remander of the paper s organzed as follows. The next secton presents the assumed computaton model. The related work s ntroduced n Secton 2. Secton 3 we formulate the computaton of slowdown factors as an nonlnear optmzaton problem. The expermental results are gven n Secton 4. Fnally, the concluson s made n Secton 5. 2 Prelmnares In ths secton, we ntroduce the necessary notaton and formulate the problem. We frst descrbe the system model followed by the related works. 2.1 System Model Ths study deals wth the fxed prorty preemptve schedulng of tasks n a realtme 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 Γ of cardnalty n, Γ = {τ 1, τ 2,..., τ n }. A perodc task τ s characterzed by a 3-tuple (C, T, D ) where each request of τ, called nstance, has an executon tme of C, a relatve deadlne D. T tme unts separate two consecutve nstances of τ (hence T s the perod of the task). Note that C s the worst case executon tme (WCET) of the task at maxmum speed, gven that t s the only task runnng n the system. The system s sad schedulable f each nstance fnshes before ts deadlne. Each task s assocated wth a deadlne D, that D may be arbtrarly large. Ths means that many nstances of the same task can be actve (n the ready queue) at the same tme. We assume that the scheduler handles tasks wth the same prortes usng a FIFO rule. Hence an early nstance of a task has prorty over a later, and must be completed before the later nstance s allowed to start. We further assume that the total utlzaton of all tasks, U, s strctly less than 1. Ths wll later be shown to be a necessary condton for the analyss.

3 3 Each task s assgned wth a prorty π and preempton threshold γ, and γ π. The smallest number s gven to the task wth hghest prorty. 2.2 Varable Speed Processors A wde range of processors lke the Intel XScale [9] and Transmeta Crusoe [10] support varable voltage and frequency levels. Voltage and frequency levels are tghtly coupled. The mportant pont to note s when we perform a slowdown we change both the frequency and voltage of the processor. Gven the mnmum frequency f mn and the maxmum supported frequency f max, we normalze the speed to the maxmum frequency to have dscrete ponts n the operatng range [η mn, 1], where η mn = f mn /f max. The slowdown factor can be vewed as the normalzed frequency. At a gven nstance, t s the rato of the scheduled frequency to the maxmum frequency of the processor. 2.3 Related Works Prevous nvestgatons on the voltage schedulng problem have focused manly on real-tme jobs runnng under dynamc-prorty schedulng algorthms such as the EDF (earlest-deadlne-frst) algorthm [4], [11], [12], [13]. For example, the problem of energy-optmal EDF schedulng has been well understood. For EDF job sets, the algorthm by Yao et al. [14] computes the energy-optmal voltage schedules n polynomal tme. Although the EDF schedulng polcy makes the voltage schedulng problem easer to solve, fxed-prorty schedulng algorthms such as the RM (rate monotonc) algorthm are more commonly used n practcal real-tme systems due to ther low overhead and predctablty [15]. Most of the earler work dealt wth general fxed-prorty task sets. Shn et al. [1] have computed unform slowdown factors for an ndependent perodc task set. Jejurka and Gupta [3] proposed an algorthm to compute a near optmal constant slowdown factor based on the bsecton method, and furthermore for the case of tasks wth varyng power characterstcs, [3] develop the computaton of near optmal slowdown factors as a soluton to convex optmzaton problem usng the ellpsod method. Yao, Demers and Shanker [14] presented an optmal off-lne speed schedule for a set of N jobs. An optmal schedule for tasks wth dfferent power consumpton characterstcs s consdered by Aydn, Melhem and Mosse [4]. The same authors [2] prove that the processor utlzaton (at maxmum speed) s the optmal slowdown factor when the deadlne s equal to the perod. Quan and Hu [16] [17] dscuss off-lne algorthms for the case of fxed prorty schedulng. Snce the worst case executon tme (WCET) of a task s not usually reached, there s dynamc slack n the system. Plla and Shn [14] recalculate the slowdown to use the dynamc slack whle meetng the deadlnes. Low-power schedulng usng slack reclamaton heurstc s studed by Aydn et al. [2] and Km et al. [12]. Schedulng of task graphs on multple processors has also been consdered. Luo and Jha [18] have consdered schedulng of perodc and aperodc task

4 4 graphs n a dstrbuted system. Non preemptve schedulng of a task graph on a mult processor system s consdered by Gruan and Kuchcnsk [19]. Zhang et al. [8] have gven a framework for task schedulng and voltage scalng of dependent tasks on a mult-processor system. They have formulated the voltage scalng problem as an nteger programmng problem. 3 Statc Slowdown Factors We compute statc slowdown factor for a system wth an underlyng fxed-prorty wth preempton threshold (FPPT) scheduler. For a task τ n FPPT, the k th nstance of τ s denoted as J k. Before Jk begn executon, the nterference caused by other tasks wth regular prorty hgher than π s possble; Once J k starts actually, only the tasks wth regular prortes hgher than γ can preempt J k. Thus the startng tme of J k play a necessary role n the analyss of FPPT, so a notaton S k s ntroduce to denote t, S k s used to represent the startng pont of J k, F k denote the fnsh tme of J k. 3.1 FPPT Scheduler The exstng schedulablty analyss for FPPT [5], [20], [21]s based on the FPPT crtcal nstant [21] whch says that f a task meets ts deadlne whenever the task s requested smultaneously wth requests for all hgher prorty tasks and the lower prorty task that contrbute the largest blockng tme to t, furthermore all of the task nstance n the level- busy perod must meets ts deadlne too, then the deadlne wll always be met for all task phrasngs. The WCRT of a task τ can be derved through the followng equatons [20]: B = max τ j Γ {C j π > π j π γ j } (1) L = B + L C j j,π j π (2) S k = B + k C + F k = S k + C + R = j,π j>γ j,π j>π max k=0,1,2,..., L ( F k ( ) 1 + Sk C j (3) ( )) 1 + Sk C j (4) (F k k T ) (5) In the course of schedulablty analyss of FPPT, as for task τ, WCRT of τ, R can be derved. At the same tme, the specfc nstance of τ whch contrbute ths WCRT can be derved too. Ths nstance s denoted as J kr. Assumng the task are ndependent, let be the slowdown factors for task τ. As we know, the allowably fnsh tme of J kr s D + k r T, thus equaton 4,3 s rewrtten as followng:

5 5 S kr D + k r T = S kr = B η b + k r C + + C + j,π j>γ j,π j>π ( ( 1 + Skr D + k r T ) Cj η j (6) ( )) 1 + Sk Cj (7) η j In the equaton 6,7, S kr s calculated n the course of schedulablty analyss, all of other terms are known already except for the slowdown factors, denoted as, = 1, 2,..., n. From the combnaton of equaton 6 and 7, the followng theorem can be derved. Theorem 1. A task set of n ndependent fxed-prorty perodc tasks, scheduled wth preempton thresholds, s feasble at a slowdown factor of for task τ f the followng relatons satsfy: D + k r T B η b + ( C ) S 1 + kr C j η j + j,π j>π, π j γ + j,π j>γ ( D+k rt ) Cj η j (8) where τ b s the blockng task whch contrbute to the longest blockng tme to τ. In the equaton 8, term ( ) D+k rt Cj η j j,π j>γ represent the workload of the tasks wth the ( regular ) prorty hgher than γ n the nterval of [0, D +k r T ], term S 1 + kr C j represent the workload of the tasks wth the regular j,π <π j γ prorty hgher than π before J kr η j these tasks can not preempt J kr start the executon. once J kr agan. The term B η b started actually, represent the blockng tme that the task wth the prorty lower than π contrbutes. The term C the workload of J kr tself. represent 3.2 Computng Slowdown Factors Power Characterstcs The number of cycles, C that a task τ needs to complete s a constant durng voltage scalng. The processor cycle tme, the task delay and the dynamc power consumpton of a task vary wth the supply voltage V DD. The power delay characterstcs of the CMOS technology [21] are as gven below. P d = C eff V 2 DD f (9) f = αk (V DD V TH ) α V DD (10)

6 6 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 of the voltage, adjustng both can produce cubc power reductons, at least n theory. Snce the tme needed to execute task τ s t = C /f, the energy consumpton of the task executng for an nterval of tme I, s E = P d I. Lnear Optmzaton Problem We formulate the energy mnmzaton problem as an optmzaton problem. The voltage and slowdown factors are normalzed to the maxmum values. We compute normalzed voltage levels for the tasks such that the condtons n Theorem 1 are satsfed. Let f R n be a vector representng the normalzed frequences f of task τ. The optmzaton problem s to compute the optmal vector f R n such that the system s feasble and the total energy consumpton of the system s mnmzed. Let P d be the normalzed energy consumpton of task τ as a functon of the normalzed frequences f (for the case of dentcal power characterstcs P d = k f 3. The total energy consumpton of the system E, a functon of the voltage vector f R n, s gven by Equaton 11. Thus, we have the followng optmzaton problem: mnmze: under the constrants: E(v) = n =1 P d C I T (11) = 1, 2,..., n D + k r T ( ) B S 1 + kr C j η j + j,π j>π, π j γ + C + ( D+k rt j,π j>γ ) Cj η j (12) η mn 1 (13) Equaton 12 ensures the feasblty when the tasks are ndependent. Equaton 13 constrants the slowdown factor to be greater than or equal to the slowdown n the ndependent mode. The normalzed slowdown factors are between the normalzed mnmum frequency η mn and 1. The optmzaton functon depends on the power characterstcs P d of the task. If P d s convex, the optmzaton functon s convex. Thus we have a convex Quadratc mnmzaton problem. 4 Experments We use randomly generated perodc task sets for our smulatons. Each task s characterzed by ts computaton tme C, ts perod T, ts deadlne D. We vary three parameters n our smulatons: (1) number of tasks totaltasks, from 5 to 15; and (2) maxmum perod for tasks maxperod, from 200 to 800; and (3)

7 7 utlzaton for task set, from 0.1 to 0.9. For any gven par of totaltasks, maxperod, and utlzaton, we randomly generate 1000 task sets, and the exprement result s the average value over these 1000 task sets. A task set s generated by randomly selectng a perod, a deadlne, and computaton tme for each of the totaltasks. Frst, a perod T s assgned randomly n the range [1, maxperod] wth a unform probablty dstrbuton functon. Then,we assgn a utlzaton U n the range [0.1/totaltasks, 2.0/totaltasks], agan wth unform probablty dstrbuton functon. The computaton tme of the task s then assgned as T U. Deadlnes are assgned just as perods, and deadlne of each task s unque. Consderng an nterval of tme I = 5, 000, the experment calculated the energy consumpton of task set n ths tme nterval. For the smplcty of descrpton, the FPPT schedulng algorthm usng slowdown factors s denoted as ES FPPT. We note that our experments are executed n the envronment of Pentum 1.8G, 786M RAM, Redhat9 ( lnux , gcc , X11, SPAK-0.3 ). Although dfferent task mght have varyng power characterstcs [4], we don t compute slowdown factors based on the task characterstcs n ths paper. k s assumed to be 1. In other words, all tasks have the same power coeffcent. (a) totaltasks=5 (b) totaltasks=10 (c) totaltasks=15 Fg.1. Average energy savngs of ES FPPT algorthm over the FPPT algorthm In the frst set of experments, we guarantee the utlzaton of task set s unchanged when the respectve slowdown factors of tasks are put n executon. Fgure 1 shows the consumed energy of ES FPPT schedulng algorthm n comparson wth the FPPT schedulng algorthm. The energy consumpton of both algorthms are collected n the tme length of 5000ms, where fgure 1(a), 1(b) and 1(c) descrbe respectvely the experment results of totaltasks s 5, 10 and 15. From the fgure, t s seen obvoulsy that the ES FPPT algorthm performs more 13% 42% energy-savng than the FPPT algorthm. ES FPPT assgns dfferent

8 8 slowdown factors for the tasks to have better energy gans. ES FPPT performs up to low utlzaton of task set better than FPPT at hgh utlzaton. Snce we compute contnuous slowdown factors and assgn them to the closest dscrete voltage levels, there s run tme slack even at worst case executon tme. The complcaton workload demand analyss s requred to mplement the dynamcally reclam the run-tme slack, whch s the future workng problem of our research. (a) totaltasks = 5, maxperod = 400, utlzaton = 0.4 (b) totaltasks = 10, maxperod = 200, utlzaton = 0.3 (c) totaltasks = 15, maxperod = 800, utlzaton = 0.2 Fg.2. Energy savngs of ES FPPT algorthm under dfferent PTAs In the second set of experments, we allow the utlzaton can change after usng slowdown factors. Fgure 2 shows the energy consumpton of the ES FPPT algorthm under dfferent preempton threshold assgnments(ptas) [21]. Wth the partal relatons between the PTAs, whch s defned n [21], It can be nferred that the smaller PTA, the more number of preemptons, and the more addtonal energy dsspaton of context swtches. It s seen that the energy gans ncrease wth the dfferent PTAs. Furthermore, Computng task slowdown factors consderng the vared PTAs, results n more energy gans. From the graph of energy gans, t s seen that the another gans under maxmal PTA are as hgh as 11% 46% over the ES FPPT algorthm under mnmal PTA, whch adds up to 2% 11% of total energy savngs. To our surprse, the consumed energy doesn t decreased steadly wth some drops n number of preemptons due to rsng PTA. Whle most of experment samples have llustrated that the energy savng maxmzed by the slowdown factors comng from maxmal PTA, there stll exst some exceptons, such as fgure 2(a) presents.

9 9 5 Concluson In ths paper, we present algorthms to compute statc slowdown factors under FPPT schedulng polcy. The suffcent constrants whch slowdown factors must satsfy to mantan the feasblty of FPPT task set are derved. Furthermore, We formulate the computaton of slowdown factors for tasks under dfferent preempton threshold assgnments as an optmzaton problem. Expermental results show that the computed slowdown factors save on an average 17% 53% energy over the known technques. The technques are energy effcent and can be easly mplemented. Ths wll have a great mpact on the energy consumpton of embedded real-tme systems. We plan to further explot the statc and dynamc slack n the system to make the FPPT algorthm more energy effcent. As a future work, we plan to compute dscrete slowdown factors for the tasks. Acknowledgment The authors wsh to express ther grattude to ChuangGuo Guo, Yan Ja for useful dscussons, and to the revewers for ther helpful comments on the paper. References 1. Youngsoo Shn, Kyoung Cho, T.S.: Power optmzaton of real-tme embedded systems on varable speed processors. In: 2000 Internatonal Conference on Computer- Aded Desgn (ICCAD 00). (2000) Hakan Ayd, Pedro Meja-Alvarez, D.M., Melhem, R.: Dynamc and aggressve schedulng technques for power-aware real-tme systems. In: 22nd IEEE Real- Tme Systems Symposum (RTSS 01). (2001) Ravndra Jejurkar, R.K.G.: Optmzed slowdown n real-tme task systems. In: 16th Euromcro Conference on Real-Tme Systems (ECRTS 04). (2004) 4. Hakan Aydn, Ram Melhem, D.M.P.M.A.: Determnng optmal processor speeds for perodc real-tme tasks wth dfferent power characterstcs. In: 13th Euromcro Conference on Real-Tme Systems (ECRTS 01). (2001) Yun Wang, M.S.: Schedulng fxed-prorty tasks wth preempton threshold. In: The Sxth Internatonal Conference on Real-Tme Computng Systems and Applcatons, Los Alamtos, IEEE Computer Socety (1999) Manas Saksena, Y.W.: Scalable real-tme system desgn usng preempton thresholds. In: 21st IEEE Real-Tme Systems Symposum(RTSS 2000). (2000) Paolo Ga, Guseppe Lpar, M.D.N.: Mnmzng memory utlzaton of real-tme task sets n sngle and mult-processor system-on-a-chp. In: 22nd Real-Tme Systems Symposum, London, England (2001) 8. ZHANA Yumn, HU Xaobo, C.D.Z.: Task schedulng and voltage selecton for energy mnmzaton. In: 39th desgn automaton conference (DMC), New Orleans LA (2002) Intel: Intel xscale processor (2000) 10. Transmeta: Transmeta crusoe processor (2000)

10 Hong Ink, Qu Gang, M.M.: Synthess technques for low-power hard real-tme systems on varable voltage processors. In: 19th IEEE Real-Tme Systems Symposum, Madrd, Span (1998) S.Km, S., T.Km: Integratng real-tme synchronzaton schemes nto preempton threshold schedulng. In: Ffth IEEE Internatonal Symposum on Object-Orented Real-Tme Dstrbuted Computng. (2002) Padmanabhan Plla, K.G.S.: Real-tme dynamc voltage scalng for low-power embedded operatng systems. In: ACM SIGOPS Operatng Systems Revew. Volume 35., ACM Press (2001) F. Yao, A.D., Shenker, S.: A schedulng model for reduced cpu energy. In: 36th Annual Symposum on Foundatons of Computer Scence (FOCS 95), IEEE Computer Socety (1995) JWS.Lu: Real-Tme Systems. Prentce Hall, Upper Saddle Rver (2000) 16. Gang Quan, X.H.: Energy effcent fxed-prorty schedulng for real-tme systems on varable voltage processors. In: Annual ACM IEEE Desgn Automaton Conference ( 38th conference on Desgn automaton ), Las Vegas, Nevada, Unted States (2001) Gang Quan, X.H.: Mnmum energy fxed-prorty schedulng for varable voltage processors. In: Desgn, Automaton and Test n Europe Conference and Exhbton (DATE 02). (2002) Jong Luo, N.K.J.: Power-conscous jont schedulng of perodc task graphs and a perodc tasks n dstrbuted real-tme embedded systems. In: 2000 Internatonal Conference on Computer-Aded Desgn (ICCAD 00). (2000) Kuchcnsk, F.G., Krzysztof: Lenes: task schedulng for low-energy systems usng varable supply voltage processors. In: The 2001 conference on Asa South Pacfc desgn automaton wth EDA Technofar Desgn Automaton Conference Asa and South Pacfc (ASP-DAC), Yokohama, Japan, ACM Press (2001) J.Regehr: Schedulng tasks wth mxed preemton relatons for robustness to tmng faults. In: 23rd IEEE Real-Tme System Symposum. (2002) Chen, J.: Extensons to Fxed Prorty wth PreemptonThreshold and Reservaton- Based Scheduln. PhD thess, Unversty of Waterloo (2005)