Int. J. Embedded Systems, Vol. 6, No. 4, College of Computer Science, Zhejiang University, Hangzhou , China

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1 It. J. Embedded Systems, Vol. 6, No. 4, Effcet algorthms for schedulablty aalyss ad prorty assgmet for fxed-prorty preemptve schedulg wth offsets Zoghua Gu College of Computer Scece, Zhejag Uversty, Hagzhou 30027, Cha E-mal: Hab Ca* Shagha Key Laboratory of Trustworthy Computg, East Cha Normal Uversty, Shagha , Cha E-mal: *Correspodg author Abstract: Fxed-prorty schedulg s the most commo schedulg algorthm used dustry practce. Imposg fxed tas release offsets s a effectve techque for mprovg schedulablty by avodg the crtcal stat whe all tass are released smultaeously. I ths paper, we address the problem of schedulablty aalyss ad prorty assgmet for a perodc tasset wth fxed-prorty preemptve schedulg, where tass have fxed offset relatoshps relatve to each other. For exact schedulablty aalyss, we preset a effcet algorthm for computg busy perods, ad obtag respose tmes of all staces of a tas τ the feasblty terval oce the prorty-level-p busy perods are determed. For prorty assgmet, we adopt Audsley s optmal prorty assgmet (OPA) algorthm, ad preset a effcet algorthm for cremetal costructo of busy perods. We also preset a effcet coservatve algorthm for schedulablty aalyss of a asychroous tasset wth much mproved accuracy compared to the schedulablty test wthout offset costrats. Performace evaluato demostrates sgfcat performace mprovemets compared to exstg algorthms terms of both computato effcecy ad aalyss accuracy. Keywords: real-tme schedulg; schedulablty aalyss; prorty assgmet. Referece to ths paper should be made as follows: Gu, Z. ad Ca, H. (204) Effcet algorthms for schedulablty aalyss ad prorty assgmet for fxed-prorty preemptve schedulg wth offsets, It. J. Embedded Systems, Vol. 6, No. 4, pp Bographcal otes: Zoghua Gu s a Assocate Professor the College of Computer Scece, Zhejag Uversty. He receved hs PhD Computer Scece ad Egeerg from the Uversty of Mchga at A Arbor Hs research area s real-tme ad embedded systems. Hab Ca s a Assocate Professor the Software Egeerg Isttute, East Cha Normal Uversty. He receved hs PhD degree from the Doghua Uversty 2008, Shagha, Cha. He receved hs BEg degree from the Natoal Uversty of Defese Techology 997, ad MS degree from the Natoal Uversty of Defese Techology Hs research terests clude pervasve computg, embedded systems ad cyber-physcal systems. Itroducto Fxed-prorty preemptve schedulg (FPPS) s the most wdely-used real-tme schedulg algorthm dustry today, supported by all commercal RTOSes. Gve a tasset wth FPPS o a u-processor, there are two problems to be solved: schedulablty aalyss to determe f the tasset s schedulable for a gve prorty orderg, ad prorty assgmet to determe a total orderg amog tas prortes to mae the tasset schedulable. A perodc tasset s sychroous f all tass are trggered depedetly from each other, ad there s o costrat o tas release offsets; asychroous or fxed-offset f tass have fxed release offsets relatve to each other. Imposg fxed tas release offsets a perodc tasset s a effectve techque for avodg the crtcal stat ad spreadg the worload over tme to mprove schedulablty. However, Copyrght 204 Iderscece Eterprses Ltd.

2 304 Z. Gu ad H. Ca eve f the crtcal stat does ot occur at tme 0 due to tas offsets, a asychroous tasset may stll experece a crtcal stat whe all tass are released smultaeously, ot at tme 0, but at some future tme stat durg ts executo. A tasset whose perods are co-prme to each other must experece a crtcal stat regardless of the tal tas offsets. For such a tasset, t s obvously useless to mpose fxed tas offsets for the purpose of mprovg schedulablty, sce t ca be treated equvaletly as a sychroous tasset for schedulablty aalyss. However, for a asychroous tasset wthout a crtcal stat, the aalyss for a sychroous tasset s a safe but pessmstc estmate of the actual WCRT. I ths paper, we preset effcet algorthms for schedulablty aalyss ad prorty assgmet for a asychroous perodc tasset wth FPPS o a u-processor va cremetal costructo of busy perods. Performace evaluato demostrates sgfcat performace mprovemets compared to exstg algorthms terms of both aalyss accuracy ad computato effcecy. Whle ths paper focuses o uprocessor schedulg, Gu et al. (204) preseted a model-checg approach to schedulablty aalyss of global multprocessor schedulg wth fxed offsets. Our wor ca also be tegrated wth schedulg algorthms that address other o-fuctoal ssues, such as low-power schedulg based o dyamc voltage ad frequecy scalg (DVFS) (Chaturved et al., 202), leaage-aware schedulg (Nu ad Qua, 203), ad mxed-crtcalty schedulg (Zhao et al., 203, 204). We leave these addtoal research topcs as possble future wor. The rest of ths paper s structured as follows: we preset bacgroud ad related wor Secto 2. I Secto 3, we dscuss exact schedulablty aalyss by computg the respose tme of every stace of a gve tas the feasblty terval. I Secto 4, we preset a effcet coservatve algorthm as a suffcet but ot ecessary codto for schedulablty aalyss of a asychroous tasset wth FPPS. I Secto 5, we coduct performace evaluato terms of both computatoal effcecy ad aalyss accuracy of exstg ad our proposed algorthms. I Secto 6, we preset coclusos. Table summarses the ey otatos used ths paper, ad Fgure llustrates the defto of some ey varables. Table N τ Notato used ths paper Number of tass the tasset, N N Tas wth dex, N, N hp(τ ) Set of tass wth prorty strctly hgher tha τ hep(τ ) Set of tass wth prorty hgher tha or equal to τ, whch s equal to {hp(τ ) τ }, assumg each prorty-level cotas a sgle tas. T D O C p τ j r j f j R j R P P BP Perod of tas τ Deadle of tas τ Release offset of tas τ WCET tas τ Prorty of tas τ (smaller umercal value deotes hgher prorty). j th stace of tas τ Release tme of job τ j Fsh tme of τ j Respose tme of tas stace τ j relatve to ts release tme WCRT of tas τ for all ts jobs Hyper-perod of the etre tasset, defed as the least commo multple of all tas perods the tasset Hyper-perod of the set of tass hp(τ ) ad τ tself Prorty level- busy perod m BP The m th prorty level- busy perod startg from tme 0 O () wr(t) wr (t) Offset of tas τ relatve to start BP ( ) Worload request fucto of all tass utl tme t Worload request fucto of tass wth prorty level- or hgher Fgure Illustrato of some of the otatos Table, assumg τ j s a lower-prorty tas tha τ Note: Upward arrows deote tas release tme stats.

3 Effcet algorthms for schedulablty aalyss ad prorty assgmet for fxed-prorty preemptve schedulg Bacgroud ad related wor 2. FPPS ad prorty assgmet We assume the followg tas model ths paper: cosder a tasset of N depedet tass {τ, τ 2,..., τ N } wth FPPS rug o a u-processor. Each tas τ has perod T ; costraed deadle D T ; ow worst-case executo tme (WCET) C ; fxed release offset O T. Tass do ot susped themselves durg executo. We assume the umber of avalable prorty levels supported by the OS to be larger tha or equal to the umber of tass the tasset, hece o two tass share the same prorty level. For a sychroous perodc tasset o a sgle-processor, there are maly two techques for schedulablty aalyss: utlsato boud test ad worst-case respose tme (WCRT) aalyss. Tae rate mootoc schedulg (RMS) for example, where tas prortes are assged reverse order of ther perods. The well-ow Lu ad Laylad utlsato boud test states that a tasset wth N tass s schedulable f the total utlsato does ot exceed N (2 /N ). Ths s a suffcet but ot ecessary codto, ad rejects some tassets that are schedulable. Utlsato boud tests are pessmstc,.e., a schedulable tasset may be determed to be oschedulable. Respose tme aalyss s a wdely-used techque for schedulablty aalyss techque. For a sychroous tasset, the crtcal stat s a tme stat whe all tass are released smultaeously, whch causes each tas to experece ts WCRT. If each tas s deadle does ot exceed ts perod (D T, ), tas τ s WCRT ca be calculated wth a recursve equato that computes the smallest possble total processor demad whe τ ad all other hgher-prorty tass are released at the crtcal stat: R = C + B + R / Tj Cj. () hp() where R s the WCRT of tas τ ; B s the worst-case blocg tme of tas τ, equal to 0 the absece of shared resources; hp() deotes the set of hgher-prorty tass tha τ. A tas s schedulable f ts WCRT does ot exceed ts deadle, ad the tasset s schedulable f all tass are schedulable. Ths s a ecessary ad suffcet codto for schedulablty of a sychroous tasset. Ufortuately, there does ot exst ay closed-form equato for calculatg the WCRT of a asychroous tasset. Leug ad Whtehead (982) proved that for a asychroous tasset wth FPPS, t s suffcet to chec the tme terval [O max, O max + 2P], called the feasblty terval, where O max s the maxmum release offset of all tass, ad P s the hyper-perod of the tasset. A tasset s schedulable f every tas stace the feasblty terval meets ts deadle. It s fact suffcet to chec a smaller feasblty terval of [ Omax, Omax + P ] for schedulablty aalyss of tas τ, where P = lcm(t,..., T ) s the hyper-perod (least-commo multple) of all tas perods wth hgher prorty tha τ ad τ tself; O s the max maxmum release offset of tass wth hgher prorty tha τ ad τ tself. For ay τ,, the schedule repeats at tervals defed by P after a tal stablsato tme of S = Omax / T T, e.g., the schedule repeats every P = P 2 tme uts wth respect to τ ; every P 2 = lcm(t, T 2 ) tme uts wth respect to τ 2, etc. Audsley (99) preseted a algorthm for schedulablty aalyss of tas τ by rug a smulato the feasblty terval [ Omax, Omax + P ]. For each stace τ of tas τ released at tme stat t the feasblty terval, the algorthm computes the total terferece delay suffered by τ from all hgher-prorty tass the tme terval [t, t + D ), whch cossts of remag terferece due to hgher-prorty tass released before t ad have ot fshed executo, ad created terferece due to hgher-prorty tass released the tme terval [t, t + D ). If the total terferece delay plus τ s WCET C does ot exceed D, the τ meets ts deadle. If all staces of τ released the feasblty terval meet ther deadles, the τ meets ts deadle [the orgal algorthm Audsley (99) checs schedulablty oly ad does ot compute tas respose tmes, but t s straghtforward to modfy the algorthm to retur the actual respose tme of each tas stace]. Besdes schedulablty aalyss, aother mportat problem real-tme system desg s prorty assgmet to acheve schedulablty. The deadle mootoc prorty assgmet polcy, where each tas s assged a prorty versely proportoal to ts deadle, s ot optmal for a asychroous tasset wth FPPS. The optmal prorty assgmet (OPA) algorthm (Audsley, 99) s show Algorthm. It wors by tryg to assg a tas to prorty level j so that the tas s feasble, startg from the lowest prorty-level N to the hghest prorty-level. Let Δ deote the set of tass wth uassged prortes. If ay tas wth uassged prorty τ Δ ca be feasbly assged the lowest avalable prorty j, the assg p = j, remove τ from A, ad move to the ext hgher-level prorty j. Durg ths algorthm, f oe of the tass Δ ca be feasbly assged the lowest avalable prorty j, the declare the tasset to be feasble wth ay prorty assgmet. For a tasset wth N tass, the OPA algorthm performs at most N(N + )/2 schedulablty tests, ad s guarateed to fd a feasble prorty assgmet f oe exsts, cotract to the ave approach of exhaustvely tryg all possble prorty assgmets wth N! schedulablty tests. I geeral, the OPA algorthm s applcable f the followg four codtos are satsfed: Codto : a tas s WCRT s depedet o the set of hgher prorty tass, but ot o the relatve prorty orderg of those tass. Codto 2: a tas s WCRT may be depedet o the set of lower prorty tass, but ot o the relatve prorty orderg of those tass. Codto 3: whe prortes of ay two tass are swapped, the WCRT of the tas beg assged a hgher

4 306 Z. Gu ad H. Ca prorty caot crease wth respect to ts prevous value. Codto 4: whe prortes of ay two tass are swapped, the WCRT of the tas beg assged a lower prorty caot decrease wth respect to ts prevous value. Algorthm Pseudo-code of the Audsley s OPA algorthm Iput: Tasset Δ = {τ, τ 2,... τ N } wth ow T, D ad O but uassged prorty for each tas τ. Output: A feasble or feasble decso, plus a prorty assgmet f feasble. for j N to do 2 prortyassged(j) = false; 3 foreach τ Δ do 4 f τ s schedulable at prorty level j the 5 p = j; 6 Δ = Δ τ ; 7 prortyassged(j) = true; 8 f!prortyassged(j) the 9 retur feasble; 0 retur feasble; 2.2 Trasacto-based tas model Tdell (994) preseted WCRT aalyss algorthms for the trasacto-based tas model wth FPPS. The system cossts of multple ed-to-ed trasactos; each trasacto cossts of multple tass; each tas has a release offset (wth possble release jtter) relatve to the trasacto start tme; there are o offset costrats betwee dfferet trasactos. Tdell developed a exact aalyss algorthm wth expoetal tme complexty ad a pessmstc tractable aalyss algorthm wth polyomal tme complexty. The trasacto-based tas model s a more expressve model that ecompasses both the sychroous tas model ad asychroous tas model as specal cases: a sychroous tasset ca be vewed as a trasacto-based tasset cosstg of multple trasactos, each cosstg of a sgle tas wth zero release jtter; a asychroous tasset ca be vewed as a trasacto-based tasset cosstg of a sgle trasacto wth perod equal to the LCM of perods of all tass, ad wth zero release jtter for all tass. The ma motvato behd the trasacto-based tas model s for aalyss of dstrbuted real-tme systems, where tas release jtters arse due to the eed to model evet-trggered precedece-costraed tas chas a dstrbuted real-tme system,.e., a dowstream tas τ j s trggered by the completo evet of ts precedg tas τ, hece τ j herts ts release offset ad jtter from τ s respose tme ad jtter. The tas model assumed ths paper ad Audsley (99) s for a u-processor system. Tas offsets are set by the desger ad eforced by the cetral OS scheduler o a sgle processor for the purpose of spreadg out the worload over tme ad mprovg schedulablty, whch justfes the statc offset ad zero release jtter assumptos. 2.3 Shared resources I ths paper, we assume the tasset s fully preemptve wth o shared resources, but our algorthms ca be easly exteded to address shared resources usg a resource sychrosato protocol such as prorty-celg protocol (PCP). The exact WCRT aalyss techque for a asychroous tasset wth o shared resources amouts to rug a fathful smulato, wth tas release offsets, the tme terval (tasset hyper-perod + max release offset). We cosder two methods of extedg the WCRT aalyss techque preseted ths paper to tag to accout blocg tme, assumg PCP s adopted: Method : Ru a fathful smulato of tas executo wth all shared resources ad actual blocg tme expereced by each tas. Method 2. Ru a optmstc smulato assumg full preemptblty (o shared resources ad 0 blocg tme); after havg ru the smulato ad obtaed the (overly-optmstc ad correct) WCRT of each tas τ, add a addtoal term of max blocg tme to τ s WCRT equal to the largest crtcal secto legth of ts lower-prorty tass. It ca be easly show that Method s usafe, whle Method 2 s correct, thas to the well-ow property of PCP that a tas ca be bloced oly oce durg each vocato by ts lower-prorty tass. Usg Method 2, t ca be easly addressed by addg a blocg tme term to a tas s WCRT obtaed wth the aalyss techques, hece the ssue of shared resources ad blocg tme s orthogoal to the ma cotrbutos of ths paper. 3 Exact respose tme aalyss ad prorty assgmet 3. Computg busy perods Defto 3.: A prorty-level- busy perod s a terval [t s, t e ) wth the followg three propertes: PL (t s ) =0 2 PL (t e ) = 0 3 PL (t) > 0 for all t (t s, t e ). where the pedg load PL (t) s the amout of processg at tme t that stll eeds to be performed for the actve jobs of tass wth hgher or equal prorty (τ j wth j ) that are released before tme t.

5 Effcet algorthms for schedulablty aalyss ad prorty assgmet for fxed-prorty preemptve schedulg 307 A wor-coservg schedulg algorthm ever leaves the processor dle whe ay tas s ready for executo. We prove the followg theorem for ay wor-coservg schedulg algorthm: Theorem : For a asychroous tasset wth N tass o a sgle processor, the prorty-level-n busy perods are the same for ay wor-coservg schedulg algorthm ad a determstc sequece of executo tmes for each tas. By a determstc sequece of executo tmes for each tas, we mea that eve though ay gve tas s executo tme may exhbt rutme varatos at dfferet vocatos, a tas has the same sequece of executo tmes over multple perods whe comparg betwee dfferet schedulg algorthms. Proof: The tas schedule s fully determstc for such a asychroous tasset wth ay wor-coservg schedulg algorthm, sce both the tas release tmes, executo tmes ad schedulg algorthm are determstc. Dfferet schedulg algorthms may produce dfferet schedules, but the CPU busy/dle status at ay tme stat t s depedet of the schedulg algorthm used as log as t s wor-coservg. We ca plot the total worload request fucto by all N tass, wr(t), as a star-case shaped curve, ad the total worload completo fucto wc(t) as a slope wth 45-degree agle. Both curves are fully determstc ad depedet of the wor-coservg schedulg algorthm used. The ed tme of ay busy terval s the tme stat whe the two curves tersect,.e., wr(t) = wc(t). The start tme of ay busy terval s the arrval tme of the frst tas stace that arrves after or at the ed tme of the precedg busy terval. Hece the prorty-level-n busy/dle perods are fully determed by the fuctos wr(t) ad wc(t), both of whch are detcal for ay wor-coservg schedulg algorthm. Therefore we coclude that the prorty-level-n busy/dle perods are the same for ay wor-coservg schedulg algorthm. Ed. Theorem states that eve though the executo trace wth each busy perod may be dfferet for dfferet schedulg algorthms or prorty ordergs, the set of busy perods are always the same for ay wor-coservg schedulg algorthm. Gve Theorem, we ca prove the followg theorem: Theorem 2: For a asychroous tasset wth FPPS ad a determstc sequece of executo tmes for each tas, the executo trace (start ad fsh tmes o the schedule tmele,.e., the Gatt Chart) of ay tas τ s oly depedet o ts posto the overall prorty orderg of all tass, ot o the relatve prorty orderg amog the set of hgher or lower-prorty tass. Proof: Tas τ ca execute oly durg the gaps betwee the prorty-level-( ) busy perods. From Theorem, we ow that the prorty-level-( ) busy perods are depedet of the relatve prorty orderg amog the set of hgher-prorty tass τ,..., τ. Therefore, τ s executo trace s depedet of the relatve prorty orderg amog ts hgher-prorty tass. Sce τ s executo s ot affected by the set of (N ) lower-prorty tass τ +,..., τ N, ts executo trace s also depedet of the relatve prorty orderg amog ts lower-prorty tass. Ed. The ave approach to computg the busy perods for a perodc tasset s to perform smulato from tme 0 to ed of the feasblty terval (O max + 2P for fxed-prorty o-preemptve schedulg, ad O max + P for FPPS), whch may be tme-cosumg f P s large. I ths secto, we preset a effcet algorthm based o solvg a set of recursve equatos. We assume that each tas s executo tme s determstc ad equal to ts WCET, although the same algorthm ca also hadle a determstc sequece of executo tmes for each tas that may be dfferet for dfferet perods, as metoed Theorem. Fgure 2 Examples for llustratg equato (2), (a) start( BP ) O (b) start( BP ) > O (a) (b) To fd the frst prorty-level- busy perod, the algorthm wors teratvely creasg tme order, startg from the frst busy perod BP. We use BP for llustrato purposes the followg explaatos. We shft offsets of tass wth hgher prorty tha ( ) to be relatve to start tme of the frst prorty-level- busy perod start( BP ): ( ( ) ) start BP O T start BP O T O = start BP > O ( ), f ( ) O start( BP ) O, f start( BP ) O Equato (2) performs the same operato as the offset refemet theorem Audsley (99), but t s reformulated wth our otato. Fgure 2(a) llustrates the case of start( BP ) O, where offset of τ relatve to start( BP ) s obtaed va a smple subtracto; Fgure 2(b) llustrates the case of start( BP ) > O, where we eed to frst fd the earest ext release tme of τ after start( BP ), (2)

6 308 Z. Gu ad H. Ca whch s ( start ( BP ) O) ( start ( BP ) O) T T relatve to O, hece T ( start( BP ) O) T relatve start( BP ). The worload request fucto wr(t) expresses the total amout of worload requested the terval [0, t) by tass τ,..., τ : t O ( ) wr() t = C T (3) We ca tur t to a recursve equato ad solve t utl the fxed-pot (wr(t) = t) to obta legth of the frst prorty-level- busy perod: ( ) le BP ( ) le BP O( ) = C (4) T Ital value of le( BP ) ca be set to the executo tme of the frst tas the busy perod after start( BP ). After obtag le( BP ), the ed of busy perod start( BP ) ca be obtaed as start( BP ) + le( BP ). To fd the ext prorty-level- busy perod, we fd rel xt, the ext earest tas release tme after ed( BP ) relatve to start( BP ), as start tme of the secod prorty-level- busy perod BP 2 : ( ) ( ) ed BP start BP O( ) relxt = m T + O ( ) (5) T Note that f multple tass are released smultaeously at tme rel xt, the a arbtrary tas amog them ca be chose to start at rel xt wthout affectg the computato result. We the shft all tas offsets to be relatve to rel xt, ad solve equato (4) to determe the legth of the ext prorty-level- busy perod. Ths process s repeated utl a user-defed maxmum schedule legth s reached, whch should be set to O max + P for FPPS. Fgure 3 llustrates equato (5) for fdg the ext release tme of τ after ed( BP ) (ote that all tme values, cludg O () ad rel xt, are relatve to the tme pot start( BP ). Fgure 3 Example for llustratg equato (5) As a example, we cosder the tasset Table 2, ad show how to use Algorthm 2 to obta legths of the frst two prorty-level-5 busy perods BP 5. From Theorem, the prorty-level-n busy perods are depedet of the prorty orderg amog tass for FPPS, hece the prorty assgmet does ot matter ad s ot show Table 2. Table 2 Tasset example Tas T C O D τ τ τ τ τ Algorthm 2 Algorthm for obtag legths of all prorty-level- busy perods. Iput: Tasset Δ = {τ, τ 2,... τ N } wth ow T, D, O, p. Output: Legths of all prorty-level- busy perods the hyper-perod Italse rel = start( BP ); xt 2 whle rel xt < O max + P do 3 Shft all tas offsets to be relatve to rel xt wth equato (2); 4 Obta legth of the ext prorty-level- busy perod wth equato (4); 5 Fd rel xt wth equato (5); Sce the frst tas stace (τ 4 ) s released at tme 0, the offset shft operato equato (2) s a ull operato, ad O = O for all τ. The worload request fucto of all fve tass from tme 0 s: wr() t wr() t 3 wr() t = wr() t 8 wr() t 0 wr() t

7 Effcet algorthms for schedulablty aalyss ad prorty assgmet for fxed-prorty preemptve schedulg 309 wth tal value wr(t) = C 4 = 3. Solvg the recursve equato yelds wr(t) = 29. The frst busy perod BP 5 s thus 2 obtaed to be [0, 29). The ext busy perod BP 5 starts at: rel xt = m 0, 20 3, , 5 0, = 30 2 Both tas staces τ 43 ad τ 53 are released at the start of BP 5 at tme 30. We use equato (2) to shft ther release offsets to be relatve to tme 30, ad obta the ew offsets O =, O2 = 3, O3 = 8, O4 = O5 = 0. We the use equato 2 (4) to obta BP 5 as [30, 37) (relatve to tme 0). 3.2 Obtag tas respose tmes from busy perods I ths secto, we prove Theorem 5 that provdes a effcet algorthm for computg respose tmes of all staces of tas τ oce all the prorty-level-p busy perods the feasblty terval are determed. Lemma 3: Gve a tasset wth FPPS, f oe or more staces of tas τ wth prorty p are released wth a prorty level-p busy perod, the ths busy perod must ed at the completo tme of the last stace of τ released wth the busy perod. Proof: We use proof-by-cotradcto. Let s deote the last stace of τ released the busy perod as τ m. Suppose the busy perod eds wth the completo tme of a tas stace τ j wth hgher prorty tha τ m, the τ m must have completed at tme t before or at τ j s release tme. Otherwse, τ m would be preempted by τ j ad complete after τ j s completo tme, so the busy perod would ed wth τ m s completo tme stead of τ j s. Accordg to Defto 3., a prorty-level- busy perod eds at the earlest tme ed(bp ) whe the processor becomes dle, ad there are o tas staces wth prorty or hgher watg to be executed that were released strctly before tme ed(bp ), τ m s completo tme t satsfes these codtos, so t must be the ed of the prorty-level-p busy perod. Ths causes a cotradcto wth our assumpto that the prorty-level-p eds wth the completo tme of the hgher-prorty tas stace τ j. Ed. Lemma 4: Cosder a perodc tasset wth FPPS, where every tas s deadle s ot greater tha ts perod (, D T ). If every stace of tas τ meets ts deadle, the ay prorty level-p busy perod must cota 0 or stace of τ. Proof: From Lemma 3, we ow that ay prorty level-p busy perod BP p ether cotas 0 stace of τ, or cotas or more staces of τ ad eds wth the completo tme of the last stace. We wll assume there are two or more staces of τ wth BP p ad the derve a cotradcto. Suppose τ m s the o-last stace amog multple staces of τ a sgle busy perod BP p, the τ m must fsh before or at the release tme of ts ext stace τ m+ wth BP p, sce f m r m + D r m + T = r,m+. Sce BP p s a cotguous terval of tme durg whch oly tass wth prorty ot lower tha p executes, the tme terval [f m, r,m+ ) must be occuped by tas staces wth prorty strctly hgher tha p. Collectvely, these hgher-prorty tas staces must start at exactly tme f m ad fsh at exactly tme r,m+. They caot be released strctly before f m, otherwse τ m caot fsh at f m. Accordg to Defto 3., f m must be the ed of the prevous prorty level-p busy perod, fm = ed( BP p ). Ths causes a cotradcto wth our assumpto that τ m s the o-last stace amog multple staces of τ the busy perod BP p. Ed. Fgure 4 shows the schedule the tme terval [0, 28] for the tasset Table 3, ad the prorty-level-3 ad level-2 busy perods. The reader ca verfy that each prorty-level-3 busy perod ( BP 3 ) deed cotas 0 or staces of τ 3, ad each prorty-level-2 busy perod ( BP 2 ) cotas 0 or staces of τ 2. Note that there are seve prorty-level-2 busy perods stead of 5, accordg to Defto 3. of the busy perod as a left-closed ad rght-ope terval. Table 3 Tasset example 2 Tas T D C O τ τ τ

8 30 Z. Gu ad H. Ca Fgure 4 Schedule tme terval [0, 28] for the tasset Table 3 wth preemptve schedulg ad prorty orderg from hgh to low (τ, τ 2, τ 3 ) Theorem 5: Gve a tasset wth FPPS, where a tas τ s deadle s ot greater tha ts perod (D T ), a tas stace τ m released wth a prorty level-p busy perod BP p meets ts deadle f ad oly f the busy perod eds before or at ts deadle,.e. ( p ) m < ( p ) ( f ( ) ) m r m D ed BP p r m D start BP r ed BP Proof: We treat the followg codto as the mplct commo precodto the followg dervatos ad omt t from the equatos: ( p ) m ( p ) start BP r < ed BP (6) Based o Defto 3., we ow that ay stace τ m released wth a prorty level-p busy perod BP p must fsh before or at the ed of the busy perod,.e., f ed( BP ), whch leads to: m ( ) p ed BP r D f r D (7) p m m m Accordg to Lemmas 3 ad 4, f τ m meets ts deadle, the t s the oly stace of τ the busy perod BP p, ad ts fsh tme cocdes wth the ed of BP : ( ) f r D f = ed BP (8) m m m p Hece: ( ) f r D ed BP r D (9) m m p m Equatos (7) ad (9), combed wth the commo precodto 6, lead to equato (5). Ed. Oce all the prorty-level-p busy perods are ow, Theorem 5 gves us a effcet techque for computg respose tme of ay tas stace of τ wth a smple subtracto operato of ed( BPp ) rm. Eve though Audsley (99) used the OPA algorthm to assg tas prortes for a asychroous tasset wth FPPS, t has ever bee formally show that the OPA algorthm s deed applcable. We show the applcablty of the OPA algorthm Theorem 6. p Theorem 6: Audsley s OPA algorthm (Algorthm ) s applcable to a asychroous tasset wth FPPS. Proof: Cosder the four codtos ecessary for the OPA algorthm to be applcable, preseted Secto. Codtos ad 2 are obvously true gve Theorem 2, sce each tas s respose tme ca be derved from the executo trace. To prove Codtos 3 ad 4, cosder two tass τ ad τ j wth p < p j,.e., τ has hgher prorty tha τ j. If ther prorty assgmets are swapped wth ew prorty assgmets p j = p ad p = pj, the some tass wth prortes p,..., p j are removed from the set of hgher-prorty tass tha τ j, hece o loger cotrbute to the prorty-level- p j busy perods. Therefore, every prorty-level- p j busy perod must have shorter or equal legth compared to the orgal prorty orderg. From Theorem 5, we ow that respose tme of each stace of τ j ca be derved as the dfferece betwee the ed of each prorty-level- p j busy perod ad the release tme of the stace of τ j that busy perod, hece t caot crease after the prorty swappg. Therefore, τ j s WCRT, the maxmum respose tme amog all ts staces wth the hyper-perod, caot crease after the prorty swappg. Smlarly, some tass wth prortes p +,..., p j are added to the set of hgher-prorty tass tha τ. Therefore, every prorty-level-p busy perod must have loger or equal legth compared to the orgal prorty orderg, ad τ s WCRT caot decrease after the prorty swappg. Ed. 3.3 Icremetal costructo of busy perods Theorem 5 gves us a effcet algorthm for computg respose tmes of all staces of tas τ, hece determg schedulablty of τ, oce all the prorty-level-p busy perods are determed. We ow preset a cremetal approach that reduces the overall rug tme of the OPA algorthm by costructg prorty-level-(p ) busy perods from prorty-level-p busy perods by removg all staces of τ from the prorty-level-p busy perods. We explot the fact that Audsley s OPA algorthm assgs prortes from the lowest to the hghest level, ad the observato that the busy perods ad executo traces of lower-prorty tass are depedet of the relatve prorty orderg amog hgher-prorty tass. We record ths formato ad reuse t to reduce computatoal load

9 Effcet algorthms for schedulablty aalyss ad prorty assgmet for fxed-prorty preemptve schedulg 3 ad mprove effcecy of schedulablty aalyss for hgher-prorty tass. Algorthm 3 shows the algorthm pseudo-code. Algorthm 3 Icremetal costructo of prorty-level-(p ) busy perods by removg staces of τ from the executo trace All prorty-level-p busy perods are stored a sorted lst L bp ; 2 Set r lp = r 0, release tme of the frst stace of τ ; 3 whle r lp maxlegth do 4 Use bary search to fd the busy perod that cotas r lp,.e., m m ( ) lp ( ); start BP r ed BP m BP L bp tas after ext s r hp = r 2 = 4. We the obta the ext 2 prorty-level- busy perod BP as the tme terval [4, 5). The ext release of τ 2 s at tme 7, ed of BP 2, hece the program breas out of the whle loop at Le 0. Now L bp cossts of two prorty-level- busy perods [, 2) ad [4, 5). We repeat ths procedure utl a user-specfed maxmum schedule legth maxlegth s reached, whch ca be set to ed of the feasblty terval [ Omax, Omax + P]. Fgure 5 Icremetal costructo of prorty-level- busy perods by removg staces of lower-prorty tas τ 2 from prorty-level-2 busy perod, (a) prorty-level-2 busy perods (b) prorty-level- busy perod obtaed by removg all staces of τ 2 from the prorty-level-2 busy perod 5 Remove m BP from L bp ; 6 m ext = start ( BP ); 7 whle true do 8 Fd the frst release tme of ay tas r hp wth hgher-prorty tha τ after ext; 9 m f r ed( BP ) the hp 0 brea; else 2 Obta the prorty-level-(p ) busy perod BP that starts at r hp, ad add t to L bp ; 3 ext = ed ( BPp ) 4 r lp = r lp + T ; p ; Whe the algorthm fshes, we have trasformed all etres L bp from prorty-level-p busy perods to prorty-level-p busy perods. As a example, cosder the partal schedule Fgure 5(a) cosstg of a prorty-level-2 busy perod BP 2. We ca costruct the prorty-level- busy perods by removg all staces of the lower-prorty tas τ 2 from the schedule to obta BP Fgure 5(b). The prorty-level-2 busy perod BP 2 cotag the frst release of τ 2 s the tme terval [0, 7), ad L bp cossts of a sgle tme terval [0, 7). r lp = r 2 = 0. We frst remove BP 2 from the lst of busy perods, so ow m L bp becomes a empty set. Set ext = start( BP2 ) = 0. The frst release of hgher-prorty tas τ after ext s r hp = r =. We obta the frst prorty-level- busy perod BP as the tme terval [, 2), ad add t to L bp. ext = ed( BP ) = 2. We go bac to the whle statemet at Le 9. The frst release tme of ay hgher-prorty (a) (b) The cremetal costructo of hgher-prorty-level busy perods by removg staces of lower-prorty tass from the executo trace meas that we must fsh aalysg respose tmes of tass wth lower-prorty tha τ before aalysg respose tme of τ. Hece ths algorthm should be used combato wth the OPA algorthm, whch tres to assg a tas to each prorty level gog from lower to hgher prorty-levels. 4 Effcet coservatve aalyss Istead of exact schedulablty aalyss dscussed Secto 3, Pellzzo ad Lpar (2005) preseted a coservatve pseudo-polyomal algorthm for respose tme aalyss of a asychroous tasset wth EDF schedulg, as a suffcet but ot ecessary codto for schedulablty. The basc dea s as follows: there s always a tal tas released at tme t that starts the crtcal busy perod, but we do ot ow whch tas t s. We buld a ew tasset Γ for each possble tal tas τ, N. Sce τ s released at the begg of the busy perod, we fx φ = 0 Γ, ad chec the busy perod startg from tme 0 stead of t, wth all the other tass release offsets adjusted accordgly. We adapt ther aalyss techque to FPPS ths secto.

10 32 Z. Gu ad H. Ca Lemma 7 from Pellzzo ad Lpar (2005) s applcable to ay asychroous tasset depedet of the schedulg algorthm: Lemma 7: Gve two tass τ ad τ j, the mmum tme dstace betwee ay release tme of tas τ ad the successve release tme of tas τ j s equal to O O Δ = + gcd ( T, Tj) (, ) j j Oj O gcd T Tj (0) where gcd(t, T j ) deotes the fucto of computg the greatest commo dvsor of T ad T j. Defto 4.: Gve tasset Γ, Γ s the tasset wth the same tass as Γ but wth offsets: O = 0 O =Δ j, j N j As a example, Fgure 6 compares the tasset Γ ad Γ wth τ as the oe fxed tas, ad the rest of the tass are shfted leftwards towards the frst stace of τ. We ca see that the tasset Γ has more pessmstc tas offset relatoshps compared to the orgal tasset Γ from the perspectve of τ, sce t s costructed by assumg the worst-case (smallest) possble separato dstaces betwee τ ad other tass all occur relatve to the frst stace of τ, whle for the orgal tasset Γ, such worst-case separato occurs relatve to dfferet staces of τ. Next, we prove a theorem that shows f τ s schedulable tasset Γ, the t must also be schedulable the orgal tasset Γ. Assume τ a s the tas uder aalyss, the we oly eed to cosder τ a ad tass wth hgher or equal prorty tha τ a whe aalysg schedulablty of τ a. Theorem 8: Gve tasset Γ cosstg of τ a ad tass wth hgher prorty tha τ a wth FPPS o a sgle processor. If τ a the tasset Γ, τ hep(τ a ) meets ts deadle the frst busy perod, the τ a s schedulable. Proof: Accordg to Theorem 5, ay gve tas stace of τ a released wth a prorty-level-p a busy perod meets ts deadle f ad oly f the busy perod eds before or at ts deadle,.e., there must be oly oe stace of τ a a prorty-level-p a busy perod. Suppose the WCRT of τ a occurs the busy perod [t, t 2 ) started by some tas, say τ. Let r j (j deotes the tas dex, ad deotes the tas stace dex) deote the ext release tme of ay τ j Γ, j after t, the we have r jl t + Δ j based o Lemma 7. I tasset Γ, the offset of τ j s equal to Δ j,.e., each tas τ j s released earler compared to τ j the schedule of Γ startg at t. Ths meas that the amout of terferece from hgher or equal prorty tass to τ a Γ must be larger or equal to that the orgal tasset Γ. Therefore, the respose tme of τ a the frst busy perod of Γ must be larger tha or equal to ts respose tme Γ. Ths leads to the cocluso that f all staces of τ a the tasset Γ, τ hep(τ a ) meet ther deadles the frst busy perod, the τ a s schedulable. Ed Next, we preset the calculato of the respose tme of τ a the frst busy perod. The legth of the frst level-a busy perod, le( BP ), s calculated as follows: ( a ) le BP a ( a ) le BP O( a) = C j () hep( a) T τ τ Fgure 6 Example tasset for llustrato of Defto 4., (a) tasset Γ, O = 0, O 2 =, O 3 = 2 (b) tasset Γ, O = 0, O 2 = r23 r24 = 0, O 3 = r3 r = 2 (a) Note: The tas perods are: T = 3, T 2 = 4, T 3 = 6. (b)

11 Effcet algorthms for schedulablty aalyss ad prorty assgmet for fxed-prorty preemptve schedulg 33 The above equato ca be solved recursvely wth the tal value equal to C, executo tme of the frst tas the busy perod. There are two cases based o the value of le( BP ): a le( BPa) O( a),.e., τ a s released later tha the ed of the frst busy perod. I ths case, τ a wll be cluded the frst busy perod of aother tasset Γ, j so we sp ths case. le( BPa) > O( a),.e., τ a s cluded the frst busy perod. Sce there must be oly oe stace of τ a the busy perod, we ca calculate WCRT of τ a as follows: Ra O ( a) Ra = Ca + Cj hp( ) T (2) τ τ j j a WCRT of τ a uder the coservatve aalyss, obtaed as follows: R a s Ra = max Ra (3) τ hep( τ ) a τ a s schedulable f Ra Da. The above coservatve approach s called the -fxed tas approach. I order to further reduce the pessmsm the aalyss, we ca geeralse Theorem 8 to -fxed tas approach. Istead of fxg the temporal posto of oly oe tas τ, we fx the postos of two or more tass, the costruct a ew tasset by mmsg the dstace of other tass relatve to the fxed tass. Cosder the 2-fxed tas approach. If τ s chose to be the frst fxed tas, ad τ j s chose to be the secod fxed tas, the we calculate all possble ext release tmes of τ j relatve to τ, ad for each case, fd the mmum dstaces of all remag tass relatve to τ (τ j ) before applyg equatos () ad (2) to calculate respose tme of τ a. The computato complexty of the -fxed tas approach s expoetal wth creasg. I the extreme case, f = N (sze of Γ), the the N-fxed tas approach s equvalet to the exact WCRT aalyss. 5 Performace evaluato The expermets are ru o a Lux worstato wth a Itel dual-core 2.4 GHz processor ad 4GB of ma memory. All tme uts are secods uless otherwse specfed. 5. Comparso of computato effcecy I ths subsecto, we focus o the computato effcecy measured terms of algorthm rug tme. We cosder the followg algorthms: [TIN]. Tdell s (994) algorthm for schedulablty aalyss of preemptve schedulg of trasacto-based tas model. I order to obta precse results wthout pessmsm, the etre tasset s coverted to a sgle trasacto before applyg [TIN]. [AUD]. Audsley s (99) algorthm for schedulablty aalyss of preemptve schedulg by rug a smulato the feasblty terval. [INC]. Our algorthm for cremetal respose tme aalyss, preseted Secto 3. [F]. Coservatve aalyss wth oe fxed tas. [2F]. Coservatve aalyss wth two fxed tass. The algorthms [TIN], [AUD] ad [INC] are exact aalyss algorthms wthout ay pessmsm, whle [F] ad [2F] are coservatve pseudo-polyomal-tme algorthms. For prorty assgmet + schedulablty aalyss, we compare [OPA+AUD], [OPA+HGL], [OPA+F] ad [OPA+2F]. (we do ot cosder [OPA+TIN] sce [TIN] has worse performace tha [INC].) I Tables 4 ad 5, we assume tas prortes are ordered creasg order of tas dces,.e., tass wth lower dces have hgher prortes. The trasacto-based tas model s a more expressve model that cludes both the sychroous tas model ad asychroous tas model as specal cases. Tdell (994) showed how a asychroous tasset ca be trasformed to the trasacto-based tas model so that hs aalyss algorthm (exact or tractable) ca be appled to compute tas WCRTs. However, ths method ca be very effcet compared to the algorthm preseted ths paper for a tasset wth a large hyper-perod ad may tas staces the hyper-perod, as show Table 4 that compares betwee [TIN] ad [INC]. As Tdell (994), we group all tass to a sgle trasacto wth perod equal to the hyper-perod of the orgal tasset, hece a tas the orgal tas model wth a smaller perod tha the trasacto perod s tured to multple tass wth dfferet offsets the trasacto-based tas model. For example, cosder τ 4 as the tas uder aalyss. Trasformg the tasset to the trasacto-based tas model yelds a sgle trasacto wth perod equal to l cm (P, P 2, P 4, P 4 ) = 60,060, cosstg of a total umber of 3,875 tass. [TIN] fshes secods, whle [INC] fshes 0.04 secods. We coclude that, eve though the more geeral aalyss algorthm [TIN] ca be appled to our more restrctve asychroous tas model, the computatoal effcecy s very bad, ad the more specalsed algorthms [AUD] ad [INC] should be appled to the asychroous tas model stead of the more geeral algorthm [TIN]. Hece [TIN] s ot cluded the rest of the performace expermets.

12 34 Z. Gu ad H. Ca Table 4 Comparso betwee [TIN] ad [INC] for schedulablty aalyss Tas T D C O lcm(t..t ) R [TIN] [INC] τ τ τ , τ , τ , >2 hrs 0.08 Table 5 Comparsos betwee [AUD], [INC], [F] ad [2F]: algorthm rug tmes (RT) for schedulablty aalyss (SA), ad prorty assgmet (PA) at each prorty level, for a 20-tas tasset Tasset parameters RT for SA (s) RT for PA (s) Tas T D C O lcm(t..t ) [AUD] [INC] [F] [2F] [O+A] [O+H] [O+H] [O+F] [O+2F] τ τ τ τ τ τ τ τ τ τ τ 3,000 2, τ 2 5,00 5,00, τ 3 5,00 5, τ 4 5,400 5, τ 5 6,000 5, , τ 6 6,500 5, , τ 7 7,000 5, , τ 8 7,000 6, , τ 9 8,500 6, , τ 20 9,500 5, , Total cumulatve rug tme (s) 2, ,622.0,042.90, Notes: For space costrats, we use [O] as shorthad for [OPA]; [O+A] as shorthad for [OPA+AUD]; [O+H] as shorthad for [OPA+HGL]; [O+H] deotes [OPA+HGL] wthout cremetal costructo of busy perods. Next, we use a larger tasset wth 20 tass for performace comparsos amog the other algorthms. The results are show Table 5. The tasset s costructed to have a relatvely large hyper-perod, ad large umber of tas staces the feasblty terval, yet a relatvely small umber of tass to ft wth space lmtatos of the paper. For schedulablty aalyss, we assume all tas prortes are gve, ad tass wth lower dex values have hgher prortes. Results dcate that [INC] s faster tha [AUD] f certa tass have relatvely large perods ad WCET values, but may be slower tha [AUD] f all tass have small perods ad WCET values. (The former case s more computatoally demadg tha the latter case, hece algorthm effcecy s more mportat.) The explaato s as follows: effcecy of [INC] s determed by the algorthm for computg busy perods based o solvg recursve equatos, whch ca be very effcet f the busy perods are log ad few whe certa tass have large perods ad WCET values, sce each recurso step may advace a large dstace o the tmele. But t may be less effcet f the busy perods are short ad may whe all tass have small perods ad WCET values, where each recurso step may oly advace a small dstace before covergece, ad a ew set of recursve equatos eed to be solved for the ext busy perod. O the other had, effcecy of [AUD] s ot depedet o the patter of busy perods. We ca also see that the coservatve aalyss

13 Effcet algorthms for schedulablty aalyss ad prorty assgmet for fxed-prorty preemptve schedulg 35 algorthm [F] s very effcet compared to the exact aalyss algorthms, ad [2F] s slghtly less effcet. For performace evaluato of prorty assgmet, we assume all tas prortes are tally uow, ad use the OPA Algorthm to fd a feasble prorty assgmet. Rug tme of the OPA algorthm s hghly depedet upo the order whch each caddate tas s tred ad checed for schedulablty at each prorty level. I the best-case, the frst caddate tas s schedulable, so oly oe tas eeds to be checed for schedulablty at each prorty level; the worst-case, all caddate tass wth uassged prorty values eed to be checed for schedulablty at each prorty level before a schedulable tas s foud. To remove ths factor of ucertaty ad have more uform performace comparso betwee dfferet techques, we adopt a modfed verso of the OPA algorthm, where all tass are always checed for schedulablty at a gve prorty level, eve after oe or more tass have bee foud to be schedulable at prorty level-. I case multple tass are foud to be schedulable, a arbtrary oe s chose from them ad assged prorty level-. [INC] shows some performace mprovemets terms of reduced rug tmes compared to [AUD] terms of schedulablty aalyss, but the performace mprovemets of [OPA+HGL] (3,622 secods) over [OPA+AUD] (,042.9 secods) are more sgfcat for the cumulatve rug tmes. The colum labelled [O+H] deotes the rug tmes wthout usg the cremetal costructo. We coclude that the cremetal busy-perod costructo brgs vsble, but lmted mprovemets to computatoal effcecy. I fact, most of the effcecy mprovemet of [OPA+HGL] ca be attrbuted to two other causes: recursve computato of busy perods stead of rug smulato over the hyper-perod 2 whe combed wth the prorty assgmet, oce the prorty-level busy perods are obtaed, schedulablty checg of tas τ by obtag ts WCRT o ths level ca be doe effcetly by fdg each busy perod that cotas a stace of τ ad obtag ts respose tme as the dfferece betwee the ed of the busy perod ad ts release tme, accordg to Theorem 5. To further evaluate the scalablty of aalyss algorthms, we preset performace comparsos betwee [OPA+AUD] ad [OPA+HGL] for a 80-tas tasset the appedx. (The rug tmes for [OPA+F] ad [OPA+2F] are cosstetly eglgble compared to the exact aalyss algorthms, so they are omtted ths example.) We have also attempted to evaluate performace wth radomly-geerated tassets, but t s dffcult to show ay deftve tred based o parameter varatos, as the algorthm rug tme s depedet o several factors, cludg umber of tass, tasset hyper-perod, umber of tas staces the hyper-perod, tasset utlsato, etc. We smply ote that the relatve performace ratos betwee dfferet algorthms Table 5 are represetatve of the other radomly-geerated tassets. The theoretcal computatoal complexty s expoetal for both exact aalyss algorthms (HGL ad AUD), sce the algorthms must exame the etre hyper-perod legth to compute tas respose tmes, whch grows expoetally wth the tasset sze the geeral case, uless tas perods are harmoc to each other. However, eve amog expoetal algorthms, there ca be large dffereces computatoal effcecy. The actual computatoal worload for a specfc tasset s hghly depedet o tasset characterstcs, ad there does ot exst ay aalytcal, closed-form fucto to express the worload terms of the umber of tass. Although the aalyss tme reducto s ot spectacular, we beleve that the modest speedups of 5-0x that we acheved ca be stll qute useful practce for the followg possble reasos: If the aalyss s used for offle desg, t ca help crease desger effcecy durg a teractve desg process, smlar to beefts brought by a fast compler ad cremetal complato. It s mportat to reduce the rug tme of schedulablty aalyss algorthm, as a few mutes of watg tme for the aalyss result s much more tolerable tha a 40-mute watg tme durg the desg process. 2 It s possble that the schedulablty aalyss (wth or wthout prorty assgmet) are performed at rutme o the target processor, stead of at desg tme o the worstato, for ole acceptace test ad admsso cotrol. 5.2 Comparso of aalyss accuracy I ths subsecto, we evaluate the effectveess of the coservatve algorthm terms of aalyss accuracy compared to the exact schedulablty aalyss algorthm. We compare the followg algorthms: -fxed tas: the coservatve algorthm troduced Secto 4 wth fxed tas 2-fxed tas: the coservatve algorthm troduced Secto 4 wth 2 fxed tass sychroous: the classc respose tme aalyss equato [equato ()] for a sychroous tasset exact aalyss: exact respose tme aalyss algorthm (ether the orgal algorthm Audsley (99) or our algorthm wth mproved effcecy troduced Secto 3). We use radomly-geerated sythetc tassets our expermets. The total utlsato of each tasset s radomly chose the rage [0.5,.0]; all tas perods are multplers of a gve value (the gcd), whch s radomly chose the rage of ether [00, 200] or [00, 500]. 2,000 tassets are geerated for each value of gcd. Each tas s CPU utlsato s radomly chose accordg to a uform dstrbuto such that the total utlsato of the tasset s equal to the desred value (from 50% to 00%). Each tas s deadle s radomly chose the rage

14 36 Z. Gu ad H. Ca [0.5 * T, 0.8 * T ]. Offset of each tas s radomly chose the rage [0, T ]. Each tasset cossts of ether sx or te tass. The results are show Fgure 7. where the x-axs s the total utlsato, ad the y-axs s the percetage of tassets determed to be schedulable wth dfferet aalyss techques. A larger acceptace percetage dcates that the aalyss techque used s more accurate. We radomly geerated 0,000 tassets for each expermet [Fgures 7(a) 7(f)]. Fgure 7 Comparso of aalyss accuracy amog dfferet algorthms (see ole verso for colours) (a) (b) (c) (d) (e) (f)

15 Effcet algorthms for schedulablty aalyss ad prorty assgmet for fxed-prorty preemptve schedulg 37 We mae the followg observatos: -fxed tas ad 2-fxed tass geerate almost detcal results all expermets,.e., t s suffcet to have oly -fxed tas, ad havg more fxed tass leads to creased computato overhead but oly margal performace mprovemets. A larger gcd value leads to a larger mprovemet of -fxed tas or 2-fxed tass over sychroous. The explaato s as follows: from equato (0), we ca see that a larger value of gcd(t, T j ) geerally (but ot always) ted to produce a larger value of Δ j. Ths tred becomes more obvous whe gcd(t, T j ) grows large, as show Fgure 8. A larger value of Δ j meas that the releases of tass the frst busy perod of Γ s postpoed more relatve to the fxed tas(s), leadg to a smaller upper boud of WCRT. Therefore, the mprovemet of -fxed tas or 2-fxed tas over sychroous becomes larger. The performace gap betwee -fxed tas or 2-fxed tas ad exact aalyss creases wth creasg tasset sze. The explaato s as follows: creasg tasset sze meas that the relatve percetage of the fxed tass the -fxed tass techque (/N) becomes smaller, hece the aalyss becomes more pessmstc. Fgure 9 shows the average WCRT mprovemets of -fxed tas over the sychroous approach for dfferet value of gcd. The umber of tass each tasset s 5. Perods of tass are radom chose [00, 500]. The value of gcd s set to be 20, 50 ad 00, respectvely. Other tas parameter settgs are the same as the prevous expermet. We ca see that the mprovemets of -fxed tas over sychroous ca be qute sgfcat. Fgure 9 Average rato of mprovemet of tas WCRTs usg -fxed tas over sychroous (see ole verso for colours) Fgure 8 How Δ j vares wth gcd(t, T j ), wth O j O = 50 (see ole verso for colours) 6 Coclusos I ths paper, we have preseted effcet algorthms for schedulablty aalyss ad prorty assgmet for a asychroous perodc tasset wth FPPS o a uprocessor. Performace evaluato demostrates sgfcat performace mprovemets compared to exstg algorthms terms of both aalyss accuracy ad computato effcecy.