Portioned Static-Priority Scheduling on Multiprocessors

Transcription

1 Portoned Statc-Prorty Schedulng on Multproceor Shnpe Kato and Nobuyuk Yamaak School of Scence for Open and Envronmental Sytem Keo Unverty, Japan Abtract Th paper propoe an effcent real-tme chedulng algorthm for multproceor platform. The algorthm a dervatve of the Rate Monotonc () algorthm, wth t ba on the portoned chedulng technque. The theoretcal degn of the algorthm well mplementable for practcal ue. The chedulablty of the algorthm alo analyzed to guarantee the wort-cae performance. The mulaton reult how that the algorthm acheve hgher ytem utlzaton, n whch all tak meet deadlne, wth a mall number of preempton compared to tradtonal algorthm. Introducton Embedded real-tme applcaton n recent year have reled on a power of multproceor platform. Wth the trend toward chp multproceng [20], proceor clock frequency can be turned down to contan power conumpton and heat generaton for embedded computng. The concern heren the guarantee of tmng contrant for recurrent real-tme computng. The well-known Rate Monotonc () and Earlet Deadlne Frt (EDF) algorthm [5], whch are optmal n unproceor chedulng, are no longer optmal n multproceor chedulng []. They rather perform far poorly dependng on properte of tak et. Therefore, alternatve chedulng technque pecfc for multproceor platform have been dcued recently. At preent, the Pfar algorthm [, 7, 8], the EKG algorthm [], and the LLREF algorthm [9] are known optmal for multproceor. The wort-cae bound on the achevable ytem utlzaton, wth the guarantee of real-tme contrant, offered by thoe algorthm are 00%. However, ther computaton complexte and chedulng overhead are controveral. Thu, more reaonable chedulng algorthm, whch are not able to acheve utlzaton bound of 00% but perform wth le computaton complexte and maller number of tak preempton, are often preferable for practcal ue. For ntance, the EDF-US[/2] algorthm [5], the EDZL algorthm [0], and the Ehd2-SIP Th work upported by the fund of Reearch Fellowhp of the Japan Socety for the Promoton of Scence for Young Scentt. Th work alo upported n part by the fund of Core Reearch for Evolutonal Scence and Technology, Japan Scence and Technology Agency. algorthm [2] are known to perform relatvely well wth mall number of tak preempton, though ther utlzaton bound are down to 50% n the wort cae. In fact, no algorthm except for the Pfar, EKG, and LLREF algorthm have ever acheved wort-cae utlzaton bound over 50%. All the algorthm ntroduced above are categorzed nto dynamc-prorty chedulng. In general, dynamc-prorty chedulng uffer from the domno-effect problem that a deadlne m of a job caue another deadlne m of a followng job. It alo ha a dadvantage of varatonal jtter n perodc executon, whch are not dered n embedded control applcaton. The prmary advantage of dynamcprorty chedulng ha been t ablty of achevng hgh utlzaton bound, however uch an advantage not lkely to tand on multproceor. Meanwhle, tatc-prorty chedulng doe not uffer from the domno-effect and perodc jtter problem, though t achevable utlzaton bound ha been nferor to dynamc-prorty chedulng. In recent year, Anderon et al. proved that tatc-prorty chedulng alo able to acheve a utlzaton bound of 50% on multproceor, though no hgher bound cannot be obtaned [3]. Therefore, th paper conder that tatcprorty chedulng may be more effcent than dynamcprorty chedulng for multproceor platform. Th paper propoe an effcent real-tme chedulng algorthm for multproceor platform. The algorthm a dervatve of the algorthm. The objectve of the algorthm to acheve hgh chedulable utlzaton,.e. ytem utlzaton n whch all recurrent real-tme tak are guaranteed to meet deadlne, wth a mall number of tak preempton. The wort-cae utlzaton bound alo acheved 50%. In addton, th paper am the theoretcal degn of the algorthm beng mplementable for practcal ue. 2 Related Work Tradtonally, there have been two approache for chedulng real-tme tak on multproceor: global chedulng and parttoned chedulng. In global chedulng, all elgble tak are tored n a ngle prorty-ordered queue, and a global cheduler dpatche the ame number of the hghet prorty tak a proceor from th queue. The relatve order of the tak prorte vare dependng on whch tak are elgble, hence a tak may mgrate among proceor. Meanwhle n parttoned chedulng, each tak

2 agned to a ngle proceor, on whch each of t job wll be executed, and proceor are cheduled ndependently. Therefore, a tak executed on a dedcated proceor and never mgrate among proceor. Lettng M be the number of proceor, the global algorthm can m a deadlne even for the cae n whch tak utlze only /M of the ytem []. Baker generalzed that a et of perodc tak, all wth deadlne equal to perod, guaranteed to be chedulable ung the algorthm, f the total utlzaton of the tak doe not exceed M( u max )/2 u mn,whereu max and u mn are the maxmum and mnmum utlzaton of every ndvdual tak repectvely [6]. Anderon et al. nvented a global chedulng algorthm called [M 2 /(3M 2)] [2], whch place the hghet prorty to the tak wth utlzaton hgher than M 2 /(3M 2) and place the prorte to the other tak. Th prortzaton boot the wort-cae utlzaton bound to M/(3M 2). Herenafter, [M 2 /(3M 2)] denoted by for mplcty of decrpton. Ramamurthy et al. propoed another global chedulng algorthm called Weght-Monotonc (WM) [2], whch extend the Pfar chedulng method to take the tatc-prorty agnment. In [3], Anderon et al. proved that the wort-cae utlzaton bound of the WM algorthm 50% that hgher than the and algorthm and no tatc-prorty algorthm can trancend th bound. However, the WM algorthm generate a large number of tak preempton due to the charactertc of Pfar chedulng. In general, parttoned chedulng approache are preferred to global chedulng approache for practcal ue, becaue no tak mgraton occur n parttoned chedulng and the chedulng problem can be reduced to a et of unproceor one once the tak are parttoned. The mot well-known algorthm -FF [], whch ue the Frt-Ft (FF) heurtc to partton the tak. Appendng a lttle complexty, the algorthm [9], whch conduct the FF heurtc after ortng the tak n decreang order of utlzaton, uually perform better than the -FF algorthm. [] another effcent algorthm that take mlar chedulablty tet and heurtc, n whch tak are ntally orted n ncreang order of perod. Oh and Baker proved that a tak et guaranteed to be chedulable by the -FF algorthm f the ytem utlzaton doe not exceed 2 % [8]. More recently, becaue % wa a lower bound and there wa tll room for mprovement, Lopez et al. preented a tghter utlzaton bound for the -FF algorthm [7]. Ung a mlar technque, Lopez et al. alo clarfed the utlzaton bound of the algorthm [6]. Anderon et al. condered ung the next-ft-rng (NFR) heurtc for the R-BOUND- MP algorthm ntead the FF heurtc, and then derved t wort-cae utlzaton bound of 50% [3]. 3 Sytem Model The ytem a memory-hared multproceor compoed of M proceor: P, P 2,..., P M. The code and data of program (tak) are hared among the proceor. The overhead of nter-proceor communcaton neglected. In other word, the cot of tak mgraton not taken nto account. Recent advancement of proceor technology have omewhat allowed uch an aumpton. For example, the reponve multthreaded (T) proceor, nvented by Yamaak [22], upport a hardware functon to wtch a context to another context n four clock cycle regardle the wtchng occur whether between proceor or wthn a proceor. In addton, there lttle pont n takng notce of the cot of every context wtch n term of chedulng algorthm, nce the cot hghly depend on proceor pecfcaton. Thu, t more varant to focu on how often context wtche occur for the dcuon of the run-tme overhead. Th paper take the number of context wtche a a performance metrc of the run-tme overhead. Th paper alo gnore the behavor of a cache ytem. The performance deteroraton of computaton due to tranent degradaton of the cache ht rato, often caued by mgraton, out of focu. Th ort of concern are turned over to the wort-cae executon tme analy. TheytemhaaetofN perodc tak, denoted by Γ= {τ,τ 2,..., τ N }.Theth perodc tak defned by τ (C, T ) where C t wort-cae executon tme and T t perod (C T ). The proceor utlzaon of τ defned by U = C /T. A tak generate a equence of job perodcally. The jth job of τ denoted by τ, j that releaed at tme r, j and t deadlne equal to the releae tme of the next job,.e. d, j = r, j = r, j T. The total utlzaton of the gve tak et defned by U(Γ) = τ Γ U. Alo, lettng Λ be any ubet of the gven tak et, the total utlzaton of the ubet denoted by U(Λ) = τ Λ U. The et of the tak executed on a proceor P x denoted by Π x. All the tak are preemptve and ndependent. Therefore, no tak ynchronze wth any tak, and make crtcal ecton uch a I/O proceng. Any job of a tak cannot be executed n parallel, whch mean that, for any and j, τ, j cannot be executed n parallel on more than one proceor. Schedulng Algorthm Th ecton propoe the Rate Monotonc Deferrable Porton () chedulng algorthm that a dervatve of the algorthm [5], wth t ba on the portoned chedulng technque [2]. The algorthm degned well mplementable for practcal ue. In the followng ubecton, the bac trategy of the portoned chedulng technque frt ntroduced. Then, the theoretcal degn of the algorthm preented. The chedulable condton and the wort-cae utlzaton bound of the algorthm are fnally analyzed.. Portoned Schedulng Technque The portoned chedulng trategy [2] formed of the tak agnng phae and the tak chedulng phae. In the tak agnng phae, each tak agned to a partcular proceor lke the parttonng approach, a long a the tak doe not caue the total utlzaton of the proceor to ex-

3 Utlzaton bound j P x k Cannot ft nto Px P x Splt nto two porton k ' j P x Fgure. Portonng example k" P x ceed t utlzaton bound. If the total utlzaton urpae the utlzaton bound, the tak vrtually plt nto two porton n the ene of utlzaton, wherea n the parttonng approach the tak jut agned to another proceor whch can receve the full utlzaton of the tak. A for the vrtually-plt tak, one porton agned to the proceor to whch the tak are beng agned, and the other porton agned to the next proceor to whch the followng tak wll be agned. Notce that vrtually-plt mean that the tak not really dvded nto two block, but t utlzaton hared on the two proceor. In th paper, uch a tak plttng defned portonng. Fgure how an example of portonng. The heght of a box n whch the name of a tak ndcated the proceor utlzaton of the tak. The example preume that τ and τ j are already agned to P x and τ k about to be agned, but f agned t caue the total utlzaton of P x to exceed t utlzaton bound. The parttonng approach n th cae jut agn τ k to another proceor, uch a P x, whle the portonng approach vrtually plt τ k nto two porton τ k and τ k. In th paper, τ k defned the frt porton of τ k and τ k defned the econd porton of τ k. Then, τ k agned to P x and τ k agned to P x repectvely. A a reult, the portonng approach obvouly mprove the utlzaton of P x compared to the parttonng approach. In the tak chedulng phae, τ k erved by τ k on P x and erved by τ k on P x. In other word, lettng C k and C k be the agned executon tme of τ k and τ k repectvely, τ k conume C k tme unt on P x and C k tme unt on P x wthn every perod T k. The executon tme of τ k and τ k are computed a follow. C k = T k (U b U U j ) C k = C k C k Note that τ k and τ k form the ame tak τ k, thereby they cannot be cheduled multaneouly, nce no job have parallelm under the aumpton. Hence, the chedulng algorthm mut be degned o that the frt porton and the econd porton of a plt tak are cheduled excluvely..2 Tak Agnng Algorthm The tak agnng algorthm of traghtforward. The algorthm aume that the gven tak et orted o that the perod of τ maller than or equal to that Aumpton: the ndex of the tak. x the ndex of the proceor. n the number of the tak agned to P x. c the executon tme of the frt porton. c2 the executon tme of the econd porton. t the perod of a plt tak. tm the mnmal perod of the tak agned to P x. U (P x ) the utlzaton bound of P x. Γ orted o that T T 2... T n.. = x = n = 2. c = c2 = t = tm = 0 3. Π x =. Ux = rmdp bound(n, T, c, c2, t, tm) 5. f U(Π x ) U Ux 6. Π x =Π x τ 7. ele f x < M 8. C = {Ux U(Π x )}T and C = C C 9. plt τ nto τ (C, T )andτ (C, T ) 0. Π x =Π x τ and Π x = {τ }. x = x 2. n = 0 3. f N. 5. ele c = C 6. return FAILURE 7. = 8. n = n 9. f N 20. go back to tep 2. return SUCCESS, c2 = C, t = T, tm = T Fgure 2. agnng algorthm of τ for any. Then, the algorthm agn the tak to the proceor equentally, and f ome tak caue the total utlzaton of a proceor to exceed t utlzaton bound, t plt the tak nto two porton. The frt porton agned to the proceor whch caued to exceed t utlzaton bound, and the econd porton agned to the next proceor to whch the followng tak wll be agned. Th procedure repeated untl all the tak are uccefully agned or no proceor reman the pare capacte. Fgure 2 how the peudo code of the agnng algorthm. When the ntalzaton are done (lne to 3), the algorthm call the rmdp bound functon, whch ndcated n Fgure 3, to calculate the utlzaton bound of a proceor P x to whch a tak τ wll be agned (lne ). The content of the rmdp bound functon,.e. the method of calculatng the utlzaton bound, partcularly decrbed n Secton 5. Once the utlzaton bound Ux calculated, the algorthm agn τ to a proceor P x, a long a the total utlzaton of P x wll not exceed the utlzaton bound (lne 5to6). Ifτ caue P x to exceed the utlzaton bound, τ gong to be plt nto τ and τ accordng to the followng procedure. At frt, the executon tme of τ and τ are calculated (lne 8). Then, τ plt nto τ (C, T )and

4 Argument: (n, T, C, C, T, T mn ). f C = 0 2. return n(2 /n ) 3. ele. L = (T T C )/T 5. U = C /T 6. R = T mn /T 7. return U n{(2 LU /R ) /n } Fgure 3. rmdp bound functon τ (C, T ) (lne 9). τ agned to P x and τ agned to P x (lne 0). The algorthm ave the value of C,C, T and T (lne ) for carryng out the rmdp bound functon n the next teraton. Fnally, the algorthm goe back to the tep (lne 20) a long a there are tll tak remanng to agn to the proceor. When all the tak are uccefully agned, the algorthm ucceeded (lne 2). Fgure depct an example of tak agnment by the agnng algorthm. Conder a tak et Γ compoed of eght tak: T T : τ (, 5), τ 2 (2, 5), τ 3 (, 8), τ (5, 0), τ 5 (3, 2), τ 6 (2, 2), τ 7 (2, 20), and τ 8 (, 20). Note that Γ already orted o that T T atfed for any where te of perod are broken arbtrarly. A tated above, the algorthm equentally agn the tak. Evaluatng the rmdp bound functon wth n =, the utlzaton bound of 00% obtaned, and hence τ can be agned to P.Next, the algorthm uppoed to evaluate the rmdp bound functon wth n = 2. Accordng to Kuo and Mok [3], the number of the tak reduced to the number of the harmonc chan n chedulng. A harmonc chan a group of the tak whoe perod have the ame leat common multple. Takng the harmonc chan nto account, the 23rd lne of Fgure 2 can be extended o that t carred out only f the perod of the next tak not ncluded n any harmonc chan. Thu, the example evaluate the rmdp bound functon wth n = due to T = T 2,andτ 2 can be of coure agned to P. τ 3 doe not have a perod wth the ame leat common multple a τ and τ 2,thenn = 2 nput to the rmdp bound functon, whle τ ha a perod wth the ame leat common multple a them, and hence n = 2 remaned to the rmdp bound functon. The reultng utlzaton bound of 3 obtaned by the rmdp bound functon for n = 2. Snce U U 2 U 3 U =.225 exceed 3, τ plt nto τ (, 0) and τ (, 0) o that U U 2 U 3 U = 25 3 and U = U U. By the ame token, the ret of the tak are clafed nto Π 2 = {τ (, 0),τ 5,τ 6 (, 2)} and Π 3 = {τ 6 (, 2),τ 7,τ 8 }, though the utlzaton bound are calculated wth takng nto account that plt tak are agned n thoe et..3 Tak Schedulng Algorthm Th ecton preent the tak chedulng algorthm of. Lke parttoned chedulng, each proceor ha t own cheduler to chedule the agned tak on the proceor. All the cheduler have the ame chedulng pol- Ub=3 ' = /0 3 = /8 2 = 2/5 = /5 Ub= ' = /2 5 = 3/2 '' = /0 Ub= = /20 7 = 2/20 6 '' = /2 P P2 P3 Fgure. agnng example ce. The followng decrpton focu on the chedulng on a proceor P x where a tak τ ubmt t econd porton τ and another tak τ k ubmt t frt porton τ k. The cheduler on other proceor perform n the ame manner, nce the ame chedulng polce are appled. The bac polcy of the chedulng algorthm that the tak are cheduled accordng to the algorthm except for the cae n whch τ ready but t correpondng frt porton τ on executon on a neghbor proceor P x. In th cae, the cheduler dpatche a tak wth the econd hghet prorty o a not to execute the frt and the econd porton of τ multaneouly. In other word, the econd porton deferred wth the hghet prorty. Therefore, the prortzaton dcplne of the algorthm not tatc n a true ene. Fgure 5 how the peudo code of the cheduler. Every tme any tak are releaed on P x,thechedule P x functon nvoked (lne to 7). Let t be uch a tme. At tme t, f the plt tak τ and τ k are releaed, the cheduler need to reet the varable t, t k, e and e k, n order to track the remanng executon tme of thoe tak. Note that τ and τ k are not allowed to conume the proceor tme of P x over the capacte C and C k repectvely, but the tak cannot recognze voluntarly that they exhaut the capacte. Thu, the cheduler need to track the remanng executon tme of thoe tak to preempt ther executon ung tmer functon. The cheduler functon proceed a follow. Frt of all, f the currently-runnng tak ether a plt porton of τ or τ j, the cheduler ave t remanng executon tme (lne to 3). Then, t elect a tak wth the hortet perod (lne 5). If the elected hghet-prorty tak τ that ha t econd porton on the proceor (lne 6), the cheduler examne whether τ currently n executon on a neghbor proceor P x (lne 7). If t n executon, the cheduler re-elect a tak wth the econd hortet perod (lne 8). Otherwe, τ gong to be dpatched and executed, o the cheduler update the tmer to nvoke the econd end functon at tme t e, whch preempt the executon of τ (lne 20). Notce that the econd end functon may not be nvoked at tme t e, nce t may be updated f τ preempted and later dpatched agan wthn the ame perod.

5 Aumpton: τ a plt tak between P x and P x. τ k a plt tak between P x and P x. t the current tme. t the lat tme at whch τ dpatched on P x. t k the lat tme at whch τ k dpatched on P x. e the remanng executon tme of τ. e k the remanng executon tme of τ k.. functon ytem tck 2. f any tak are releaed on P x 3. f τ releaed. e = C and t = t 5. f τ k releaed 6. e k = C k and t k = t 7. call chedule P x 8. functon chedule P x 9. f P x dlng 0. go to tep 5. f τ currently runnng 2. e = e (t t ) 3. ele f τ k currently runnng. e k = e k (t t k ) 5. let τ j be a tak wth the hortet perod 6. f τ j refer to τ 7. f τ n executon on P x 8. let τ j be a tak wth the econd hortet perod 9. ele 20. update the tmer to nvoke p2 fnhed at t e 2. t = t 22. ele f τ j refer to τ k 23. update the tmer to nvoke p fnhed at t e k 2. t k = t 25. f τ k n executon on P x 26. nvoke chedule P x on P x 27. dpatch and execute τ j on P x Fgure 5. chedulng algorthm Tme t e jut the earlet tme at whch τ can exhaut C. A long a the remanng executon tme of τ tracked, τ guaranteed not to overrun the capacty C on P x.after updatng the tmer, the cheduler mut ave the current tme a the lat dpatched tme of τ for trackng t remanng executon tme. By the ame token, f the elected tak τ k that ha t frt porton on the proceor (lne 22), the cheduler update the other tmer to nvoke the frt end functon at tme t e k, whch preempt the executon of τ k, and ave the current tme a the lat dpatched tme of τ k (lne 23 to 2). Unlke the cae of τ, f the correpondng econd porton of τ k currently n executon on P x (lne 25), the cheduler need to nvoke the cheduler functon that operatng on P x to let t rechedule the tak on P x (lne 26), nce the econd porton mut be deferred whle the correpondng frt porton executed. Fnally, the cheduler execute the elected tak on P x (lne 27). The job fnhng functon are ndcated n Fgure 6. When any job except for thoe of the plt tak are completed, the job call the job end functon (lne to 3). The frt end and econd end functon are called only through the tmer. Thoe functon call the cheduler to rechedule. functon job end 2. remove the caller tak from a ready et 3. call chedule P x. functon frt end 5. remove τ r from a ready et 6. call chedule P x 7. f τ k ready on P x 8. nvoke chedule P x on P x 9. functon econd end 0. remove τ from a ready et. call chedule P x Fgure 6. Job fnhng functon P P2 P Fgure 7. chedulng example the tak (lne 3, 6, and ). Only when τ k conume the capacty C k on P x but ha not conumed the capacty C k on P x, the cheduler nvoke the cheduler operatng on P x to execute τ k on P x,nceτ k ha the hghet prorty on P x but ha been deferred. Fgure 7 ndcate how the three tak et of Π = {τ (, 5), τ 2 (2, 5), τ 3 (, 8), τ (, 0)}, Π 2 = {τ (, 0), τ 5 (3, 2), τ 6 (, 2)} and Π 3 = {τ 6 (, 2), τ 7(2, 20), τ 8 (, 20)}, are cheduled by the algorthm. Snce Π doe not nclude the econd porton of a plt tak, the tak et cheduled completely accordng to the polcy. Meanwhle, Π 2 nclude τ and t nterfered by t correpondng frt porton τ. At tme t = 0, τ cheduled on P 2 and then completed at tme t = wthout any nterference from τ, becaue τ cheduled at tme t = on P. A for the econd job of τ, on the other hand, τ cheduled at tme t = 0 on P 2 but preempted at tme t = 3, becaue τ cheduled at th tme on P. Hence, τ 5 cheduled ntead of τ and τ later reumed when τ completed at tme t =. The thrd job of τ alo ha the ame tuaton. In th cae, there are no ready tak on P when τ preempted at tme t = 23. Therefore, the tme lot left dle. Although Π 3 alo nclude the econd porton τ 6,thetakonP 3 can be cheduled wthout any retrcton wthn the example, nce τ 6 and τ 6 are never overlapped n chedulng. 5 Schedulablty Analy Th ecton provde the chedulablty analy of the algorthm to guarantee the wort-cae performance. The analy ha the followng aumpton. In the tak agnng phae, ome tak τ plt and t econd porton τ agned to a proceor P x. Then, the followng n

6 ' 2 n- n S Tn-T Tn- 0 Tn Fgure 8. Cae n whch τ wthn T and T n T T2 executed twce tak, denoted by τ,τ 2,..., τ n for mplcty of decrpton, are alo agned to P x. Note that the frt porton of a plt tak not a concern, becaue the cheduler treat t n the ame manner a the non-plt tak. The goal of the analy to derve a formula leadng to the chedulable condton of P x for any x. Accordng to the analy [5], the mnmal chedulable utlzaton for n tak occur for the cae n whch all the tak are releaed at the ame tme wth the followng relaton: T < T n < 2T, C = T T ( n ) and C n = 2T T n. Thu, the mnmal chedulable utlzaton fornon-pltn tak n the algorthm alo occur for th condton, thoe tak are cheduled accordng to the algorthm. Now therefore, the wort-cae phang of a concern. For a fxed value of U, the remanng utlzaton of P x obvouly mnmzed when τ ubmt a many job a poble. That occur for the cae n whch an arbtrary job of τ deferred a much a poble and τ t followng job are executed wthout any preempton a oon a they are releaed. Such wort-cae phang clafed nto three cae, hown n Fgure 8, Fgure 9 and Fgure 0. The dfference between the three phae that τ executed () twce wthn T and T n n the phae, () three tme wthn T and T n n the phae 2, and () twce wth n T and three tme wthn T n n the phae 3. The analy executed more than three tme wthn T or T n, nce the condton of the wort-cae phang nclude T < T n < 2T. For mplcty of decrpton, the fracton of T and T defned R = T /T and that of the perod of two conecutve tak τ and τ defned R = T /T henceforth. Notce that k, R R 2 R k = T k /T. Alo t neceary to have the mnmum lack S n the fgure to obtan the doe not need to conder the cae n whch τ chedulable utlzaton. Snce τ ha the hghet prorty on the proceor, t only deferred by τ cheduled on the neghbor proceor. Therefore, the latet fnh tme of τ r,k C C = r,k C = r,k (T C ). A a reult, the mnmal lack can be expreed by S = T C. 5. Cae of Phae The analy begn wth the mplet cae n whch τ executed twce wthn T and T n a hown n Fgure 8. Under th relaton, the executon tme of the n tak are denoted a follow. C = T T ( n ) C n = T 2C n j= C j = 2T 2C T n Hence, the reultng utlzaton wrtten a Equaton (). U = C T C T C 2 T 2 C n = U n = n = U = U T n T T 2T 2C k T n T = n = T T 2 R T n ) ( C T n T T n 2 ( ) U R n () R R 2 R n In order to mnmze U over R, the above expreon partally dfferentated wth repect to R. U = 2 ( ) U R R R 2 ( n j R j ) Now, U mnmzed when each R atfe the followng equaton where P = R R 2 R n. ) R P = 2 ( U ( n ) R That, U mnmzed when all the R have the ame value. { )} /n R = R 2 = = R n = 2 ( U R By ubttutng the value of each R to Equaton (), the utlzaton bound U b obtaned a follow. Here, let K = 2( U /R ) due to lmtaton of pace. U b = U (n K )K/n K n n /n = U nk/n K /n K /n n ( 2( U ) /n = U n The above expreon mple that U b moreover mnmzed by reducng the value of R. Accordng to Fgure 8, the condton below mut be atfed. n C = T 2C S = T C = Dvdng by T, the range of R acqured a follow. R 2U U R 2U The condton of T T lead to R. Hence, U b mnmzed when R = max{, 2U }. Fnally, U b decrbed by Equaton (2) where R = max{, 2U U }. U b = { )} /n U n 2 ( U R (2) R )

7 ' S ' S T T 2 T2 2 T2 n- Tn- k Tk n 0 C''TS Tn Fgure 9. Cae n whch τ tme wthn T and T n executed three k n- n Tk Tn- 0 Tn Takng the lmt a n, the wort cae derved. )} lm U b = U n {2 ( ln U (3) R Equaton (3) a monotoncally-ncreang functon wth repect to R.SnceT T, R never le than. Hence, Equaton (3) mnmzed to Equaton () wth R =. Û b = U ln{2( U )} () Conequently, the abolute mnmum value of the utlzaton bound become Û b = 0.5 wth U = 0.5 andu =. Note that th bound gven only for the tuaton n whch T = T = T 2 = = T n and C = C 2 = = C n = Cae of Phae 2 The analy move on the cae n whch τ executed three tme wthn T and T n a hown n Fgure 9. The executon tme of the n tak are defned a follow. C = T T ( n ) C n = T 3C n j= C j = 2T 3C T n Hence, the reultng utlzaton wrtten a Equaton (5). n U = U = U = n = T T R ) (2 3C T n T T n 2 3U R R R 2 R n n (5) The value of R that mnmze Equaton (5) can be calculated by the ame tep n Secton 5.. That, the utlzaton bound obtaned by the followng expreon. U b = U n (2 3U R ) /n Notce that the above expreon a monotoncallyncreang functon wth repect to R. Hence, t mnmzed by reducng the value of R.Inordertofndthemnmum value of R,thevalueofT mut be explored frt. Fgure 9 lead to T a follow. T = T 2C S = 2T 2C C Fgure 0. Cae n whch τ executed twce wthn T and three tme wthn T n Dvdng by T,thevalueofR acqured a follow. R = 2U U 2 Hence, U b now expreed by Equaton (6). ( U b = U 3U ) /n n 2 2U U 2 ( U = U ) 2U /n n 2U (6) U 2 Takng the lmt a n, the wort cae appeared. ( U lm U b = U n ln 2U ) 2U U (7) 2 In order to fnd the value of U mnmzng U b, Equaton (7) dfferentated wth repect to U U b U = 2U 2 (0 5U )U 2U 2 5U 2 (U 2U )(2U U 2) Fnally, the value of U Equaton (8).. that mnmze U b obtaned by Û = 5U 0 9U 2 60U 8 (8) Snce U 0, the abolute mnmum value of the utlzaton bound become Û b 52 wth U 6 and U =. Snce the prevou cae ha a lower utlzaton bound of 50%, th cae not the wort cae. 5.3 Cae of Phae 3 The fnal cae the analy conder a combnaton of the prevou two cae. Snce τ executed twce wthn T whle executed three tme wthn T n, the executon tme of the n tak are denoted a follow. C = T T ( k ) C k = T k C T k C = T T (k n ) C n = T 2C n j= C j = 2T C T n

8 Then, the reultng utlzaton wrtten a Equaton (9). U = U = U k = T T T 2T C T n T n k = (n ) k = U = T T U k R U k C k T k n =k n =k R n =k T T T T T ) (2 C T T T n 2 U R R R 2 R n (n ) (9) Unfortunately, t too complcated to explore the value of R that mnmze Equaton (9) wth repect to two varable of n and k. Therefore, let the value of k be fxed frt. U k decrbed by U k = R k C /T k due to C k = T k C T k, Equaton (9) can be tranformed a follow. n U = U = R C T k 2 U R R R 2 R 3 R n n The above expreon mple that Equaton (9) obvouly mnmzed for k = due to T T 2 T n. Therefore, Equaton (9) reduced a follow. n U = U U 2R U R (n ) R R R 2 R 3 R n =2 In order to mnmze U, the above expreon dfferentated wth repect to R a follow, where 2 n. U R = 2R U R R R 2 n j R j That, U mnmzed when all the R have the ame value. ( 2R U ) /(n ) R 2 = R 3 = = R n = R R Now, the mnmum value of U decrbed a follow. U = ( U 2R U ) /(n ) (n 2) R R 2R U ( R R 2R ) U /(n ) (n ) R R ( = U U 2R U (n ) R R ) /(n ) C k = T k C T k and k = lead to R = U U /R. Alo, the value of R that mnmze the above expreon max{, 2U U } a n Secton 5.. Fnally, the utlzaton bound U b decrbed by Equaton (0) where R = max{, 2U U } and m = n. { U b = U 2R U } /m U m R (U ) U (0) Takng the lmt a n, the wort cae appeared. { lm U b = U n U 2R U } ln R (U ) U () By the ame token a Secton 5., the mnmum of Equaton () expreed by Equaton (2). ( Û b = U 2 U ) U ln U (2) U Seekng the value of U and U that mnmze Û b,theabolute mnmum bound obtaned Û b 0.5 wth U = /2 and U = 0. Th bound occur only for the tuaton n whch T = T = T 2 = = T k, T k = T k2 = = T n = T C and C = C 2 = = C n = 0. Comparng the analyzed three cae, the wort-cae utlzaton bound of each proceor fnally derved 50%. That, the utlzaton bound of the entre ytem alo 50%. 5. General Cae The tak are orted o that T T, hence T a known value when τ plt. In other word, R a known value when τ plt. Therefore, the wort-cae not necearly preumed, but the utlzaton bound can be calculated by ether Equaton (2), Equaton (6), or (0). The analy now proceed to conder the general cae. Equaton (5) can be rewrtten a follow. n U = U = T T ) (2 C T T Equaton (9) can be alo rewrtten a follow. U = U n = T T ) (2 C T n T T n n 2C (3) T n T n T n T k C () T n Notce that T n /T k n Equaton () never exceed 2, becaue τ mut be executed twce wthn T k and three tme wthn T n. Hence, Equaton (5) alway maller than Equaton (9) wth repect to the ame et of {T, T, T 2,..., T n }.Th fact mple that the utlzaton bound for the cae n whch τ executed L tme wthn T and T n alway maller than that for the cae n whch τ executed F < L tme wthn T and executed L tme wthn T n. Therefore, the analy need to concern only the cae n whch τ executed L tme wthn T n for the general cae. Referrng to Fgure 9, L can be decrbed by the followng expreon. Tn C S Tn T C L = = T T Fnally, the utlzaton utlzaton for the general cae obtaned by Equaton (5). U b = U n (2 LU R ) /n (5) The value of Equaton (5) 50% wth L = 2, U = and U = 0.5. Hence, the wort cae contaned. Lettng

9 U = 0, the analy alo contan the cae n whch there no tak that ha the econd porton on P x. In fact, th cae lead to Equaton (6) whch the well-known utlzaton bound of the algorthm [5]. 6 Smulaton Stude U b = n(2 /n ) (6) Th ecton evaluate the effectvene of the algorthm n term of the chedulablty and the number of tak preempton. The utlzaton bound wa proved n the prevou ecton, and hence the algorthm can be compared wth the tradtonal algorthm theoretcally. However, n order to etmate the performance of the algorthm truly, the uffcent numberof tak et wth dfferent properte mut be ubmtted to the algorthm, nce the utlzaton bound of the algorthm actually vary dependng on gven tak et. The number of tak preempton are alo domnated by the charactertc of the gven tak et. The evaluaton compare the algorthm wth the tradtonal -baed algorthm: -FF,, -NFR,,, and WM. 6. Expermental Setup The mulaton etmate the chedulablty of an algorthm a follow. Every ytem utlzaton U y rangng from 30% to 00%, 000 tak et wth dfferent properte, whoe ytem utlzaton are all U y equally, are generated and ubmtted to the algorthm. Then, the ucce rato, defned by the followng expreon, meaured. the number of uccefully cheduled tak et the number of cheduled tak et Algorthm havng hgh ucce rato are etmated to offer hgh chedulable utlzaton. The defnton of a uccefully-cheduled tak et depend on a chedulng algorthm. For the, -FF,, and R- BOUND-MP-NFR algorthm, a tak et ad to be uccefully cheduled f all the tak can be agned to the proceor, nce thoe algorthm are degned o that no tak wll m the deadlne once they are uccefully agned to the proceor. For the, and WM algorthm, on the other hand, a tak et ad to be uccefully cheduled f all tak are cheduled wthout mng any deadlne, nce the theoretcal utlzaton bound of thoe algorthm are very pemtc. Meanwhle, the mulaton etmate the number of preempton for an algorthm by calculatng t average number, defned by the followng expreon. the total number of preempton n the cheduled tak et the number of cheduled tak et Each mulaton characterzed by the four parameter: M, U max, U mn and U total. M the number of the proceor. U max and U mn are the maxmum and mnmum value of the proceor utlzaton of every ndvdual tak n a gven tak et. U total refer to the total proceor utlzaton of the tak. The ytem utlzaton defned by U y = U total /M, whch range from 0% to 00%. Although many combnaton of the parameter can be condered, the mulaton attempt the followng combnaton due to the lmtaton of pace. The ytem utlzaton determned wthn the range of [30%, 00%]. Mot of the extng algorthm can uccefully chedule a tak et wth a ytem utlzaton below 30%, hence ytem utlzaton below 30% are removed. A for the number of the proceor, the mulaton prepare the three et: M =, M = 8, and M = 6. The target ytem of th reearch, uch a humanod robot, would make ue of multcore proceor havng uch number of core. In thoe ytem, mple actvte can be realzed wth only lght tak (tak wth low utlzaton), wherea enhancng the qualty of the actvte requre heavy tak (tak wth hgh utlzaton). Thereby, the mulaton prepare the two et of U mn and U max :(U mn, U max ) = (0.0, 0.) and (U mn, U max ) = (0.0,.0). 6.2 Schedulablty Reult Fgure how the ucce rato for each algorthm wth repect to tak et n whch the utlzaton of every ndvdual tak range from wthn [0.0, 0.]. The R- BOUND-MP-NFR algorthm abbrevated a R-BOUND- MP for mplcty. The chedulable utlzaton of the algorthm are around 70 73%. The -FF, - FFDU and algorthm are alo compettve, achevng chedulable utlzaton around 65 67%, though the algorthm lghtly outperform the other. Therefore, the mulaton demontrate that the parttoned chedulng approache have lttle performance dfference to lght tak et n whch tak have low proceor utlzaton. The and algorthm generate the ame chedule for the cae wth U max = 0. < M/(3M 2). It remarkable that the and algorthm perform much better than the and parttoned chedulng algorthm. Snce the chedulable utlzaton of the and algorthm are domnated by the maxmum utlzaton of every ndvdual tak, thoe algorthm offer excellent performance. However, remember that the tmng contrant cannot be guaranteed by the and algorthm, f the ytem utlzaton exceed M/(3M 2) [2]. Thu, thoe algorthm are not dered n afe ytem. The WM algorthm acheve chedulable utlzaton around 90%, whch are much better than the other algorthm. Such a uperorty of the WM algorthm derved from the charactertc of Pfar chedulng, whch can make an optmal chedule. Unfortunately, the WM algorthm not mplementable for practcal ue, nce t may nvoke a cheduler every quanta. Fgure 2 how the ucce rato for each algorthm wth repect to tak et n whch the utlzaton of every ndvdual tak range from wthn [0.0,.0]. Note that th rangng generate heavy tak. Unlke the prevou reult n whch only lght tak are generated, the algorthm clearly outperform the other algorthm except for the WM algorthm. The chedulable utlzaton of the

10 M=, U mn =0.0, U max =0. M=, U mn =0.0, U max =.0 Succe Rato 0. -FF 0.2 WM 0 Sytem Utlzaton Succe Rato 0. -FF 0.2 WM 0 Sytem Utlzaton M=8, U mn =0.0, U max =0. M=8, U mn =0.0, U max =.0 Succe Rato 0. -FF 0.2 WM 0 Sytem Utlzaton Succe Rato 0. -FF 0.2 WM 0 Sytem Utlzaton M=6, U mn =0.0, U max =0. M=6, U mn =0.0, U max =.0 Succe Rato 0. -FF 0.2 WM 0 Sytem Utlzaton Succe Rato 0. -FF 0.2 WM 0 Sytem Utlzaton Fgure. Succe rato (U max = 0.) Fgure 2. Succe rato (U max =.0) algorthm are around 70 73% a well a the prevou reult. Meanwhle, the -FF, and R-BOUND- MP algorthm have chedulable utlzaton around 50 60%, that, the outperform about 0 20% over the parttoned chedulng algorthm. Therefore, the mulaton how that the effectvene of portoned chedulng lkely to appear wth tak et n whch ome tak have hgh utlzaton. The and algorthm, on the other hand, offer very low chedulable utlzaton. Hence, the mulaton evnce that the performance of thoe algorthm are truly affected by heavy tak. However, t nteretng reult that the algorthm retan hgh ucce rato n hgh ytem utlzaton. It even trade the ucce rato wth the algorthm n ytem utlzaton over 80%. A tated above, the algorthm doe not conduct a chedulablty tet n the mulaton, wherea the, -FF, and algorthm carryng out chedulablty tet may reject tak et that can be n fact uccefully cheduled. A a reult, the four algo- rthm have lower ucce rato than the algorthm n hgh ytem utlzaton. Recall that the tmng contrant cannot be guaranteed by the algorthm, f the ytem utlzaton exceed /M. It urprng that the algorthm perform much wore than the algorthm. It can be condered that the algorthm dobey the prortzaton, thereby deadlne are more lkely to be med, though t mprove the wort-cae chedulablty. The WM algorthm tll offer good performancen the preence of heavy tak n exchange for a great deal of computaton complexty. 6.3 Preempton Reult For calculaton of tak preempton, the mulaton nterval are et the maller of the leat common multple of the tak perod n the gven tak et and Then, for each algorthm, the average number of tak preempton n the 000 tak et calculated every ytem utlzaton.

11 Number of Preempton M=, U mn =0.0, U max =0. -FF Number of Preempton M=, U mn =0.0, U max =.0 -FF 0.9 Sytem Utlzaton 0. Sytem Utlzaton Number of Preempton M=8, U mn =0.0, U max =0. -FF Number of Preempton M=8, U mn =0.0, U max =.0 -FF Sytem Utlzaton 0. Sytem Utlzaton Number of Preempton M=6, U mn =0.0, U max =0. -FF Number of Preempton M=6, U mn =0.0, U max =.0 -FF 0.9 Sytem Utlzaton 0. Sytem Utlzaton Fgure 3. Tak preempton (U max = 0.) Fgure. Tak preempton (U max =.0) Then, the number of tak preempton for each algorthm relatve to that for the algorthm calculated only only for the cae n whch both the target algorthm and the algorthm have a chedulable utlzaton of 00%. The WM algorthm contently generate more than ffty tme a many preempton a the algorthm, hence t not ncluded n the reult. Fgure 3 how the number of tak preempton for each algorthm relatve to that for the algorthm wth repect to tak et n whch the utlzaton of every ndvdual tak range from wthn [0.0, 0.5]. The algorthm caue lghtly more preempton than the -FF, and algorthm, but they are almot compettve. The addtonal preempton occur n the algorthm for the cae n whch the frt porton of plt tak are dpatched when ther correpondng econd porton are n executon. Snce the utlzaton of the tak are relatvely low n the dcued mulaton, the frt and the econd porton of plt tak have le chance to be overlapped. A a reult, the number of preempton n the algorthm not ncreaed very much. The and algorthm generate about.5.7 tme a many preempton a the algorthm. In thoe algorthm, one global cheduler manage all the tak, and hence the prorty order of the tak are more lkely to be changed compared to the and parttoned chedulng algorthm. A a reult, the and algorthm grow the number of preempton. Fgure how the number of tak preempton for each algorthm relatve to that for the algorthm wth repect to tak et n whch the utlzaton of every ndvdual tak range from wthn [0.0,.0]. The preence of heavy tak expand the performance dfference between the algorthm and the parttoned chedulng algorthm. In the algorthm, preempton occur every tme the frt porton of plt tak are dpatched whle ther correpondng econd porton are n executon. Snce the tak are lkely to have hgh utlzaton, uch preemp-

12 ton are lkely to happen, and hence the number of preempton n the algorthm are booted up, compared to the parttoned chedulng algorthm n whch uch preempton never occur. The reultng number of preempton n the algorthm are tme a many a thoe n the parttoned chedulng algorthm, though they are much fewer than the and algorthm. The mpact of the performance dfference n the number of preempton to the ytem depend on the rato of the proceor tme conumed by the tak executon and the proceor tme conumed by the tak preempton. For example, the algorthm acheve chedulable utlzaton about 0% hgher than the algorthm, whle t ncur about.8 tme a many preempton. Therefore, the algorthm may be better than the algorthm, f the cheduler conume more than about % of the overall ytem tme for the tak preempton. The ytem degner hould take th fact nto account. A long a th paper mulated, the algorthm eem better than the algorthm, becaue the cheduler hardly conume % of the ytem tme for the tak preempton, conderng the pecfcaton of the current proceor. 7 Concluon Th paper preented the algorthm that combne the portoned chedulng technque and the Rate Monotonc algorthm. The theoretcal decrpton gave that the algorthm well mplementable, nce t ncur only a lttle mplementaton n addton to the parttoned algorthm. The chedulablty analy derved that the wortcae utlzaton bound of the algorthm 50%. The mulaton tude demontrated that the algorthm uccefully cheduled the tak et wth hgher ytem utlzaton than the tradtonal -baed algorthm, wthout generatng many preempton. Bede, t guaranteed by the theory that the degree of tak mgraton for the algorthm uppreed o that at mot M tak occur mgraton and each of them mgrate between the retrctve two proceor. 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