Real-time scheduling for systems with precedence, periodicity and latency constraints

Transcription

1 Real-tme schedulng fo systems wth pecedence, peodcty and latency constants lana ucu, Rémy Kock, Yves Soel () INRI Rocquencout, BP e hesnay edex, Fance llana.cucu@na.f, yves.soel@na.f (2) ESIEE, té Descates - BP 99-2, Bd Blase Pascal 9362 Nosy-le-Gand edex, Fance kock@esee.f bstact Fst we pesent the man esults concenng, n the one hand systems wth peodcty constants and deadlnes, and n the othe hand systems wth pecedence constants and deadlnes, n both cases fo one computng esouce. Then, we gve a model n ode to state clealy the poblem fo schedulng systems wth pecedence, peodcty and latency constants. In ode to solve ths poblem we gve a nonpeemptve, off-lne schedulng algothm whch uses n tun an algothm of latency makng. We demonstate the optmalty of the schedulng algothm, and afte povng the equvalence of the notons of latency constant and deadlne, we extend ths latte algothm fo schedulng ealtme systems wth pecedence, peodcty constants and deadlnes fo one computng esouce. Keywods: algothm, schedulng, optmalty, eal-tme, peodcty, latency, pecedence, deadlne. Intoducton In schedulng theoy of eal-tme systems, we have two man nteests: peodc systems and systems wth pecedence constants. Even f both felds ae sepaately ch n esults

2 n the lteatue, to ou best knowledge thee ae few esults whee both aspects ae teated togethe. The pupose of ths pape s to popose, n the one hand a model whch deals wth both aspects, and n the othe hand to solve the poblems of fndng a schedule (f thee s one) whch satsfes the peodcty, latency and the pecedence constants of the system, n the case of one pocesso. The pape stats wth notatons used to pesent the man esults fo peodc systems and fo systems wth pecedence constants. The next secton pesents the model used to descbe the poblem to be solved. The next secton pesents the algothm fo ths poblem wth the demonstaton of ts optmalty. The pape ends wth a concluson and futhe eseach. 2 Notatons and esults In ode to clealy dstngush the specfcaton level and ts assocated model we ae manly nteested n, fom the mplementaton level, we use the tem opeaton nstead of the commonly used task too closely elated to the mplementaton level. Fo the same eason we use the tem opeato nstead of pocesso o machne. The esults often found n the lteatue ae gven accodng to thee paametes: ] [], whee gves the numbe of opeatos and specfes f they ae of the [ same type (homogeneous machne) o not, the opeaton chaactestcs (peod, deadlne, computaton tme, pecedence constants) and the optmalty cteon (a elaton that allows us to compae possble solutons of the poblem). Fo an opeaton, we (defned fom the stat e- may specfy a computaton tme, a peod, a deadlne the of the peod, ethe of the schedule), a elease tme, a stat tme and a jtte!#%$')( wth *,+-+.+/ and,+0 (see fgue ). T + T s D s D + + Fgue : Basc eal-tme model Fo the poblem wth peodc opeatons and wthout pecedence constant [/ /], u and ayland gve the RMS algothm [2], mpoved by ehoczky and Sha (ntoducton of blockng ) [3] and by udsley (ntoducton of jtte by holstc analyss,[]) and the EDF algothm fo [/.+/ /]. The pecedence constants ae gven by a patal ode on the executon of the opea- means that B cannot stat befoe s executed) that may be epesented by tons (325 a dected acyclc gaph. The patal ode assocated to ths gaph defnes a potental paallelsm [5] on the set of opeatons. Fo example n fgue 2, and may be executed

3 n paallel on dffeent opeatos f we ae n the case 068, wheeas and must be executed sequentally. B D Fgue 2: Pecedence constants gaph In the case of systems wth pecedence constants whee all the opeatons have the same elease tme [/ pecedence/ mnmze the maxmum lateness], awle gves the fst-to-last-ule whch s optmal [6, ]. If the opeatons do not have the same elease tme we obtan a NP-had poblem [8], but f peempton s allowed the poblem s polynomal [9]. lso, fo the poblem [/ pecedence, D/] Blazewcz gves an polynomal soluton [0]. In the geneal case of systems fo whch all the opeatons may have pecedence, latency and also peodcty constants, to ou best knowledge, thee s no esult fo the poblem wth one opeato. 3 The model and the poblem to solve Each opeaton may belong to a pecedence constant,.e. belongs to a pa defnng the patal ode, o/and each opeaton may have a peodcty constant. Moeove, an opeaton may be epeated leadng to seveal nstances of the same opeaton ethe spatally wthout pecedence constant between consecutve epettons, defnng a local potental paallelsm, o tempoally wth pecedence constants between consecutve epettons. ctually, the eal-tme system nteacts wth the physcal envonment [], theefoe the gaph of pecedence constants s a patten tempoally nfntely epeated [5]. If one consde only the patten tself, accodng to ts patal ode the fst opeatons ae called nputs and the last opeatons ae called outputs. These opeatons coespond to espectvely sensos and actuatos. Fo example n the patten of fgue 3, s epeated tempoally thee tmes wheeas s epeated spatally two tmes, s the nput opeaton, and, :9 ae the output opeatons. ctually, the nfnte epetton of the patten nduces an nfnte epetton of all the opeatons. In ode to smplfy ths complex gaph model, that we call not factozed gaph, we defne a fnte epettve pecedence constant when fntely epeated (spatally o tempoally) opeatons ae n elaton of pecedence constant. Ths allows to epesent n

4 * * 2 2 B 2 B th epetton (+)th epetton Fgue 3: Not factozed gaph the model only one nstance of each epeated opeaton wth ts numbe of epettons, and to descbe synthetcally wth a functon code the exstng edges between two dffeent opeatons possbly belongng to dffeent epettons, o between the same opeatons belongng to dffeent consecutve epettons. ll these edges ae nta-patten. Due to the nfnte epetton of the patten, two nput opeatons o two output opeatons, belongng to two consecutve pattens, may be n elaton of pecedence that we call nfnte epettve pecedence constant. It s usually the case when we have a unque senso fo an nput and/o an unque actuato fo an output nstead of an nfnte numbe of sensos and/o actuatos. These edges ae nte-patten. The esultng gaph s sad factozed. FEGHIHJ%K NM Fo two opeatons and epeated tempoally o spatally, we have ;=<!>@?B : OEGHIHJQP) D D 2 * such as: V R$%TS@U WYX f an edge stats fom the $[Z\ epetton of ;=<!>@?B and ends on the S]Z\ epetton of othewse whee K s the numbe of epettons of and P fo. It s obvous to demonstate that the numbe of the possble edges between all the epettons of and s bounded by K_^`P (we may have at most P edges fom evey ). Fo an opeaton tempoally epeated K OEGHIHJQKcNM FEGHIHJ%K tmes, we have ;a<!>g?b : D D * such as: e$%fsg'hg ;=<!>@?d fo S,0$ and $UjkK othewse If thee s no edge between successve epettons of, the functon ;a< >@?b specfed. When the functon ;=<!>@?d s not s specfed the numbe of the possble edges between successve epettons of s K_, othewse *. Fo an nput (esp. output) opeaton whch s tempoally nfntely epeated, and also fntely tempoally epeated K tmes (nsde the patten), we have an edge l n the factozed gaph, coespondng to the nfnte pecedence constant n the not factozed

5 o * o gaph. If the ;a< >@?b has been defned fo then we have an edge between the last epetton bm and the fst epetton belongng to the next patten. If no ;a< >@?b has been defned, then we have an edge between and the belongng to the next patten, fo all the epettons, $U FEGnHHHaQK D. 3 codeb codeb B 2 code Fgue : Factozed gaph It s obvous to demonstate that a factozed gaph contans cycles, f and only f, the not factozed gaph contans cycles. Fom the not factozed gaph pesented n fgue 3 we obtan the factozed gaph FEGQpGqM llustated n fgue whch has the followng ;=<!>@? functons: ;=<!>@?B : $s FEGOpG um FED D D 2 * wth ;a<!>g?d (,)=, D, ;=<!>@?`t : %$T $v FE@ D D 2 * wth ;a< >@?`t = D and the functon ;=<!>@?d gven by the geneal defnton of the functon ;=<!>@? fo an opeaton tempoally epeated n tmes (n ths case, Kswp ). The opeatons and have an edge l. Remak 3. In the case whee we have n the factozed gaph two opeatons wth the same numbe K FEGHIHJ%K xm of epettons and the functon ;=<!>@?B : OEGHIHJQKc D D 2 * such that R$%TS@U g ;=<!>@?B f $ ys othewse n ode the smplfy the model, the functon ;=<!>@? wll not be specfed fo ths patcula case of epettve pecedence constant. lso, when the numbe of epettons of an opeaton s equal to, we do not menton t n the factozed gaph. Fnally, n ode to complete the model, we gve the two types of the eal-tme constants whch we want to be satsfed by the system, namely the latency and the peodcty. Defnton 3. fo two dffeent opeatons and belongng to the patten nfntely epeated, we say that the pa has a latency constant z when the opeatons have to be scheduled such that { :{ ` }+wz. Remak 3.2 we must have a dected path statng wth and endng wth f has a latency constant, because t only concens opeatons whch ae n elaton of pecedence constant. We call fst n the latency constant and last n the latency constant. We denote by ~ the set of all the pas of opeatons havng a latency constant. n opeaton may belong to seveal pas of opeatons havng dffeent latency constants. and

6 Œ Defnton 3.2 fo two consecutve epettons say that the fst epetton of. has a constant of peodcty f ` G ƒ and of the same opeaton, we $ (. We denote by N Remak 3.3 We assume that the peodc opeatons ae scheduled stctly wth the exact peodcty constants. Thus, t amounts to not allow any jtte. lso, we assume that all the peodcty constants ae multples of each othe. Wthout any loss of genealty [2], we assume that all the opeaton chaactestcs,.e. the fxed computaton tme (exactly known), the peodcty and the latency constants, ae defned as multples of a clock tck (tme s dscete). ftewads the values of the gven chaactestcs ae mplctly multpled by. ˆE ŠE In the example pesented n fgue 3 we have, :{ and, and Œ the followng eal-tme constants to be satsfed: the peodcty constants and % {, the latency constants: z Ž9 :9 * and z. The peodcty constant of can be defned because has a fnte epettve pecedence constant between ts thee successve epettons, and ths pecedence constant s completed by the nfntve epettve pecedence constant. The peodcty constant of can be defned only thanks to the nfnte epetton of the patten. The opeaton can not have a peodcty constant because of the potental paallelsm of and 9 whch does not mpose an ode of executon between and :9 each tme the patten s epeated. Then we can not have the noton of consecutve epettons necessay to defne the peodcty of. The poblem to be solved s the followng: fo the consdeed system modeled by a gaph wth epettve pecedence constants (epettve gaph), and wth latency and peodcty constants, we must fnd a feasble schedule. That s to say, a schedule whch gves the stat tme fo all the opeatons and whch satsfes the eal-tme constants fo the poblem: [/ epettve gaph/ peodcty and latency constants]. lgothm Ths secton s oganzed as follows: we pesent the algothm fo the poblem [/ epettve gaph/ peodcty and latency constants] and we show that t s optmal (f thee s a feasble schedule fo the poblem, the algothm wll fnd t). In ode to solve the poblem, we fst gve an algothm of latency makng fo the vetces of the patten, and then usng these latency maks we gve the schedulng algothm whch s a non-peemptve, off-lne algothm. Ths latte algothm s appled to the nfntely epeated patten.. lgothm of latency makng 8f 'O We denote by the dected acyclc gaph of opeatons wth epettve pecedence constants, whee s the set of vetces and ˆ M the set of edges, and œ by such that a dected path statng wth and endng wth.

7 ² l z Gven that some pas of opeatons have a latency constant, the goal of the algothm s to assgn to each opeaton a numbe. Ths numbe ndcates f an opeaton, be- wth a latency constant, wll be executed afte the executon of longng to a pa due to the exstence of a dected path between and, and no dected path between and. If fo an opeaton, thee ae seveal opeatons satsfyng ths popety, then the numbe wll be the smallest value of these latency constants. We call ths numbe P_ž q( ' l (l s a natual numbe bgge than all the natual numbes). The latency maks evolve dung the algothm of latency makng, whch s appled to the patten. emma. If a pa of opeatons has a latency constant z and f thee s an opeaton wth, then { :{ +z s satsfed!. Poof If has a latency constant, then and ˆ+!{. Because we have! h+ `. Then we have! +5 + `{ whch means that because s aleady scheduled when becomes schedulable, the stat tme of the opeaton s not affected by the stat tme of. lgothm Intalzaton: If has a latency constant z then P_ž and P_ž, othewse P_ž y P_ž ª l. Moeove, f belongs to seveal pas of opeatons havng dffeent latency constants, then P_ž U P_$eK «F { Ā± z, and P_ž l. We denote by ² the wokng-set and let ~., we have thee possbl- Step : fo b ³, ² and fo each opeaton! tes: (a) f, then accodng to emma., P_ž UwP_ž ; (b) f and, then P_ž U0P_$eKµRP_ž FQP_ž (c) f and, then P_ž 'wp_ž. The pa s emoved fom ². Step 2: f ² then goto step, othewse the algothm stops. % ; We apply the algothm of latency makng to the patten of the example depcted fgue. t the begnnng of the algothm, we have the wokng-set ² d9 ¹F` O :9 The table gves the values of the functons P_ž fo each opeaton of the patten. The fnal values of the functons P_ž ae gven by the last lne of ths table. These ae the values that wll be used by the schedulng algothm...2 Schedulng algothm The schedulng algothm tansfoms the patal ode assocated to the gaph n a (one of the possble) total ode satsfyng the constants. Ths algothm s appled to the n-

8 z z z «l * P_ž % P_ž P_ž P_ž P_ž % P_ž d9 bº :9 Intalzaton l l l l 9 0 % l l l l 9 0 B9 :9 l l Table : Results gven by the algothm of latency makng fntely epeated patten. We denote by ² the wokng-set, by!» the stat tme of the last opeaton that was scheduled, by :» ts computaton tme, and by ¼ the set of all the opeatons whch have a peodcty constant. Dung the algothm ² contans all the schedulable opeatons,.e. opeatons the pedecessos of whch ae aleady scheduled. We note that between two opeatons whch schedulable n the same tme thee s no path. Implctly, evey tme an opeaton s scheduled!» (esp. :» ) changes ts value nto the stat tme (esp. computaton tme) of ths opeaton. lgothm Intalzaton: ² NĀ½ and ¾ TÀeÁ J Ã! and» Ä» *. Step (opeaton wth latency constants): ² such that P_ž then `» Ä» such that P_ž / P_$eK P_ž O {Å næç, we emove t fom ², all the opeatons whch became schedulable ae added to ², and go to Step 5. Step 2 (opeaton wthout peodcty and whch ae not fst n a latency constant): x ² such that thee s no wth ~ and ¼, then!» Ä», we emove t fom ², all the opeatons whch became schedulable ae added to ², and go to Step. Step 3 (opeaton whch ae fst n a latency constant): 5 ² such that thee s wth 0 ~, then!» Ä» wth Ç P_ž]È «O É n± and næ z, we emove t fom ², all the opeatons whch became schedulable ae added to ², and go to Step 5. Step (opeaton wth peodcty fo whch each fst epetton s not aleady scheduled): 0P_$[K we have `» Ä» such that =ÆqÊÌË and Í not aleady scheduled!. Go to Step 6. Step 5: f ² Ïм wth Step 6: we seach an opeaton aleady scheduled, then go to Step. ² Ï ¼ fo whch ts fst epetton s scheduled

9 ¼ ² z and fo whch we have: N» Ä» wp_$[k {G Ñ næ}êäë and {GÍ aleady scheduled `{G whee system s not schedulable and the algothm stops. s the last epetton scheduled of. If we fnd seveal opeatons {» Ä» then the Step (opeaton wth latency constants): f G ]! ÏªÒ such that» Ä» : q+0 G N fo the opeaton found pevously, then we have» Ä» wth: P_ž Ì P_$[K P_ž OD É næ and DÓÔDÕaÖ the scheduled opeaton s emoved fom ², all the opeatons whch became schedulable ae added to ², and go to Step 6. Step 8 (opeaton wth peodcty fo whch each fst epetton s not aleady scheduled): f G ² Ïм wth Ä not scheduled and `» Ä» : ª+- G N fo the opeaton found pevously, then we have the scheduled opeaton s emoved fom ² ae added to ², and go to Step 6. Step 9 (opeaton wth peodcty): we have `»» Ä» wth Å wp_$[k É næ)êìë and Ó ÔDÕ Ö!É, all the opeatons whch became schedulable Ä» such that ` G ƒ fom ², all the opeatons whch became schedulable ae added to ², and go to Step 6., the scheduled opeaton s emoved Remak. the schedulng algothm neve stops unless the peodcty constants can not be satsfed n the Step 6,.e. n ths case, the system s not schedulable. We apply the schedulng algothm to the example of the fgue fo whch we have. The begnnng of the esults ae gven n the table 2.! emma.2 The peodcty constant s a patcula case of a latency constant. onsequently, the peodcty constant s a stonge constant than the latency constant. Poof If f F $}( has a latency constant z, then! Ø ƒ Ä Ø ƒ by zµú, we have Ø ƒ has a peodcty constant we have G ƒ N $ 0( D. If the pa Ä G ƒ +Ùz. fte eplacng z :+ÛzµÚ. Then we notce that the equalty expessng the peodcty constant s ncluded n the nequalty expessng the latency constant. may be exe-. But, $U)(. Remak.2 The latency constant of the pa f cuted afte the executon of, anytme dung the nteval ` f has a peodcty constant the stat tme of mples that Ä G must be equal to ` G Ä G Remak.3 We denote fou types of opeatons to be scheduled, dung the algothm, as follows: ( ) opeatons wth P_ž l ; (D9 ) opeatons wth P_ž l and thee s not wth _ ~ and

10 l * E Þ p * Œ ² `» Ä» Step used cton F! 0 0 Step ƒ!d9 Œ!d9 0 2 Step 9 B9!bº!bº 5 2 Step 9 a Ü `*! a! 0 2 Step 8 a F `{! a F :9! :9 2 Step a ` ƒ! a :9! :9 3 2 Step 9 ƒ!d9 :9!d9 :9 5 2 Step ` ]Ý!Þ!d9 E!d9 2 Step 9 Ý!bº EŒ!bº 20 2 Step 9 a Ü! a E! 25 2 Step 9 a F `{! ^`^`^ :9 0 2 Step 6 chosen 5 2 Step 6 chosen 0 2 Step 6 chosen 2 Step 6 chosen 3 2 Step 6 chosen 5 2 Step 6 chosen 2 Step 6 chosen 20 2 Step 6 chosen 25 2 Step 6 chosen 2 Step 6 chosen ¼ ; (]º ) opeatons (ß ) opeatons Table 2: Results gven by the schedulng algothm wth P_ž wth P_ž and G l and q wth ¼. Note that the opeatons of do not have any constant, and they ae scheduled only f thee s no moe opeaton wth constants among the schedulable opeatons. Theoem. Fo a system of opeatons wth pecedence and peodcty constants and wthout latency constant, a schedule obtaned accodng to the nceasng ode of the peodcty constants of the fst nstances of the schedulable opeatons, s feasble f and only f t s feasble accodng to the deceasng ode of the peodcty constants of the fst nstances of the schedulable opeatons. Poof We demonstate the equvalence by double mplcaton. Fst, we demonstate that f the system s schedulable accodng to the nceasng ode of the peodcty constants of the fst nstances of the schedulable opeatons, then the system s, also, schedulable accodng to the deceasng ode of the peodcty constants of the fst nstances of the schedulable opeatons. We can demonstate the mplcaton fo the two fst nstances of the opeatons wth the smallest values of the peodcty constants and then by mathematcal nducton the esult can be genealzed to all the fst nstances of the schedulable Ì ~ ;

11 z S S S g g j z S 6 $ j S S $ 6 $ opeatons. We denote by and the two opeatons wth the smallest values of the w$ peodcty constants and we suppose Å 3+à{. We have ` ƒ N 6á, ys and smlaly,!{ â!{ ƒ { 6ã. Because the system s schedulable, we have S}j3$ d `{ â% and Ä :{y+än. Moeove, because t s schedulable accodng to the nceasng ode of the peodcty constants of the fst nstances of the schedulable opeatons, we have:!{ ƒ ƒ Ä så ª$ ` G ` ƒ N æs!{ â ƒ Ä { ÐS S ÐS {ä N and Ä {/6 S Ð( Fom the pevous elatons, we have N. We demonstate by contadcton that we obtan also a feasble schedule f we schedule the opeatons accodng to the deceasng ode of the peodcty constants of the fst nstances of the schedulable opeatons. Indeed, n ths case, we have: ƒ `{ ƒ :{qå G `{ â `{ ƒ `{ ƒ Ð$ æs :{ N { Theefoe, we assume that we do not obtan a feasble schedule,.e. We have also: Ð$ S :{ N Å{ {æ6 N Ä Ûç whch s n contadcton wth smla to the fst one. åèä :{x6 ésê $T :{/+ÛN. The second pat of the demonstaton s Theoem.2 The schedulng algothm.2 s optmal (f thee s a feasble schedule, the algothm wll fnd t). Poof We consde thee cases:. the system has only pecedence and latency constants. Then we schedule a system that may have opeatons of types, D9 and º (Remak.3). t the begnnng of the algothm we have two possbltes: (a) thee ae opeatons of type among the schedulable opeatons. It means that along the algothm only the Step schedules the opeatons, and ths, accodng to the nceasng ode of the maks. We pove the optmalty of ths Step by contadcton. Fo ths, we assume that the opeatons ae scheduled accodng to the deceasng ode of the maks,.e. fo two opeatons and f P_ž ÄjP_ž, then!{. We consde the latency constant e e ÌëP_ž wth and z (c.f. algothm of latency [ì: makng, Step, b)). lso we consde the latency constant z wth eì 'wp_ž and z. Fnally, we obtan É ` whch s n [ Äj eì contadcton wth the fact that z. `{ â.

12 j j l l j j (b) thee s no opeaton of type among the schedulable opeatons. It means that the schedule s obtaned by usng the Step 2 o the Step 3, untl an opeaton of type becomes schedulable. The Steps 2 and 3 schedule an opeaton of type D9 befoe an opeaton of type Dº. When the opeaton s of type ]º, then G such that the pa b ~ and because P_ž : l, we have. It means that thee ae two possbltes fo the schedule, ethe ` `{ `, ethe!{ `. We obseve that n both cases, the stat tme of opeaton does not modfy the value of `{, and consequently the latency constant. So, t does not matte what s the schedulng of the opeatons of type D9 and º. Once an opeaton of type ]º s scheduled, opeatons of type become schedulable and, untl the algothm stops only the Step schedules the opeatons. Hence we ae n the case (a), and we saw pevously that the choce made by Step s optmal. 2. the system has pecedence and peodcty constants and no latency constants. Then we schedule a system that may have only opeatons of (Remak.3). In ths case because P_ž v ², the schedule s found only by usng the steps, 8 and 9. These steps schedule the schedulable opeatons n the nceasng ode of the peodcty constants. Indeed, due to the theoem. t s equvalent to schedule the fst nstances of the opeatons, wth peodcty constants, accodng to the nceasng o to the deceasng ode of the peodcty constants. Once the fst nstances of all the opeatons wth peodcty constants wee scheduled, the schedulng algothm only have to schedule the opeatons accodng to the peodcty constants. 3. the system has the thee constants: pecedence, peodcty and latency. t the begnnng of the algothm we have two possbltes: (a) thee ae opeatons of type among the schedulable opeatons. n opeaton wth P_ž may have o not a peodcty constant. The peodcty constant must be satsfed, only afte the fst epetton of the opeaton s scheduled. s soon as Step scheduled the fst epetton of an opeaton wth peodcty constant, the emanng opeatons ae scheduled only by usng the Steps 8, 9 and 0. So, untl the fst epetton of an opeaton wth a peodcty constant s scheduled, only the Step s used. We have two possbltes:. no fst epetton of an opeaton wth a peodcty constant was yet scheduled. It means that we have to satsfy only latency constant and we ae n same stuaton as.(a), and we saw that the choce made by Step s optmal.. the fst epetton of an opeaton wth a peodcty constant was scheduled, so we have a peodcty constant to satsfy. fte calculatng the stat tme of the opeaton such that N» Ä» wp_$ek {@ =Æ ÊÌË and {GÍ aleady scheduled!{@ Å{ `» Ä»

13 l (Step ), the emanng opeatons ae scheduled by usng the Step 8, 9 and 0. The Steps 8 and 9 seach fo opeatons, among the schedulable opeatons, that may be scheduled befoe the calculated stat tme. Because we aleady chose the fst peodcty constant to be satsfed, we have only latency constants to meet lke n the case.(a), and we saw that ths choce of the opeatons wth the smallest P_ž s optmal. If thee s no opeaton wth P_ž ä whch may be scheduled befoe the calculated stat tme, then t does not matte how the emanng opeatons ae scheduled befoe the calculated stat tme. Fnally, the Step 0 schedules the opeaton at the calculated stat tme. (b) thee s no opeaton of type among the schedulable opeatons. It means that the schedule s obtaned by usng the Steps 2, 3 and untl an opeaton wth peodcty and/o type becomes schedulable. The Step 3 s used only f thee s no moe opeaton of s soon as the Step 3 scheduled an opeaton, opeatons of type become schedulable and the Step wll neve be used. If the Step schedules an opeaton, then only the Steps 8, 9 and 0 wll schedule the emanng opeatons. So, the Steps 2, 3 and schedule ethe the opeatons of befoe schedulng opeatons of type Dº, ethe the opeatons of befoe schedulng opeatons of type ]ß. We saw n. (b) that t does not matte how the opeatons of type G9 and ]º ae scheduled. n opeaton of has no constant to satsfy, and an opeaton of type ß once ts fst epetton s scheduled, has ts peodcty constant to satsfy. Hence, fo opeatons of and ß, we educe the numbe of opeatons whch must be scheduled when thee ae peodcty constants to be satsfed. Once the Step 3 o the Step s used, we ae n the case o..3 onsequence of solvng ou poblem We emnd that we denote by í the deadlne of an opeaton defned fom the begnnng of the schedule mplyng that! )+0,. Theoem.3 The latency constant z deadlne of opeaton. of a pa Poof We demonstate the equvalence by double mplcaton. Fst, we demonstate that a latency constant z ~ may be ex- Ž ~, then of a pa d pessed as the deadlne of opeaton, once s scheduled. Because ` we have `{æ+0z :{. If we denote by í{ the expesson z we obtan!{æ+î{. Theefoe, as soon as ~ s equvalent to the s scheduled, the value of ï{ s known. Second, we expess the deadlne of an opeaton as a latency constant z. Because has a deadlne {, then `{æ+î{. We denote by the fst opeaton whch was scheduled, then ` *. Because the deadlne s defned fom the stat of the schedule, we obtan that!{ +î{. Moeove, because all the opeatons wthout pedecessos q 'w may be the fst scheduled opeaton, we have { ` x+î{ Å wth Ò?`; ` ^J! :{,

14 ² ² «««. So, n ode to satsfy the deadlne of, we defne seveal latency con- U 'w í{ fo each such that Ò?`; and. and stants z Due to ths theoem, we may use the same schedulng algothm n ode to solve the poblem [/ epettve gaph/ peodcty constants and deadlnes], but n ths case the functon P_ž of the opeaton must be equal to the P_$eK {Å n± î{. The followng algothm of deadlne makng allows to obtan these maks: lgothm Intalzaton: ²,, then P_ž U NĀ½ and ð!ñaá JÂ Ã Ò?`;,, othewse P_ž ' %P_ž îòp_$[kep_ž s the wokng-set. If l. aqp_$ek has a deadlne Step : Fo { ð`ñ=á and ²!. We add to ² the opeatons fo whch all the pedecessos has been emoved fom ². Step 2: Repeat Step untl ² ë. Remak. ecusvely, each opeaton nhets the deadlne of ts successos. t the end of ths latte algothm, the mak of an opeaton may be equal to ethe ts nheted deadlne o ts ntal deadlne. Theefoe, the algothm does not modfy the maks of the opeatons wthout successos. Ths algothm s, also, appled to the patten. ó P_ž F 5 oncluson and futhe eseach The pape gves a model based on gaphs n ode to state clealy the poblem of schedulng eal-tme systems wth pecedence, peodcty and latency constants fo one computng esouce. Because usually the poblem of schedulng systems wth peodcty constants and deadlnes, and the poblem of schedulng systems wth pecedence constants and deadlne, ae teated sepaately, we defne the noton of latency constant. By povng ts equvalence wth the noton of deadlne whch s commonly used, we mege these two domans of eseach. We gve an optmal algothm fo solvng ou poblem, and n ode to pove the optmalty of ths algothm we need to demonstate that n the case of systems wthout latency constant, t s equvalent to schedule the fst nstances of the opeatons wth peodcty constants, accodng to the nceasng o to the deceasng ode of the peodcty constants. These esults have as consequence the extenson of the algothm fo schedulng eal-tme systems wth pecedence, peodcty constants and deadlnes fo one computng esouce. Pesently we ae seachng fo a condton establshng f a eal-tme system wth pecedence, peodcty and latency constants, s schedulable. Moeove, when ths condton s not satsfed, we plan to study f ntoducng peempton allows to fnd a feasble

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