Real-Time Sysems and Limiing Even Sreams: A General Model

Transcription

1 Limiing Even Sreams: A General Model o Describe Dependencies in Disribued Hard Real-Time Sysems Seffen Kollmann, Karsen Albers and Frank Slomka Ulm Universiy Insiue of Embedded Sysems/Real-Time Sysems {firsname.lasname}@uni-ulm.de Inernal Repor Absrac This paper inroduces a common approach o describe dependencies in disribued hard real-ime sysems. Taking dependencies ino accoun will improve schedulabiliy analysis. Thereby, we idenify a new kind of dependency no considered in relaed work. The new model can be also applied o differen kinds of dependencies including ask offses allowing an absrac and generalized analysis. Finally, we show how his approach leads o igher bounds of his analysis. 1. Inroducion Requiremens of embedded sysems increase from generaion o generaion. A car, for example, ges more sofware funcionaliy in each generaion which resuls in more complex embedded sysems. New cars have up o 70 ECUs conneced by several busses. Mos funcions have real-ime consrains such as he engine managemen sysem and he ABS-sysem. For a wide accepance of real-ime analysis by indusrial sofware developers igh bounds o he real wors-case response imes of asks are imporan. A successful approach is he consideraion of dependencies of chained ask-ses. Previous work considers differen kinds of dependencies. Muual exclusion of asks, offses beween ask simulaion, ask chains or asks compeing for he same resource are some examples for such dependencies. Bu he inegraion of many oher ypes of dependencies are sill open. Especially, missing is a holisic model for dependencies as new absracion layer. The idea is o separae he calculaion and he concree ype of dependencies from he remaining real-ime analysis. In oher words he calculaion of he dependencies is orhogonal o he real-ime analysis. This paper presens a new holisic model o inegrae differen ypes of dependencies in real-ime analysis. We show how his general model can be inegraed ino he schedulabiliiy analysis of fixed-prioriy sysems. We also ouline he calculaion of wo oally differen kinds of dependencies in order o show he flexibiliy. This paper is organized as follows: In chaper 2 an overview of he relaed work is given. Chaper 3 describes he problem which will be explored. The model is defined in chaper 4. Chaper 5 defines he limiing even sream model and how i can be used o describe dependencies. The resuling real-ime analysis is presened in chaper 6. A case sudy in chaper 7 shows he significance of our approach. The work closes wih he conclusion. 2. Relaed Work Mos considered relaed work uses he periodic even model wih jier. The analysis for disribued sysems was inroduced by Tindell and Clark [15]. In his holisic schedulabiliy approach, asks are considered as independen and so for each ask he wors-case response ime is calculaed separaely. This idea has been improved by he ransacion model [14] which allows he describion of saic offses beween asks. Guierrez e al. [4] exended his work o dynamic offses so ha he offse can vary from one job of a ask o anoher. Furhermore hey inroduced an idea abou muual exclusion of asks [5] which is based on offses beween asks. Since Guierrez e al. have considered simple ask chains, Redell has enhanced he

2 idea o ree-shaped dependencies [13] and Pellizoni e al. applied he ransacion model o earlies deadline firs scheduling in [12]. Henia e al. used he SymTA/S approach o exended he idea of he ransacion model in order o inroduce iming-correlaions beween asks in parallel pahs in disribued sysems [6]. This idea has hen been improved in [7]. Furhermore in [8] i has been shown ha maximum inpu jier and wors-case response ime mus no occur during he same job leading o a relaxaion of he response imes in a sysem. A scalable and modular approach o analyse realime sysems is he real-ime calculus (RTC) as described by Wandeler in [16]. Using he RTC i is no possible o describe dependencies like offses, muual exclusion of asks or compeing-based dependencies. Because of he lack of generaliy or exacness he RTC and he SymTA/S approach are merged in [10]. In his paper he auhors do no consider ypes of dependencies like muual exclusion of asks or compeingbased dependencies. A furher dependency considering he simulaneous occurrence of evens is considered by Kollmann e al. in [9]. In his paper he dependency is direcly included ino he real-ime analysis and no calculaed separaely as in his paper. 3. Conribuion In figure 1 a ypical disribued sysem is depiced. The sysem consiss of wo CPUs and one BUS. We assume fixed prioriy scheduling on each resource. The sysem has eigh asks execued on he processors and he bus. The prioriies are assigned as described in he figure. The simulaion of he asks is represened by even sreams Θ. Θ A - Θ 1 Θ B Θ C τ 1 prioriy=high τ 2 prioriy=middle τ 3 prioriy=low offse based dependency Θ D Θ E Θ F τ 4 prioriy=high τ 5 prioriy=low Θ G Θ H - Θ 2 τ 6 prioriy=low τ 7 prioriy=middle Θ τ K 8 prioriy=high CPU1 BUS1 CPU2 compeing based dependency Figure 1. Example of a disribued sysem having several differen dependencies In order o analyse he sysem from figure 1 i is necessary o calculae he wors-case response ime for Θ I Θ J each ask. The common way o do his is o assume ha all asks and even sreams describing he simulaion in a sysem are independen. This means ha evens can occur during a wors-case response ime analysis in heir maximal densiy, because he conex of he sysem is no considered. The resul of a real-ime analysis is ha he inerference beween asks is always maximal and leads o very pessimisic resuls. To ge igher response ime bounds, we inroduce wo kinds of dependencies: The firs dependency is he compeing based dependency describing he siuaion ha asks execued by he same componen compee for his componen. Such a compeiion has he effec ha cerain evens can no occur in he same densiy when he asks are assumed no o be independen as i is he case. For example, Θ G and Θ H in figure 1. The second one is an offse based dependency describing ha evens from differen even sreams mus occur ime-shifed o each oher. Consider, for example, he even sreams Θ B and Θ C in figure 1. I is assumed ha a correlaion beween he even sreams exiss. This has a direc impac on he successive asks and even sreams shown in [13]. The purpose of inroducing his dependency is o show he generaliy of our approach. These wo inroduced dependencies lead o igher bounds for he real-ime analysis, I is desirable o include boh dependencies ino i. In previous work no holisic model as general approach o describe dependencies beween asks is exising. 4. Task and Even Model In his secion we inroduce he model necessary for he real-ime analysis discussed in secion 6. Task Model: Γ is a se of asks on one resource Γ = {τ 1,...,τ n }. A ask is a 4-uple wih τ = (c +,c,φ,θ). c + is he wors-case execuion ime, c is he bes-case execuion ime, φ is he prioriy for he scheduling (he lower he number he higher he prioriy) and Θ defines he simulaion of he ask by an even sream. Le τ i j be he j-h job/execuion of ask τ i. We assume ha each job of a ask generaes an even a he end of is execuion o noify oher asks. Even Sream Model: The even sream model gives an efficien general noaion for he even bound funcion. Definiion 1: ([1],[2],[3]) The even bound funcion η(, Θ) gives for every inerval an upper bound on he number of evens occurring from he even sream Θ in any inerval of he lengh. So we can se up he following lemma:

3 Lemma 1: ([3]) The even bound funcion is a subaddiive funcion, ha means for each inerval, : η( +,Θ) η(,θ) + η(,θ) (1) Proof: η(,θ), η(,θ) reurn he maximum number of evens possible wihin any or. The evens in + have o occur eiher in or in. Therefore he condiion holds. Definiion 2: An even sream Θ is a se of even elemens θ. Each even elemen is given by a period p and an offse a (θ = (p,a)). In cases where he wors-case densiy of evens is unknown for a concree sysem an upper bound of he densiies can be used o describe he even sream. I is possible o model any even sequence. Only hose even sequences for which he condiion of subaddiiviy holds are valid even sreams. Corollary 1: ([3]) The even bound funcion for an even sequence Θ and an inerval is given by: η(,θ) = θ Θ a θ aθ p θ (2) As he inverse funcion we define he following inerval funcion which gives o a number of evens and an even sream he minimum inerval in which hese evens can occur: Corollary 2: ([3]) The inerval funcion for a number of evens and a Θ is given by: + (n,θ) = min{ η(,θ) = n} (3) Some examples of even sreams can be found in [1]. 5. Limiing Even Sreams wih Dependencies To exend he previous discussed model of embedded real-ime sysems we will inroduce he limiing even sreams in his secion. Definiion 3: The limiing even sream is an even sream which defines he maximum occurrence of evens for a se of even sreams. The limiing even sream is defined as Θ = (Θ, Θ). Θ describes he limiing even sream and Θ represens he se of even sreams for which he limiing even sream holds. The limiing even sream fulfills he condiion: η(,θ) η(,θ i ) Θ i Θ Example 1: If no correlaions beween even sreams are defined hen Θ = ( Θi Θ Θ i, Θ). Example 2: Figure 1 gives an example for a limiing even sream. Assume Θ B = Θ C = {(20,0)} and an offse of 10.u. beween hese wo even sreams. The cumulaed occurrence of evens can be described by he limiing even sream: Θ = ({(20,0),(20,10)},{Θ B,Θ C }). If we consider he even sreams as independen we ge wo evens in an inerval = 5. Bu he limiing even sream describes how many cumulaed evens can occur in an inerval. Wih his dependency we ge only one even in he inerval = 5. Nex we define how a limiing even sream can be calculaed. Definiion 4: Le β : n be a limiing inerval funcion which assigns a minimal ime inerval from a given number of evens in dependency from a given relaionship of even sreams Θ := {Θ 1,...,Θ n }, hen a limiing even sream Θ can be deermined by: Θ := ν( Θ, β(n)) Noe ha ν( Θ, β(n)) and β(n) are absrac formulaions which mus be concreely formulaed for he differen ypes of dependencies Compeing Based Dependencies In his secion we inroduce a new kind of dependency. We call i compeing based dependency. In figure 1 his kind of dependency beween asks is exemplarily depiced. The asks τ 4 and τ 5 are execued by he same resource. Which means ha hey compee for he resource. In relaed work during he analysis of he asks τ 4 or τ 5 he ougoing even sreams Θ G and Θ H are considered independenly, for example in [15]. 1) 2) τ 4 τ 5 τ 4 τ 5 4,x 4,x c- 4,x+1 c - +1 Δ 1 c - 4,x+1 Δ' 1 - c +1 Improvemen ' Figure 2. Example of a limiing even sream describing a compeing dependency Le us consider he gan-diagrams in figure 2. Arrows above he ime line represen incoming evens. Arrows under he ime line represen evens generaed by he ask. In par one of he gan-char he case is considered of non compeing asks. The firs jobs of he asks are scheduled in he way ha he wo ougoing evens can occur almos simulaneously. The

4 nex evens are produced as soon as possible afer he firs even of he firs job. In he independen case he nex wo evens can occur also simulaneously. Bu his is no possible, since he jobs mus be execued ask afer ask, because τ 4 and τ 5 are execued by he same processor. This is depiced in he lower gan-char which describes he correc occurrence of he evens. Because of he ask inerference i is no sufficien o consider he ougoing even sreams independenly from each oher. This inerference can be modeled by a limiing even sream. As figure 2 illusraes wo cumulaed ougoing evens can be generaed simulaneously. This is based on he fac ha he ask wih he higher prioriy inerrups he second ask jus before i finishes. The resul is ha he wo evens occur almos simulaneously. This can also be applied on n asks wih he resul ha n evens can occur simulaneously. For one ask we can conclude ha a leas (n 1) c execuion demands mus be execued in order o generae n evens. To calculae he limiing even sream we have o deermine he minimal disance beween n evens by formulaing he limiing inerval funcion for compeing based dependencies. Lemma 2: Le Γ R be a subse of m asks sharing he same processor and N = {(n 1,...,n m ) : m i=1 = n} he se of disribuions of n evens, where each ask τ i Γ R produces n i evens, hen he limiing inerval funcion is given by: ( β(n)= min max (n 1,...,nm) N ( max i=1,...,m ( + (n i,θτ i )), ( m i=1 (n i 1) c τ i ))) (4) Proof: Assume ha n evens can occur in a smaller disance han in he assumpion. This would mean ha one of he combinaions of he minimum resuls in a shorer disance. Consequenly, he inerval funcion + (n i,θ τi ) or he sum m i=1 (n i 1) c τ i delivers a shorer disance. Assume ha he inerval funcion + (n i,θ τi ) delivers a shorer disance and herefore he evens occur in a shorer disance han in he even sream definiion. Bu his is a conradicion according o he even sream definiion. Therefore he sum over he bes-case execuion imes mus occur in a shorer disance. This can only occur when one of he considered execuion imes is smaller han he one from he assumpion which is a conradicion since we already assume he bes-case execuion imes for all asks. We inroduce an algorihm using lemma 2 which delivers for n evens he minimum inerval in which hey occur. The algorihm in figure 3 can be used in conjuncion wih he mehods inroduced in [9] o implemen his funcion. In his paper a normalisaion for even sreams is inroduced in order o calculae he + (n,θ) efficienly. 1 β ( n ) { 2 n i N ; 3 e old = ; 4 f o r ( ( N n i = n) ) { 5 = max{ + (Θτ i,n i )} 6 e = 0 ; 7 f o r ( τ i τ i Γ R ) { 8 e = e + (n i 1) c τ i ; } 9 e old = min( e old,max( e, )) ; } 10 reurn e old ; } Figure 3. Calculaion of he inervals of limiing even sreams for compeing asks The ouer loop ieraes over all combinaions considered by he minimal operaion of lemma 2 (line 4 o 10). Line 5 considers all inervals of each even sream as i is done by max( + (n i,θ τi )) of lemma 2. The inner loop (line 7 o 9) calculaes he minimal disance produced by he bes-case execuion imes like m i=1 (n i 1) c τ i of lemma 2. Finally, he minimum of all inervals is deermined and he minimal inerval in which n cumulaed evens can occur is reurned (line 10). As menioned above o calculae he concree even sreams via ν( Θ, β(n)) we refer o [9] Offse Based Dependencies In order o show he generaliy of our new approach we adap he problem abou offses inroduced in he ransacion model by [12]. We will only consider saic offses beween ask simulaion as an example, however his approach covers also dynamic offses. Lemma 3: For wo sric periodic asks τ 1 and τ 2 wih an offse a we only have o calculae he minimum disance a beween evens of τ 1 and τ 2. This minimum disance is calculaed by a = min(mod(a,x),mod( a,x)) using he greaes common divisor x = gcd(p τ1, p τ2 ) of he periods of he asks. Leading o he limiing inerval funcion: β(n)=min( + (n,{(p τ1,0),(p τ2,a )}), + (n,{(p τ1,a ),(p τ2,0)})) (5) Proof: For a deailed explanaion see [12]. Now we can direcly se up he even sream via ν( Θ, β(n)): In he case of mod(a,x) mod( a,x) he limiing even sream is Θ = ({(p τ1,0),(p τ2,a )},{Θ τ1,θ τ2 }). In he case of mod( a,x) < mod(a,x) he limiing even sream is

5 Θ = ({(p τ1,a ),(p τ2,0)},{θ τ1,θ τ2 }). For more han wo asks he approach represened in [12] can be adaped o calculae he limiing inerval funcions. 6. Real-Time Analysis wih Limiing Even Sreams In order o use he limiing even sreams i is necessary o adap he new concep o he real-ime analysis, especially he wors-case response ime analysis. We have o deermine how grea he wors-case conribuion of asks in an inerval is when limiing even sreams are considered. Lemma 4: The maximal conribuion of asks in an inerval occurs when he ask wih he maximum wors-case execuion occurs as much as possible, hen he ask wih he second greaes execuion ime as much as possible up o he ask wih he smalles worscase execuion ime unil he limiing even sreams prohibis he occurrence of furher evens. Proof: Assume ha here is anoher disribuion han he one given by he assumpion. Therefore, i mus exis a leas one even which does no follow he paern in he assumpion. In order o increase he conribuion of he asks, he even mus rigger a ask whose wors-case execuion ime is greaer han assumed. Bu his is a conradicion, since we already assume for all asks wih greaer wors-case execuion imes he maximum number of invocaions. The response ime analysis was inroduced by Lehoczky e al. [11] and is defined as follows: Definiion 5: If he condiion τ Γ : r + (τ) d τ holds, he ask se is feasible and he real-ime analysis is successful. The wors-case response ime of a ask considering even sreams can be calculaed by: r + (τ)=max k N {r+ (k,τ) + (k,θ τ ) r + (k 1,τ)> + (k,θ τ )} c + τ k=0 r + (k,τ)= min{ =k c + τ + τ HP η(,θ τ ) c + τ } k 1 (6) } {{ } calchpsaic The amoun of execuions produced by higher prioriy asks can be calculaed by he even bound funcion muliplied by he wors-case execuion ime. By means of a fixed-poin ieraion he wors-case response ime can be calculaed for every job k. To ake he limiing even sreams ino accoun lemma 4 is used. To implemen τ HP η(,θ τ ) c + τ he algorihm in figure 4 was developed. The res of equaion 6 is unmodified. The algorihm has as parameers he inerval which is considered, k he job number of he ask 1 / / The i n e r v a l which i s c o n s i d e r e d 2 / / k Number o f c a l l s o f τ 3 / / τ The a s k under a n a l y s i s 4 / / Θ all A l l n e c e s s a r y l i m i i n g e v e n s r e a m s 5 / / Γ HP S e o f τ : φ τ > φ τ 6 c a l c H P S a i c (,k,τ,θ all,γ HP ) { 7 e = 0; 8 s o r ( Γ HP ) ; / / i j : c τi c τ j ; 9 10 ( Θ Θ all ) : η[θ] = η(,θ Θ ) ; ( Θ Θ all Θ τ Θ Θ ) : η[θ] = η[θ] k ; 11 f o r ( τ j Γ HP ) { 12 n = η(,θ τ j ) ; 13 m = min ( η[θ] Θ τ j Θ Θ ) ; 14 e = e + min ( n,m) c + τ j ; 15 f o r ( ( Θ Θ all Θ τ j Θ Θ ) { 16 η[θ] = η[θ] min ( n,m) ; } } 17 reurn e ; } Figure 4. Calculaion of he conribuion of higher prioriy asks for he wors-case response ime wih limiing even sreams under analysis, τ he ask which is explored, Θ all he se of he necessary limiing even sreams and Γ HP conaining all asks having a higher prioriy han τ. The algorihm sors he asks by heir wors-case execuion imes (line 8) and sores for every limiing even sream he maximum amoun of evens which his sream allows wihin (line 9). The number of invocaions of he ask under analysis τ mus be subraced from he corresponding limiing even sreams (line 10). In a loop (line 11 o 16) all higher prioriy asks are considered. The ask wih he greaes wors-case execuion ime is considered firs according o lemma 4. The algorihm deermines he maximum amoun of invocaions for he ask by he even sream of he ask (line 12) and he bound of he even sream if one exiss (line 13). The minimum of hese are used o calculae he maximum conribuion of he ask wihin (line 14). The second loop (line 15 o 16) reduces he corresponding limiing even sreams by he used evens (line 16). Therefore he loops disribue he amoun of evens of he limiing even sreams over he asks according o lemma 4. This leads o he worscase conribuion of higher prioriy asks wihin. Noe, ha he complexiy of he response ime analysis is sill pseudo-polynominial. The complexiy o calculae he limiing even sreams depends on he kind of he dependency which is considered. To calculae he problem of compeing-based dependencies can become challenging, because of is combinaorial complexiy. The analysis, however, is no affeced by his problem. So i is suggesive o find upper bounds for he limiing even sreams o improve he runime performance.

6 7. Case Sudy The significance of his new approach is shown by he following case sudy. The sysem o explore is depiced in figure 1 and described in chaper 3. Table 1 gives he parameers for he sysem and able 2 he even sreams. We have chosen his sysem, because i is easy o follow and i shows he new mehodology in he whole. CPU 1 τ 1 τ 2 τ 3 c c φ Θ Θ A Θ B Θ C CPU 2 τ 6 τ 7 τ 8 c c φ Θ Θ D Θ G Θ H BUS 1 τ 4 τ 5 c c φ 1 2 Θ Θ E Θ F Table 1. Parameers of he disribued sysem which is depiced in figure 1 To calculae he even sreams of he sysem, we use he approach given in [9]. The resuling even sreams in he sysem are shown in he able 2. Thereby, we compare he even sreams calculaed wih dependencies versus ones wihou dependencies. A saic offse of 100.u. beween he even sreams Θ B and Θ C is assumed. Θ wih dependencies wihou dependencies Θ A {(100,0)} {(100,0)} Θ B {(200,0)} {(200,0)} Θ C {(200,0)} {(200,0)} Θ D {(,0),(100,90)} {(,0),(100,90)} Θ E {(,0),(200,150)} {(,0),(200,150)} Θ F {(,0),(200,140)} {(,0),(200,70)} Θ G {(,0),(200,100)} {(,0),(200,100)} Θ H {(,0),(,80),(,160),(200,310)} {(,0),(,80),(,160),(,240),(200,370)} Θ I {(,0),(,30),(,60),(100,130)} {(,0),(,30),(,60),(,90),(100,165)} Θ J {(,0),(200,55)} {(,0),(200,55)} Θ K {(,0),(,70),(,150),(200,300)} {(,0),(,70),(,150),(,130),(200,360)} Table 2. All even sreams of he disribued sysem. The resuls are compued wih as well as wihou he approach. To deermine he ougoing even sreams wih dependencies i is necessary o calculae he limiing even sreams of he sysem. We consider only wo limiing even sreams Θ 1 and Θ 2. Θ 1 describes he offse beween Θ B and Θ C. Θ 2 describes he compeing based dependency beween Θ G and Θ H. Θ Θ Θ Θ 1 {(200,0),(200,100)} {Θ B,Θ C } Θ 2 {(,0),(,0),(,80),(,130),(,210),(200,300),(200,310)} {Θ G,Θ H } Table 3. Resuls of he calculaed limiing even sreams Afer calculaing he even sreams, we have a closer look on he improvemens in he analysis of he sysem. A firs, some even sreams and he improvemen of he densiy in he sysem are considered. This is depiced in able 4 and figure 5. n Θ F Θ F imp. Θ H Θ H Imp. Θ I Θ I imp % 0 0 0% 0 0 0% % % % ,58% % % ,96% ,5% ,7% ,45% ,45% ,26% ,44% ,71% ,69% ,14% ,38% ,11% ,22% ,61% ,26% ,54% ,68% ,31% ,02% ,27% ,9% Table 4. This shows he improvemen of he approach on he even sreams Θ F,Θ H and Θ I. Θ shows he inervals wih he dependency, Θ shows he inervals wihou he dependency, improvemen is given in % Reducion of Densiy [%] #Evens ΘF ΘH ΘI Figure 5. This figure shows he improvemens of he inervals in percen of he even sream Θ F,Θ H and Θ I. The dependencies have no only an influence on he densiy of he even sreams, bu also a direc influence on he wors-case response imes. The worscase response ime of he ask τ 3 has been reduced from 150.u. o 80.u. This means ha he resul of he analysis wih dependencies is in his case 46,66% igher compared o he analysis wihou dependencies. The ask τ 6 has a wors-case response ime wihou dependencies of 255.u. and wih dependencies of 205.u., which is a reducion of he wors-case response ime of 19,6%. This synheical example shows ha dependencies can improve he real-ime analysis. Thereby we have

7 shown how easy differen dependencies can be combined in a general approach. 8. Conclusion We have shown he possibilily o achieve a holisic model for ask dependencies in disribued real-ime sysems. The new approach has been applied o fixedprioriy sysems. We have shown by wo kinds of dependencies how hese can be described by he new defined limiing even sreams. Thereby, a new kind of dependency has been inroduced. Wih he effec, ha we have cu he complexiy of he dependencies from he real-ime analysis. Finally, a case sudy has been conduced o show he improvemens of he approach. Despie he example is synheical, i has been shown ha our concep works for differen kinds of dependencies. In he fuure we will show how more kinds of dependencies can be inegraed by his new model and how he limiing even sreams can be propagaed hrough he sysem. Furhermore, he inegraion of he limiing even sreams ino approximaive realime analysis like he real-ime calculus [16] or he hierarchical even sreams [1] o improve he runime performance is also an aim. References [1] Karsen Albers, Frank Bodmann, and Frank Slomka. Hierarchical even sreams and even dependency graphs: A new compuaional model for embedded real-ime sysems. In ECRTS 06: Proceedings of he 18h Euromicro Conference on Real-Time Sysems, pages , Washingon, DC, USA, IEEE Compuer Sociey. [2] Karsen Albers and Frank Slomka. An even sream driven approximaion for he analysis of real- ime sysems. In ECRTS 04: Proceedings of he 16h Euromicro Conference on Real-Time Sysems, pages IEEE, July [3] Klaus Gresser. An even model for deadline verificaion of hard real-ime sysems. In Proceedings of he 5h Euromicro Workshop on Real-Time Sysems, [4] J. C. Palencia Guierrez and Michael Gonzalez Harbour. Schedulabiliy analysis for asks wih saic and dynamic offses. In RTSS, page 26 ff, [5] J. C. Palencia Guierrez and Michael Gonzalez Harbour. Exploiing precedence relaions in he schedulabiliy analysis of disribued real-ime sysems. In IEEE Real-Time Sysems Symposium, pages , [6] Rafik Henia and Rolf Erns. Conex-aware scheduling analysis of disribued sysems wih ree-shaped askdependencies. In DATE 05: Proceedings of he conference on Design, Auomaion and Tes in Europe, pages , Washingon, DC, USA, IEEE Compuer Sociey. [7] Rafik Henia and Rolf Erns. Improved offse-analysis using muliple iming-references. In DATE 06: Proceedings of he conference on Design, auomaion and es in Europe, pages , 3001 Leuven, Belgium, Belgium, European Design and Auomaion Associaion. [8] Rafik Henia, Razvan Racu, and Rolf Erns. Improved oupu jier calculaion for composiional performance analysis of disribued sysems. In Proceedings Workshop on Parallel and Disribued Real-Time Sysems, March [9] Seffen Kollmann, Karsen Albers, and Frank Slomka. Effecs of simulaneous simulaion on he even sream densiies of fixed-prioriy sysems. In Specs 08: Proceedings of he Inernaional Simulaion Muli-Conference. IEEE, June [10] Simon Kuenzli, Arne Hamann, Rolf Erns, and Lohar Thiele. Combined approach o sysem level performance analysis of embedded sysems. In CODES+ISSS 07: Proceedings of he 5h IEEE/ACM inernaional conference on Hardware/- sofware codesign and sysem synhesis, pages 63 68, New York, NY, USA, ACM. [11] John P Lehoczky. Fixed prioriy scheduling of periodic ask ses wih arbirary deadlines. In Proceedings of he 11h IEEE Real-Time Sysems Symposium, pages , December [12] Rodolfo Pellizzoni and Giuseppe Lipari. Improved schedulabiliy analysis of real-ime ransacions wih earlies deadline scheduling. In RTAS 05: Proceedings of he 11h IEEE Real Time on Embedded Technology and Applicaions Symposium, pages 66 75, Washingon, DC, USA, IEEE Compuer Sociey. [13] Ola Redell. Analysis of ree-shaped ransacions in disribued real-ime sysems. In ECRTS 04: Proceedings of he 16h Euromicro Conference on Real-Time Sysems (ECRTS 04), pages , Washingon, DC, USA, IEEE Compuer Sociey. [14] Ken Tindell. Adding ime-offses o schedulabiliy analysis. Technical repor, Universiy of York, Compuer Science Dep, YCS , [15] Ken Tindell and John Clark. Holisic schedulabiliy analysis for disribued hard real-ime sysems. Microprocessing and Microprogramming, 40: , April [16] Erneso Wandeler. Modular Performance Analysis and Inerface-Based Design for Embedded Real-Time Sysems. PhD hesis, ETH Zurich, Sepember 2006.