Efficient Feasibility Analysis for Real-Time Systems with EDF scheduling*

Transcription

1 Effcent Feasblty Analyss for Real-Tme Systems wth EF schedulng* Karsten Albers, Frank Slomka epartment of omputer Scence Unversty of Oldenburg Ammerländer Heerstraße Oldenburg, Germany {albers, Abstract Ths paper presents new fast exact feasblty tests for unprocessor real-tme systems usng preemptve EF schedulng. Task sets whch are accepted by prevously descrbed suffcent tests wll be evaluated n nearly the same tme as wth the old tests by the new algorthms. Many task sets are not accepted by the earler tests despte them beeng feasble. These task sets wll be evaluated by the new algorthms a lot faster than wth known exact feasblty tests. Therefore t s possble to use them for many applcatons for whch only suffcent test are sutable. Addtonally ths paper shows that the best prevous known suffcent test, the best known feasblty bound and the best known approxmaton algorthm can be derved from these new tests. n result ths leads to an ntegrated schedulablty theory for EF. 1. ntroducton The analyss of the tme behavor of embedded real-tme systems s essental for the automaton of the desgn process. A formal verfcaton whch guarantees all deadlnes n a real-tme system would be the best. Ths verfcaton s called feasblty test. Three dfferent knds of tests are avalable: Exact tests wth long executon tmes or smple models [2], [3], [11]. Fast suffcent tests whch fals to accept feasble task sets, especally those wth hgh utlzatons [9], [12]. Approxmatons, whch are allowng an adjustment of performance and acceptance rate [1], [8]. For many applcatons an exact test or an approxmaton wth a hgh acceptance rate must be used. For many task sets a fast suffcent test s adequate. Ths paper proposes two new tests for preemptve EF schedulng, whch have a performance comparable wth the suffcent tests for those tasks sets whch can be recognzed by these suffcent tests and outperforms the exstng exact tests by orders of magntude. These algorthms are tested and compared usng a large amount of randomly generated task sets and some examples from lterature. n Secton 2 the analyss model s ntroduced, Secton 3 gves an ntroducton to the exstng feasblty tests and * The research descrbed s supported by the eutsche Forschungsgemenschaft under grant SL 47/1-1 the concepts needed further n ths paper. t contans the frst contrbuton of ths work: the proof of equvalence between the suffcent test gven by ev [9] and the superposton approxmaton descrbed n [1]. t s shown that the test by ev s a specal case of the superposton approach. Secton 4 presents as the man contrbuton of the work the new fast suffcent and necessary feasblty tests and also the proof for dervng the feasblty bound out of these tests. n Secton 5 the new test algorthms are evaluated followed by a concluson n Secton Model We consder the sporadc task system consstng of a set of tasks { 1,..., n }. Each task s descrbed by an ntal release tme (or phase) φ, a relatve deadlne (measured from the release tme), a worst-case executon tme and a mnmal dstance (or perod) T between two nstances of a task. An nvocaton of a task s called a job, and the k th nvocaton of each job s denoted,k. n the followng only the synchronous case s consdered, so the frst jobs of all tasks are released smultaneously. Ths s a common assumpton whch also leads to a suffcent test for the asynchronous case [14]. A good suffcent condton for the asynchronous case s proposed n [13]. t s based on the tests mproved n ths paper. The specfc utlzaton of a task s the part of the capacty whch s needed for executng ths task ( U() T ). The utlzaton U of the system s the sum of the specfc utlzatons of all tasks. We consder the unprocessor feasblty test. The schedulng s done usng earlest deadlne frst (EF) whch s known to be optmal [12]. To keep the explanatons n ths paper smple the sporadc task model s used. The new tests can be extended to more advanced task models. Especally the extenson for the event stream model [11] s easy by followng the defntons proposed n [1]. 3. Background and related work The feasblty test for unprocessor systems usng EF wth deadlnes smaller then the perods of the tasks s Proceedngs of the esgn, Automaton and Test n Europe onference and Exhbton (ATE 5) /5 $ 2. EEE

2 known to be o-np-hard [3]. There are only feasblty tests wth pseudo-polynomal complexty avalable so far. t s unknown whether the problem has polynomal complexty. n the followng an ntroducton to the relevant feasblty tests of the lterature s gven. An overvew s gven n [14] Suffcent tests For a restrcted verson of the gven model wth the condton T for each task, Lu and Layland [12] proved that the task system s feasble f the utlzaton U Test by ev Recently ev [9] proposed a test whch allows task systems wth deadlnes smaller than perods wth a reduced complexty: ef. 1: Test ev [9]: A task system, arranged n order of non-decreasng relatve deadlnes s feasble usng EF schedulng f n and k k ( 1 k n) T mn( T, ) T 1 k 1 Unfortunately ths test s only suffcent. t allows a fast evaluaton and acceptance of task sets n many cases Processor demand test n [3] Baruah et al. proposed an exact test for task systems wth deadlnes shorter than ther perods. The man dea was to calculate the maxmum demand of all tasks wthn a tme nterval and compare t wth the avalable capacty of the processor. The demand s the cumulated worst-case executon tme of the relevant jobs. The maxmum demand for an nterval can be found f all tasks are released synchronous at the begnnng of and only those jobs are consdered whch have both ther release tme and ther absolute deadlne wthn. ef. 2: emand bound functon bf() [2]: The maxmum cumulated executon requrement of jobs havng both request tme and deadlne wthn nterval dbf(, ) t s possble to splt the demand bound functon nto demand bound functons for each sngle task. dbf(,) s the cumulated executon requrement of only one task. The avalable capacty of executon tme n an nterval s exactly the length of the nterval. Testng the demand bound functon for all ntervals aganst the capacty leads to an exact test. To make the test tractable t s possble to calculate a maxmum test nterval ( max ) whch s an upper bound (feasblty bound) and t s only necessary to test the demand bound functon for ntervals lower than max. ef. 3: Processor demand test [2]: A task system s feasble f and only f dbf(,) for all max ( U ( 1 U) ){ max( T )}. t s only necessary to check those ntervals where the value of dbf(,) s changed. The absolute deadlnes of all jobs determne these ntervals. Other values for max are dscussed n Secton 4.3. The test has a pseudo-polynomal complexty f the consdered utlzaton s bound by a constant. The problem T T s that the runtme of the algorthm depends not only on the utlzaton but also on the rato of the dfferent perods and deadlnes n the task set [1]. f the task set contans tasks wth small perods and tasks wth large perods, the runtme can become qute large. See the experments gven n Secton 5 for more detals Approxmaton by superposton Recently, several approxmatons have been proposed to solve the complexty problem of the demand bound approach: One by hakraborty et al. [8] and the superposton approach [1]. They brdge the gap between the fast but only suffcent test of ev and the necessary but slow processor demand test. The algorthms are only suffcent, but the degree of suffcency can be adjusted. The man dea of the superposton approach s to test only a selectable lmted amount of test ntervals for each task and use an approxmaton to compensate the remanng test ntervals. ef. 4: Approxmated task demand bound functon dbf (,) [1]: An upper bound for the task demand bound functon consderng only the jobs up to the selectable maxmum test nterval m() exactly. dbf' (, ) The approxmaton s usng the specfc utlzaton of the task. n the specal case of consderng only the frst job of each task, the maxmum test nterval s equal to the deadlne of the task. The approxmated demand bound functon s a superposton of the approxmated task demand bound functons. ef. 5: Approxmated demand bound functon [1] dbf (,): dbf' (, ) dbf' (, ) One advantage of ths approxmaton s that t s not necessary to calculate dbf (,) for each job separatly. Lemma 1: Super poston test: A task system s feasble f dbf (,) for all max ef. 6: SuperPos(x) : The super poston test whch calculates at the maxmum the frst x jobs of each task exactly (test level x) s called SuperPos(x) A hgher level of the superposton test leads to a hgher acceptance rate but also to a longer executon tme of the test. Fgure 1 shows measured acceptance rates usng dfferent SuperPos(x). The superposton test s a suffcent test wth a selectable error. t can be regarded as a seres of percentage task sets feasble dbf( m(), ) ( m() ) > m() dbf(, ) m() Utlzaton (%) ev SuperPos(2) SuperPos(3) SuperPos(4) SuperPos(5) SuperPos(6) SuperPos(7) SuperPos(8) SuperPos(9) SuperPos(1) Processor emand Fgure 1: Results for dfferent values for SuperPos(x) Proceedngs of the esgn, Automaton and Test n Europe onference and Exhbton (ATE 5) /5 $ 2. EEE

3 ev bf l 2 ev bf Fgure 2: Approxmaton by ev for two tasks suffcent tests, whch gets better wth declnng error Superposton vs. ev An nterestng pont s the relatonshp of the superposton test to the best prevously known suffcent test gven by ev [9]. Lemma 2: Relatonshp ev: A task set recognzed as feasble by the test of ev s also recognzed as feasble by SuperPos(1). n other words, the test by ev s only a specal case of the superposton approach. Proof: onsder the test SuperPos(1) (Lemma 1). Remember that n ths specal case the maxmum test nterval for each task s equal to ther deadlne ( m ( ) ) as ths s the endng tme for the frst job of the task. The test can be expressed as follow: ( () { ( )} ( T ) ( ) ( ) T T ( ) ( ) ( ) Ths test has to be preformed only for the frst job of each task whch has a deadlne equal to the deadlne of the task. Now consder the test of ev (efnton 1). t can be expressed as: ( ( 1 k ) k k k T k ---- mn( T, ) T T 1 1 k k k k T T T 1 1 k k k + k ---- T T 1 1 Ths test also has to be performed for the deadlne of each task. Settng k n the above equaton results n SuperPos(1). Ths proves Lemma 2. The relatonshp between the test by ev and the superposton approach allows to nclude the extensons of the test by ev Fgure 3: Approxmaton by ev for a sngle task descrbed n [9] nto the super poston approach. The extensons concern practcal relevant ssues lke swtchng tme, prorty celng protocol, self-suspenson and lmts for the number of prortes Superposton vs. real-tme calculus n the concept of real-tme calculus [7] based on the network calculus [6] the demand and the capacty of a system are descrbed by arrval and servce curves. The dea of these functons s the same as n the processor demand test, apart from that the capacty s not regarded as bsectng lne but also as a functon. The dea behnd the real-tme calculus s to defne mathematcal operatons on the arrval and servce curves to solve the feasblty test problem. By defnton the curves are unlmted. Therfore the equatons behnd ths concept are unfortunatly expensve to compute n the case of general arrval and servce curves [7]. To make the concept practcable a pecewse lnear approxmaton wth up to three staght lne segments s proposed. The error of ths approxmaton s unkown. By usng the results of Secton 3.5 t s possble to calculate a lower bound on the approxmaton error of the approxmated real-tme calculus. Fgure 4 shows the realtme calculus approxmaton of one task. Frst the approxmaton of a smple perodc task wth two lnes s shown (a), second the approxmaton of a bursty task s outlned (b). n ths case three lnes are needed for a good approxmaton. omparng Fgure 3 wth Fgure 4 a) shows a close relatonshp between the approxmaton of the real-tme calculus and the test gven by ev. The real-tme calculus approxmaton s a bt worse than the test gven by ev because of the lmted number of curves n the approxmaton of the real-tme calculus. As shown n the last secton the test by ev s equal to SuperPos(1) of the superposton approach. The man dfference between the test by ev and the real-tme calculus s the specfcaton of bursts whch can be only expressed by the real-tme calculus. By usng event streams t s easy to analyse burst by the superposton test [1]. However, ths leads to a hgher complexty then the test by ev because of each element of the burst has to be handled as a seperate element of the event stream. l 3 l 2 l 1 l 1 l 2 a) perodc task b) task wth burst Fgure 4: Real-tme calculus approxmaton Proceedngs of the esgn, Automaton and Test n Europe onference and Exhbton (ATE 5) /5 $ 2. EEE

4 4. mproved feasblty tests Approxmated feasblty tests are only suffcent. Ths s a problem because not all scheduable task sets are accepted by ths tests. Even f the degree of suffcency s selectable the tests fals to recognze feasble task sets (Fgure 1). Even worse, choosng a hgh degree of suffcency leads to a test wth long executon tme, whle choosng a low degree leads to a bad acceptance rate for the feasble task sets. The dea s to use dfferent levels of approxmaton, startng wth a fast approxmaton (SuperPos(1)) and swtch to a slow one just as necessary ynamc error test Ths dea leads to a fast test for all those task sets whch the suffcent tests could recognze. The algorthm for the new test s shown n Fgure 5. The test starts at the level SuperPos(1). Ths level allows all tasks to be approxmated after ther frst job. The test runs at ths level untl ether the test succeeds or the approxmated demand (dbf ) exceeds the actual test nterval ( act ) and the task set s not feasble by ths approxmaton. n ths case t s necessary to rse the level and wthdraw the approxmatons for those tasks whch would not be approxmated usng the new level ( rev ). t s not necessary to recalculate the whole test because the values calculated by the frst approxmaton can be reused. Lemma 3: f dbf (,G) for an nterval than dbf(,g) for all ntervals between and next. next s the next test nterval after whch s not approxmated. The prove of ths lemma s vsualzed n Fgure 6 and can also be followed usng Lemma 4. Therefore f the test fals at next, the prevous test guarantees all ntervals whch are smaller than next, even f they are approxmated. ALGORTHM ynamcerror F U > 1 not feasble max mnmum feasblty nterval; :testlst.add( ι,t + ) WHLE (testlst {} AN act < max), act testlst.getnext() dbf dbf + + ( act - old ) * Uready WHLE (dbf > act ) F (ApproxLst {}) NOT FEASBLE ncrease level; j rev EN WHLE F ( act < Testboarder() testlst.add(, act + T ) ELSE Uready Uready + / T ApproxLst.add(); old act EN WHLE FEASBLE : Uready Uready - j, / T j, dbf dbf - app( act, j ) testlst.add( j, Nextnt( act, j )) Fgure 5: ynamc Test bf approxmated next Fgure 6: Possble proven test ntervals { bf Lemma 4: Let, next be two consecutve test ntervals for dbf (). f we assume t exsts a for whch < + < next and dbf( + ) > + apples then the approxmated demand bound functon holds dbf () > [1]. Proof: See Secton 3.2 n [1]. For all tasks for whch the approxmaton s revsed due to the swtch of the level the next test nterval followng act has to be nvestgated: Lemma 5: Followng test nterval Nextnt(, ) T + Ths nterval s added to the lst of test ntervals. t s further necessary to reduce the approxmated demand by the overestmated approxmaton costs for these revsed tasks. These costs are gven by app. Let rev be the set of all tasks for whch the approxmaton s revsed: Lemma 6: Overestmated approxmaton costs app(, rev ) rev f dbf act the new level s suffcent, otherwse t has to be ncreased. f no task s approxmated before ncreasng the level the demand bound functon exceeds the capacty and the test fals. We propose to double the level at each step whch lmts the amount of steps to n max. Ths s small compared to the total amount of test ntervals. Only task sets whch cannot be evaluated usng a fast level wll need a long evaluaton tme. These task sets would not be accepted by exstng suffcent tests lke the test by ev. Task sets accepted by the test by ev are accepted by ths test runnng completly on the level SuperPos(1). Ths condton holds for all levels of the superposton test. t s not necessary for a good average case performance of the analyss to combne suffcent and necessary tests. The maxmum level for the dynamc test can be lmted. The result s a test wth a strctly lmted worst-case run-tme and a good average case run-tme All approxmated test espte that the proposed test outperform the exstng approaches by orders of magntude, t can be mproved even further. Especally for task sets contanng very small and very large tasks, the dynamc test could swtch soon nto hgher levels. onsderng many test ntervals may result n an unnecessary effort. t could be the case that only a few test ntervals are crtcal and that t s possble to approxmate the others. Ths leads to a new test. The algorthm for ths test can be found n Fgure 7. The goal of the new algorthm s to reduce the number of consdered test ntervals further. Fgure 7 shows the algorthm. nstead of usng fxed test bounds for each task, approxmaton s done as much as possble. The frst test ntervall resultng out of the frst job of each task s nserted nto testlst. Testlst s processed n ascendng order. All the followng test ntervalls are approxmated frst. Only f a test fals for an test nterval test, the approxmaton of the demand bound functon of each task s step by step revsed untl the test ether succeds or no task s approxmated any more. The revson s done by replacng the approxmated costs of the task by ther real cost. Because of quttng the approxmaton n ths nterval t s necessary to add one T T Proceedngs of the esgn, Automaton and Test n Europe onference and Exhbton (ATE 5) /5 $ 2. EEE

5 ALGORTHM AllApprox F U > 1 not feasble : testlst.add(,t + ) WHLE (testlst {}), test testlst.getnext() dbf dbf + + ( test - old ) * Uready WHLE (dbf > act ) F (ApproxLst {}) not feasble, ApproxLst->getAndRemoveFrstTask; Uready Uready -, /, dbf dbf - app( test,, ) testlst.add(,, Nextnt( test,, )) EN WHLE Uready Uready + / T ApproxLst.add(); EN WHLE feasble Fgure 7: All approxmated test addtonal test nterval for each revsed task nto testlst. These addtonal test ntervals can be calculated usng Lemma. Note that t s only necessary to calcultate the nterval followng test because the approxmaton has already verfed all test ntervals smaller than test. The algorthm termnates f t s possble to approxmate all tasks successfull n one test nterval or f the test fals. f the ntal test nterval s accepted for each task wthout generatng new test ntervals, the behavour and the performance of the test s equal to the test gven by ev [9] Feasblty bound A feasblty bound descrbes the upper lmt for all ntervals whch are necessary to be tested. t s mportant to fnd a short one because ths lmts the effort for the feasblty test. A good overvew of the dfferent feasblty bounds can be found n [14]. Baruah et al. [3] defned such a bound for the processor demand test, whch s part of efnton 3. Another one can be derved from the busy perod condton [14]. George et al. [1] defne a bound whch s smaller than the bound gven by Baruah et al: T T george U The new all approxmated superposton approach delvers a new feasblty bound. t s reached (and no further test nterval s needed) f for a test nterval the dfference between the demand bound functon and the capacty allows the approxmaton of all tasks. A task system wth a utlzaton lower 1% can never exceed the capacty beyond ths nterval. Usng Lemma 6 the sum of the demand bound functon and the approxmaton errors can be calculated: app(, ) + dbf(, ) Ths gves the new feasblty bound: sup sup f the condton sup max s used, t can be followed: sup sup Usng U T leads to the followng bound: ( 1 T ) sup mn ( max, ) 1 U For the case that all ths bound s the same as the bound gven by George et al., for the other cases ths new bound s lower than ths bound. Ths relatonshp allows a better understandng of the bound and leads to a ntegrated feasblty theory for EF. The man contrbuton of ths result s that t s not necessary for the new test to calculate and check the feasblty bound by George et al. The superposton bound s tghter and s checked n an mplct way. t mght be nterestng to calculate the busy perod of the system as outlned n [14], because ths bound mght be tghter than the superposton bound under some condtons. alculatng ths bound however has exponental complexty and may need more effort than the test wthout ths bound. 5. Experments We have generated a large number of random task sets to evaluate the performance gan due to the new tests. As a metrc we counted the test ntervals checked by each of the algorthms. Ths metrc s common to evaluate feasblty tests [5], [13] and shows exactly the behavor of the dfferent algorthms. The generaton of the random task sets follows the unform dstrbuton proposed by Bn [4]. n Fgure 8 a test was performed usng 18, task sets wth a utlzaton between 9% and 99% (hgh utlzatons are hard to test). t shows the average and the maxmum teratons needed for the tests. The sze of the task sets vared between 5 and 1 tasks usng a unformed random dstrbuton. They had an average gap of 2%, 3% and 4%. The gap descrbes the dfference between deadlne and perod. The szes of perods are also equally dstrbuted, the rato between the maxmum and the mnmum perod was of no concern for ths experment. The result of the experment s that the all approxmaton test needs 1 to 2 tmes less teratons than the processor demand test, whch goes up to 2 tmes less effort consderng the maxmum amount of teratons. Fgure 9 shows the result of another experment whch nvestgates the effect of the rato T max /T mn. Therefore the test was performed wth a rato varyng from 1 to 1,,. The results are presented usng an exponental scale. Such hgh ratos could for example be possble f system nterrupts and the schedulablty overhead are defnnded as tasks. For each value 4, tasks set where generated randomly wth a sze between 5 and 1 tasks and an average gap between 1% and 5% and an utlzaton varyng between 9% and 1%. The frst graph shows the maxmum measured effort needed for the processor demand test whch ncreases up to more than 5 mllon teratons. The effort of the other values s vsualzed n the second graph. Also there are some varatons due to the random values especally for the dynamc test. The measured effort of both new tests are far below the effort needed for the old test. The maxmum effort measured for the dynamc test was about 9,. For the all approxmaton test t s about 3, teratons. Another nterestng pont s that the effort doesn t depend on the rato of the perods. Ths becomes even more obvous when nvestgatng the average effort needed. 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6 teratons teratons teratons teratons Maxmum effort for dfferent utlzatons ynamc Processor emand All approxmated Utlzaton (%) Average effort for dfferent utlzatons 7 ynamc Processor emand All Approxmated Utlzaton (%) Fgure 8: Results wth dfferent utlzatons 6M 5M 4M 3M 2M 1M Max executon effort usng dfferent values for Tmn / Tmax ynamc Processor emand All Approxmated Tmax / Tmn Max executon effort usng dfferent values for Tmn / Tmax 1 ynamc Processor emand All Approxmated Tmax / Tmn Fgure 9: Results wth dfferent values for T mn / T max the all approxmaton test t vares between 12 and 116 teratons, as opposed to 465 to 955,53 for the processor demand test. Tab. 1 contans some results for task sets comng from real examples. The task set of Burns and of the modfed set of Ma & Shn can be found n [1], the Generc Avoncs Platform example (GAP) n [14] and the other two task sets n [11]. The amount of tasks are small (7 to 21 tasks) but even for such small examples the new tests need between 5 and 1 tmes less teratons than the processor demand test. The run-tme overhead of one teraton ot the new tests s small compared to both alternatve algorthms, the processor demand test and the test by ev. Only the approxmaton leads to a some more effort. ompared to the algorthm of ev the overhead can be completely elmnated. Test ev yn. All Appr. Proc. em. Burns ,112 Ma & Shn FALE GAP ,228 Gresser 1 FALE Gresser 2 FALE Table 1: teratons for example task graphs 6. oncluson We proposed two new feasblty test for EF. They are exact and outperform the exstng tests by orders of magntude. Task sets whch are recognzed by the exstng suffcent test or approxmatons can be evaluated by the new test wth a comparable effort. We proved that the new concept ntegrates the processor demand test, the feasblty bound by George et al. and the suffcent feasblty test by ev. 7. References [1] K. Albers, F.Slomka. An Event Stream rven Approxmaton for the Analyss of Real-Tme Systems, EEE Proceedngs of the 16th Euromcro onference on Real-Tme Systems, pp , 24. [2] S. Baruah,. hen, S. Gornsky, A. Mok. Generalzed Multframe Tasks. The nternatonal Journal of Tme-rtcal omputng Systems, 17, 5-22, [3] S. Baruah, A. Mok, L. Roser. Preemptve Schedulng Hard-Real-Tme Sporadc Tasks on One Processor. Proceedngs of the Real-Tme Systems Symposum, , 199. [4] E. Bn, G. Buttazzo. Basng Effects n Schedulablty Measures. EEE Proceedngs of the 16th Euromcro onference on Real-Tme Systems, pp , 24 [5] E. Bn, G. Buttazzo. The Space of Rate Monotonc Schedulablty. Proceedngs of the 23th EEE Real-Tme Systems Symposum, 22. [6] J.Y. Le Boudec, P. Thran. Network alculus - A Theory of determnstc Queung Systems for the nternet. LNS 25, Sprnger Verlag, 21. [7] S. hakraborty, S. Künzl, L. Thele. A. Herkersdorf, P. Sagmester. Performace Evaluaton of Network Processor Archtectures: ombnng Smulaton wth Analytcal Estmaton, omputernetworks, Vol. 41, No. 5, pp , 23. [8] S. hakraborty, S. Künzl, L. Thele. Approxmate Schedulablty Analyss. 23rd EEE Real-Tme Systems Symposum (RTSS), EEE Press, , 22. [9] M. ev. An mproved Schedulablty Test for Unprocessor Perodc Task Systems. Proceedngs of the 15th Euromcro onference on Real-Tme Systems, 23. [1] L George, N. Rverre, M. Spur. Preemptve and Non-Preemptve Real-Tme Un-Processor Schedulng. Rapport de Recherche RR-2966, NRA, France, [11] K. Gresser. Echtzetnachwes Eregnsgesteuerter Realzetsysteme. ssertaton (n german), V Verlag, üsseldorf, 1(286), [12]. Lu, J. Layland. Schedulng Algorthms for Multprogrammng n Hard Real-Tme Envronments. Journal of the AM, 2(1), 46-61, [13] R. Pellzzon, G. Lpar. A New Suffcent Feasblty Test for Asynchronous Real-Tme Perodc Task Sets. Proceedng of the 16th Euromcro onference on Real-Tme Systems, pp , 24. [14] J.A. Stankovc, M. Spur, K. Ramamrtham, G.. Buttazzo. eadlne Schedulng for Real-Tme Systems EF and Related Algorthms. Kluwer Academc Publshers, Proceedngs of the esgn, Automaton and Test n Europe onference and Exhbton (ATE 5) /5 $ 2. EEE