Response Time Analysis of Asynchronous Periodic and Sporadic Tasks Scheduled by a Fixed-Priority Preemptive Algorithm

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1 Response Tme Analyss of Asynchronous Perodc and Sporadc Tasks Scheduled by a Fxed-Prorty Preemptve Algorthm Manuel Coutnho EDISOFT /IST-UTL manuel.coutnho@edsoft.pt José Rufno FCUL ruf@d.fc.ul.pt Carlos Almeda IST-UTL cra@comp.st.utl.pt Abstract Real-tme systems usually consst of a set of perodc and sporadc tasks. Perodc tasks can be dvded nto two classes: synchronous and asynchronous. The frst type does not defne the task frst release, contrary to the second. Hence, synchronous perodc tasks are assumed to be released at the worst nstant: the crtcal nstant. The schedulablty test s reduced to check a sngle executon of the task under analyss. The ntegraton of sporadc tasks s also straghtforward: they are treated as a perodc task wth maxmum arrval frequency. On the other hand, asynchronous perodc tasks requre a test for each release n the hyperperod and the ntegraton of sporadc tasks s not trval: the worst release nstant s unknown a pror. However, they do not assume that the tasks are released at the worst nstant. Ths paper presents a new schedulablty analyss method based on the Response Tme Analyss (RTA) to determne the worst response tme of both asynchronous perodc and sporadc tasks, scheduled by a fxed-prorty preemptve algorthm wth general deadlnes. It also presents another method that enables the ntroducton of a user confgurable degree of pessmsm, reducng the hyperperod dependency. 1. Introducton Tradtonally, real-tme system desgners choose to mplement both perodc and sporadc tasks due to ther deter- Rua Qunta dos Medronheros - Lazarm, Apartado 382, Caparca, Portugal. Tel: Fax: Insttuto Superor Técnco - Unversdade Técnca de Lsboa, Avenda Rovsco Pas, Lsboa, Portugal. Tel: Fax: Ths work was supported by EU and FCT, through Project POSC/EIA/56041/2004 (DARIO) WWW Page - Faculdade de Cêncas da Unversdade de Lsboa, Campo Grande - Bloco C8, Lsboa, Portugal. Tel: Fax: Ths work was supported by EU and FCT, through Project POSC/EIA/56041/2004 (DARIO) and through the FCT Multannual Fundng Programme. Ths work was partally motvated by the research results of Project AIR, supported by ESA (European Space Agency) through the ITI program, ESTEC Contract 19912/06/NL/JD. mnsm and responsveness. Perodc tasks are used typcally for process control (e.g. atttude control n space systems) whereas sporadc tasks provde very fast responses to external events (e.g. arbag n automotve applcatons). Two consecutve releases of a perodc task are separated by a well-known fxed tme nterval: the task perod. On the other hand, sporadc tasks have a more non determnstc behavour: two consecutve releases are separated by a wellknown mnmum tme nterval. Perodc tasks can be dvded nto two man classes: synchronous and asynchronous 1. Ths dfference relates to defnton (or lack of) of the frst release nstant. Sporadc tasks usually react to external events so there s lttle advantage n specfyng ther frst release tme. The frst release nstant s undefned n synchronous perodc tasks. Hence, the schedulablty analyss has to consder the worst possble release: the crtcal nstant. Therefore, only one release has to be studed, allowng pseudopolynomal tests to produce necessary and suffcent condtons [11, 12]. The ntegraton of sporadc tasks wth synchronous perodc tasks s straghtforward: they are treated as synchronous perodc tasks, released at a crtcal nstant and wth perod equal to ts mnmum nter-arrval tme (maxmum frequency). On the other hand, the frst release nstant s defned n asynchronous perodc tasks. These are more dffcult to analyze snce the worst release nstant s unknown apror. The determnaton of suffcent and necessary schedulablty condtons of asynchronous perodc tasks n a preemptve fxed-prorty schedulng algorthm has been shown to be co-np-complete [13]. In fact, an analyss through the task hyperperod s requred to determne that all releases meet ther deadlne [1]. Snce the hyperperod ncreases exponentally wth the number of tasks, specally those wth prme perods, ths analyss can be heavy, even wth few tasks. Furthermore, sporadc tasks are more dffcult to analyze snce the worst release nstant s also unknown. 1 Asynchronous perodc tasks are sometmes referred as concrete perodc tasks n the doman of non-preemptve schedulers [10].

2 However, asynchronous perodc tasks have an mportant advantage over the synchronous counterpart: they are not released at the crtcal nstant. Hence, ths generc methodology may be used to accurately represent real settngs [16, 19, 18]. In addton, there can co-exst tasks that never nterfere wth each other, even wth dfferent perods. The use of asynchronous an perodc task model allows an ncrease of CPU usage and can guarantee hgher system responsveness thus avodng the costs of buyng and certfyng faster equpment [16]. The hyperperod dependency s also present n the tmetrggered archtecture, whch to some degree demonstrates the practcablty of ths type of analyss. However, even though these systems generally allow a hgher CPU usage, they have a low responsveness to external events. Ths paper proposes a method to analyze the schedulablty of systems comprsng asynchronous perodc and sporadc tasks wth general deadlnes (deadlnes less than or equal to the perod), scheduled by a fxed-prorty preemptve algorthm. As far as our knowledge can say, the schedulablty analyss of sporadc tasks together wth asynchronous perodc tasks has not been addressed yet n the lterature. The proposed methods follow the Response Tme Analyss (RTA) approach, determnng the response tmes of all perodc releases throughout the system hyperperod. The determnaton of the worst response tmes also allows a greater nsght to the amount of jtter assocated to a perodc task and the number of mssed deadlnes (assumng the response tme s larger than the deadlne but smaller than the perod). In control systems ths s useful because, f not carefully bounded, the jtter can potentally degrade the system performance [17]. The worst release nstant caused by the asynchronous task set, the asynchronous crtcal nstant, s unknown a pror. Ths paper presents a method that determnes a set of canddates to the asynchronous crtcal nstant to analyze the schedulablty of sporadc tasks. Snce many real-tme embedded systems have large desgnng/mplementng perods, the heavy schedulablty analyss requred s stll manageable n most systems, as seen n the tme-trggered archtecture usage cases. However, f the hyperperod s too large, another method s presented that produces a faster result at the expense of the ntroducton of a varable degree of pessmsm. 2. Related work The schedulablty analyss has commonly used the crtcal nstant as a means to produce very fast results. In fact, n systems where the release nstants are not known apror derve necessary and suffcent condtons, as the RTA [11]. Ths analyss gves the response tme of a gven task renderng the schedulablty test trval: the response tme must be smaller than the deadlne. Attempts to remove the crtcal nstant assumpton by defnng release nstants for perodc tasks have produced some results. Perhaps the most wdely known result s produced by the offset system model [3, 15, 4, 20, 14]. In ths model, there s a set of perodc transactons, each one composed by a set of tasks (wth the same perod as the transacton and normally related by a precedence order). The tasks are released at a fxed tme nterval after the transacton has begun but the release of the transacton s unknown. As a consequence, tasks nsde the same transacton have lttle or no nterference wth one another. However, tasks n dfferent transactons are analyzed assumng they are released at the worst nstant. Furthermore, tasks nsde a transacton must share the same perod. As a consequence, the hyperperod s typcally small (smaller number of tasks wth dfferent perods). Note also that ths offset analyss can be made a partcular case of asynchronous perodc tasks wth the addtonal defnton of the release nstants of the transactons. Regardng asynchronous perodc tasks, schedulablty analyss has been addressed n [1] and n [6, 8]. In [1] t s presented an optmum prorty assgnment algorthm for fxed-prorty schedulers wth O(n 2 ) complexty. Ths algorthm requres a suffcent and necessary schedulablty test (also known as a feasblty test) whch s also presented. However, ths analyss determnes the nterference of each ndvdual hgher prorty task release, whch requres a sortng procedure and a hgh number of teratons. In [5] the author treats both asynchronous perodc and sporadc tasks, but only provdes a suffcent test when consderng sporadc tasks. Both [6] and [8] determne the response tme n dynamc prorty schedulers wth arbtrary deadlnes (deadlnes can be larger than the perod), whereas fxed-prorty schedulng s brefly addressed n [7]. However, ths work can be smplfed when appled to fxed-prorty systems. Other work has addressed offset-free systems, characterzed by allowng the scheduler to choose the best (frst) release nstants of the perodc tasks. In [9] near-optmal prorty and release assgnments are produced usng a heurstc and usng the schedulablty test descrbed n [1, 2]. Fxed-prorty non-preemptve schedulers have been consdered n [10], where t s demonstrated that the complexty of the schedulablty analyss s NP-hard n the strong sense, but t dd not provde a schedulablty test. 3. System Model The system model consdered n ths paper ntegrates both perodc and sporadc tasks, scheduled by a fxedprorty preemptve algorthm. It s assumed that the tasks are ndependent, co-exst n a sngle processor and the con-

3 text swtchng overhead s null. Each task has a unque prorty but perodc and sporadc can ntertwne ther relatve prortes, that s, the prorty of a gven sporadc task can be n between the prortes of two other perodc tasks. Concernng only the perodc task set, t s commonly referred as an asynchronous task set wth general deadlnes. In ths model, the perodc tasks are defned by the parameters Γ = {c,t,r,d } where c represents the Worst Case Executon Tme (WCET), T the task perod, r the release tme of the task and D the relatve deadlne from the task release. The condton of general deadlnes specfes that the deadlne s smaller than or equal to the perod. Hence, the parameters are condtoned to 0 c D T and 0 r. A job s a partcular executon of a perodc task. Smlarly, the sporadc task set s characterzed by τ m = {e m,mit m,h m } where e m s the WCET, MIT m s the Mnmum Inter-Arrval Tme between consecutve task releases and H m s the relatve deadlne from the task release. Smlarly, the deadlne s nferor to the mnmum nter-arrval tme, hence 0 e m H m MIT m. Both tasks sets are ordered by ncreasng prorty, that s, task Γ has a hgher prorty than task Γ +1 and τ m has also a hgher prorty than τ m+1. Addtonal Defntons R - Response tme (nstant) of the synchronous perodc counterpart of the asynchronous perodc task Γ l r = r +lt - Release nstant of the l th job of the asynchronous perodc task Γ l R - Response nstant of the l th job of the asynchronous perodc task Γ R m - Response nstant of the sporadc task τ m l R l r - Response tme of the l th job of an asynchronous perodc task Γ Λ - Hyperperod of the asynchronous perodc task Γ 4. Response Tme Analyss for Synchronous Tasks The Response Tme Analyss (RTA) for synchronous perodc tasks and sporadc tasks s a well-known fast method that produces necessary and suffcent schedulablty condtons [11]. Ths secton presents the essental detals of how ths method works, snce t s used as a startng pont of the analyss descrbed latter on. In synchronous perodc tasks, the worst nstant where the tasks can be released s at a crtcal nstant: all hgher prorty tasks start at the same nstant. Hence, the response tme of the synchronous counterpart of Γ,gvenbyR,s the soluton of the equaton Fgure 1. Iteratons of the Response Tme Calculaton, Equaton (2) R = c + 1 R j=1 T j c j (1) Due to the non-lnear celng operator x, ths equaton can not be easly solved from drect analyss. Instead, an teratve fxed-pont method s used to determne ts value R (0) = c R (n+1) = c + 1 j=1 R (n) (2) c j T j The basc phlosophy behnd ths teratve method s to ncrementally ncrease the response tme based on the hgher prorty task demanded workload. Each teraton gves the nstant where the workload request has been fulflled. However, between the last teraton and the present one, new workload may have been demanded. Hence, the algorthm contnues to update the new workload demanded untl t stops ncreasng. At ths pont the method halts, where R (n+1) = R (n) = R. The schedulablty test s reduced to check f R D. If so, then the task s schedulable. Fgure 1 shows an example of how the teratons progress untl the response tme s found. The ntegraton of sporadc tasks s straghtforward wthn ths model: they are treated as perodc tasks, released at the crtcal nstant and wth perod equal to ther MIT. As such, there s lttle dfference analysng synchronous perodc or sporadc tasks. 5. Response Tme Analyss of Asynchronous Perodc Tasks Ths secton provdes a new method to determne the response tme of all asynchronous perodc jobs. For smplcty of exposton, sporadc tasks are not yet consdered. The ntroducton of the release nstant of perodc tasks has to be frstly reflected nto the workload functon. Hence, the workload requested n the nterval [0,t[ by the tasks Γ 1,...,Γ s gven by

4 w (t)= j=1 t r j T j c j (3) 0 where the x 0 operator gves the max{0, x }. Ths operator s ntroduced to reflect that the number of tmes a task s nvoked cannot be lower than zero. The response nstant of an asynchronous perodc task released at l r s obtaned by addng ts executon tme, c, and the nterference from the hgher prorty tasks, I 1 ( l r, l R ), to the nstant t was released, l r l R = l r + c + I 1 ( l r, l R ) (4) The nterference from hgher prorty tasks s the only unknown element. Ths nterference can be dvded nto the nterference remanng at l r, desgnated by I b 1 (l r ),and the nterference that arses after the task release untl ts response nstant, I a 1 (l r, l R ). I 1 ( l r, l R )=I b 1 (l r )+I a 1 (l r, l R ) (5) The term I b 1 (l r ) reflects the nterference from hgher prorty tasks requested before the release nstant. In [1], ths term s calculated by searchng backwards all the hgher prorty task releases n the nterval [ l 1 r + D, l r [ plus the workload remanng at l 1 r + D. Ths set of task releases s then sorted chronologcally and ndvdually analyzed. Instead, our approach requres the determnaton of the last dle nstant before l r, desgnated by L 1 ( l r ).Thslast dle nstant s defned as the last nstant where all hgher prorty tasks prevously demanded workload has been fulflled. Hence, from L 1 ( l r ) to l r the system s processng the hgher prorty tasks. The remanng nterference s easly found by subtractng the workload demanded wth the workload processed by the hgher prorty tasks n the nterval [L 1 ( l r ), l r [. I 1 b (l r )=w 1 ( l r ) w 1 (L 1 ( l r )) }{{} workload demanded ( l r L 1 ( l r )) }{{} workload processed (6) As wll be shown later on, the calculaton of the last dle nstant follows a method smlar to the RTA, so t does not analyze each hgher prorty task pror release ndvdually, nor does t requre a sortng operaton. The term I 1 a (l r, l R ) accounts for the total workload request by the hgher prorty tasks after the release, that s, n the nterval [ l r, l R [. In [1] the author determnes all the hgher prorty tasks releases n the nterval [ l r, l r + D [. Ths set of releases s ordered chronologcally and ndvdually analyzed. The nterference found at l r + D s used to calculate the remanng nterference for the next job, I 1 b (l+1 r ), as prevously stated. Our approach determnes the nterference after the release untl the response nstant, correspondng to the nterval [ l r, l R [ and t s gven by I 1( a l r, l R )=w 1 ( l R ) w 1 ( l r ) (7) Jonng equatons (4), (5), (6) and (7), the response nstant s gven by l R = L 1 ( l r ) + c + w 1 ( l R ) w 1 ( L 1 ( l r )) The smallest soluton (larger than l r ) of ths equaton gves the response nstant of a task released at l r. The equaton starts from the last dle nstant, as opposed to l r n equaton (4). It then adds the task executon tme to the workload demanded by the hgher prorty tasks n the nterval [L 1 ( l r ), l R [. Ths equaton has to be verfed for all jobs of Γ released nsde the hyperperod. In [1] the hyperperod s proven to correspond to the nterval [max(r 1,...,r ),max(r 1,...,r )+LCM(T 1,...,T )[ where the LCM functon corresponds to the Least Common Multple. The hyperperod s the mnmum nterval n whch the task set starts repeatng tself. Note that t starts after a ntal transent phase. Ths ntal transent phase does not need to be analyzed snce there s less nterference than the worst case (not all tasks have been made actve). Hence, the hyperperod can start at any nstant after max(r 1,...,r ) as long as t mantans the same length. Due to the sporadc task nterference, we begn the hyperperod at S = max(r 1,...,r )+T Λ =[S,S + LCM(T 1,...,T )[ (9) Ths new nterval has the same length and starts after the transent phase, so t mantans the same propertes. Secton 6.2 explans the reasons for the change made to the hyperperod. In short, t s necessary to analyze the nterval ] l 1 r, l r ] to determne l R, so ths nterval must not contan a transent phase. The mnmum and maxmum l such that l r Λ are gven by l mn = S r l max = T S +LCM(T 1,...,T ) r T 1 The schedulablty analyss can be summarzed n the condton (8) { l: l r Λ l R l r D, Γ s schedulable l: l r Λ l R l r > D, Γ s unschedulable (10)

5 w (t)= j=1 1 + t r j c j (11) T j 0 Fgure 2. Extended Workload Request 5.1. Last Idle Instant Calculaton The response tme analyss presented requres the determnaton of the last dle nstant of the hgher prorty perodc tasks for a gven tme, L (t). Even though prevous work n [1] analyzes the small nterval [ l 1 r + D, l r [ to determne the remanng workload at l r, t also has to add the remanng workload at l 1 r + D. So, n essense, t has to analyze the whole tme sequence snce the beggnng of tme to determne the remanng workload at l r. In fact, all prevous work regardng asynchronous systems requred determnng the schedule snce the beggnng of tme [1, 5]. Our method also has to analyze the nterval [0,t[ but, nstead of searchng every ndvdual hgher prorty task release as [1], our approach follows a method closely related wth the fast teratons of the RTA algorthm. Our am s to extend the workload functon wth the knowledge of the dle perods: w ext (t), Fgure 2. When the system s n an dle state (at the current prorty level) the extended workload s equal to t (nstant t 1 n Fgure 2), so w ext (t) t gves the remanng workload at each nstant. By trackng the extended workload functon, the dle perods can be determned and the last dle nstant found. In summary, when the system s non dle (computng), the method advances n tme smlarly to the RTA method; when t s n dle, a new functon, ρ (t), determnes the next computng nstant. The transton between an dle nstant to a computng nstant requres the determnaton of the workload requested at a partcular nstant. To do ths, the auxlary functon w (t) s ntroduced where the non-lnear operator x denotes the floor functon and x 0 = max{0, x }. The functon w (t) gves the workload request n the nterval [0,t]. Notce that ths nterval s closed at both ends. Hence, the workload requested at t can be gven by w (t) w (t). The method advances through the computng perods through a fxed-pont method. Let ϒ(t) be the total dle tme untl a gven nstant. At each teraton, the fxed-pont method gves the nstant where the workload demanded has been fulflled. Snce from the last teraton to the present one new workload may have been demanded, the method contnues to update the new workload untl t stops ncreasng w (n+1) = j=1 w (n) T j r j c j + ϒ(t) (12) When w (n+1) = w (n) the method stops and w (n) s equal to the nstant where the computng perod has ended. The dagram n Fgure 4 shows how ths teratve method fnds the next pont computng nstant. After fndng the nstant where the workload as been fulflled, whch also corresponds to an dle nstant, the functon ρ (t) gves the next computng nstant, whch corresponds to the closest task release ( t r j ρ (t)= mn j=1... T j 0 ) T j + r j 0 (13) Beng t the last computng nstant of a computng perod, the dfference between ρ (t) and t gves the next dle perod. By sequentally addng these dle perods to the workload request functon we get the extended workload functon, Fgure 2. w ext (t)=w (t)+l (t) w (L (t)) }{{} total dle tme (14) An teratve method s thus bult by sequentally determnng the computng and dle perods. The transton from an dle to a computng perod s accomplshed by addng the workload request at that nstant, gven by w (ρ (t)) w (ρ (t)). The pseudo-code that mplements ths method s llustrated n Fgure 3. An example of how ths pseudo-code works s shown on Fgure 4. As llustrated, the algorthm adds to the workload the dle tme perods so as to follow the extended workload request functon.

6 functon L (t){ f( = 0)return t; // there are no hgher prorty tasks last dle nstant = 0; total dle tme = 0; terator = 0; whle(true) { last terator = terator; terator = w (terator)+total dle tme; f(terator > t) return last dle nstant; f(last terator = terator) { // arrved at an dle nstant f(terator t ρ (terator)) return t; last dle nstant = ρ (terator); total dle tme += ρ (terator) terator; terator = ρ (terator)+w (ρ (terator)) w (ρ (terator)); }}} Fgure 3. Pseudo-code to determne L (t) 5.2. Response Instant Calculaton Fgure 4. Iteratons of the functon L (t) Lke the classc RTA teratve method, the response nstants can also be found teratvely. Hence, the response nstants of Γ, l R, are determned usng the fxed-pont method { l R (0) = c + l r l R (n+1) = c + w 1 ( l R (n) )+L 1 ( l r ) w 1 (L 1 ( l r )) When l R (n+1) = l R (n) the response nstant has been found Applcaton Example An example of how ths method can be appled s presented n Table 1 and the frst tme sequences are shown n Fgure 5. The Example 1 has a 98,67% CPU usage and c D = 282,54%. Notce also that the larger hyperperod, Λ 10, s qute large, above 60 mllon tme unts. Ths was made to show that even for very large hyperperods, the amount of tme requred by the analyss s stll managable. The schedulablty method presented thus far has very demandng processng tmes 2, so the analyss followed an mproved method, dscussed n Secton 7.1. The Table 1 compares the number of teratons of the presented method wth the one descrbed n [1] 3. As depcted, the number of teratons requred s smaller by up to 95% and the performance gan ncreases as the hyperperod becames larger. From Table 1 we can see that the response tme calculated assumng a crtcal nstant, R, renders four tasks unschedulable: Γ 2,6,7,8. Note n partcular that Γ 2 s deemed 2 A rough estmaton gves more than hours to analyze Γ Ths result may vary somewhat wth the mplementaton snce the algorthm n [1] requres a sortng procedure. Fgure 5. Schedulng Example 1 for the tasks Γ 1,...,Γ 8. The symbol denotes the task release whereas denotes the response nstant schedulable from Fgure 5. In fact, ths task never suffers nterference from the hgher prorty task Γ 1. For the tasks Γ 7,8, we show that ther response tme s always nferor to the deadlne by analysng every job n the hyperperod: Fgure 6. Followng a smlar approach, task Γ 6 s also deemed schedulable. Note that to determne f the task s schedulable the determnaton of R (or other, even faster schedulablty test) can be suffcent n most cases. There are, however, scenaros where the classc schedulablty analyss gves too pessmstc results. Ths analyss can be taken n these stuatons and when there s enough tme. Ths method provdes the worst response tme for all jobs and therefore allows a better vew as to how much response jtter s ntroduced. For example, as seen on Fgure 5, Γ 2 never suffers the nterference of Γ 1. Hence ts jtter s only bounded by ts mnmum and maxmum executon tme, [c 2,c 2], makng t an almost jtter-free task.

7 System Parameters Schedulablty Results Γ c T r D R max (l R l ) r LCM(T 1,...,T ) Analyss Tme (s) # Iteratons Improvement over [1] Γ % Γ % Γ % Γ % Γ % Γ % Γ % Γ % Γ % Γ % Table 1. Parameters of Example 1. The schedulablty analyss followed the mproved method descrbed n Secton 7.1, Fgure 13 (a) (b) (c) Fgure 6. Response tme of each job n the hyperperod for the tasks: (a) Γ 7 ;(b)γ 8 ;(c)γ 8 wth D 8 = 90 In a more general case, the determnaton of the worst jtter requres two analyss: one wth the mnmum executon tmes, c 1,...,c ; the other wth the worst executon tmes, c 1,...,c. The nterval between the best response tme of the frst case wth the worst response tme of the latter gves the largest jtter. If, for example, c = c, then Fgure 6 can tell us the jtter of tasks Γ 7 and Γ 8. Ths analyss also covers systems that support occasonal mssed deadlnes. For example, by lowerng D 8 to 90 tme unts, there are some jobs that do not fulfll the deadlne, as depcted n Fgure 6. Note however, that snce the responses are all lower than T 8, f one job msses ts deadlne, the next job wll not suffer any temporal nterference. If a job has a response tme hgher than ts perod, than ths analyss does not suffce and a model extenson (to ncorporate arbtrary deadlnes, for example) s needed. Snce the knowledge of L 1 ( l r ) s requred to determne the response tme for each job nsde a hyperperod, the overall complexty to determne f a gven task Γ s schedulable s O(ELCM(T 1,...,T ) 2 ),wheree s the average number of steps requred to calculate the next dle nstant snce the prevous one. The algorthm s dependent on the quadratc power of the LCM because, when calculatng L 1 ( l r ), the method starts from the begnnng for every job. For a LCM(T 1,...,T )/T number of jobs, the number of total teratons requred s E 1 + E ELCM(T 1,...,T )/T = O(E LCM(T 1,...,T ) 2 ). In Secton 8.1 we wll descrbe how to make ths O(ELCM(T 1,...,T )). 6. Integraton of Sporadc Tasks One major advantage of usng event-drven systems wth a combnaton of perodc and sporadc tasks comes from the hgh responsveness and low resource consumpton. Suppose a hgh prorty sporadc task wth a very low deadlne (H ) but wth a very large MIT: the sporadc task wll execute seldomly havng a low mpact on the remanng tasks schedulablty. A pure tme-trggered system, on the other hand, would need a hgh frequency pollng task to provde such responsveness. The synchronous approach assumes that all tasks,

8 whether perodc or sporadc, are released at the same nstant: crtcal nstant. However asynchronous perodc tasks are released at fxed nstants whch may not concde wth the crtcal nstant. Therefore t s necessary to know the worst possble nstant where a sporadc task can be released: asynchronous crtcal nstant. It s also necessary to account for the nterference produced by the sporadc tasks. Hence, assumng that all sporadc tasks are released at t = 0, the workload demanded by the tasks τ 1,...,τ m,gvenbyω m (t), s equal to m t ω m (t)= e k (15) k=1 MIT k 0 If the sporadc tasks are released at t 0 then ther workload functon s ω m (t t 0 ). As wll be shown latter on, the worst nstant comes when all sporadc tasks are released at the same nstant Asynchronous Crtcal Instant Ths secton presents a method to determne the asynchronous crtcal nstant, κ, mposed by the hgher prorty asynchronous perodc tasks. Ths nstant corresponds to the release nstant where low prorty sporadc tasks are released to produce the worst response tme. Observng Fgure 7, whch llustrates the functon w ext (t) t, we conclude that the asynchronous crtcal nstant can only be n the begnnng of the computng perods/end of the dle perods. Ths can easly be proved by assumng the contrary: f the asynchronous crtcal nstant s n the dle perod before, then the response nstant wll also be sooner by at least the same amount, Fgure 7; f t s n the computng perod after, then the response nstant s equal but, snce t started later, the response tme s smaller, Fgure 7. Hence, we need only concern wth the start of the computng perods because these wll gve the canddates for the asynchronous crtcal nstant. The worst release nstant for the lower prorty sporadc tasks s therefore one of these canddates. To determne the canddates for the asynchronous crtcal nstant we need a method to determne the next dle nstant, ϕ (t). Ths new method s very smlar to the teratons of the L (t) algorthm. As specfed n Fgure 8, the functon ϕ (t) receves a work nstant. Ths s the frst work nstant of the computng perod. The frst teraton adds the work demanded at that nstant. The followng teratons are very smlar to the other methods, addng at each teraton the new workload demanded. An example of the ϕ (t) teratons s shown n Fgure 9. Fgure 8 also descrbes a method, κ (t mn,t max ), that captures all canddates nsde the nterval ]t mn,t max ]. Ths method s ether calculatng the next work nstant, usng Fgure 7. Asynchronous Crtcal Instants Canddates ρ (t), or the next dle nstant, usng ϕ (t). When the work nstant found s greater than t max the method stops. The complexty of determnng the asynchronous crtcal nstant canddates n Λ s smlar to the determnaton of the last dle nstant: O(ELCM(T 1,...,T )) Response Tme Analyss of Asynchronous Perodc Tasks The worst nterference that sporadc tasks can produce to the lower prorty perodc tasks can be determned wth knowledge of the asynchronous crtcal nstant canddates. Suppose a perodc job released at l r and that τ K s the lowest prorty sporadc task wth hgher prorty than Γ. In Fgure 10 s llustrated the hgher prorty perodc tasks remanng workload at each nstant, w ext (t) t. There are four canddate nstants between l 1 r and l r : κ (1),...,κ (4). From Fgure 10 t s clear that the worst nterference that hgh prorty sporadc tasks nduce nto the low prorty perodc tasks happens when they are released at the crtcal nstant canddates. If the sporadc tasks are released earler, then they wll be executed (snce the system s dle). If they are released after the canddate, then the nterval where they can cause nterference s smaller.. Ths release nstant removes the dle perod between κ (2) and l r, makng one sngle computaton perod. In fact, f there are dle perods between the release of the sporadc tasks and l r then that cannot be the worst release nstant snce all the workload has been fulflled at one nstant and hence wll not be added to the next computng perod (take example of the The worst release nstant n Fgure 10 s κ (2) release at κ (1) n Fgure 10). Snce from the worst release nstant to l r there cannot exst dle perods, f the worst release s before l 1 r then the l 1 job could not be executed and would mss ts deadlne. Hence, assumng that all prevous jobs fulfll ther deadlne,

9 functon κ (t mn,t max ){ next work nstant = L (t mn ); work nstants = {}; whle(true) { next dle nstant = ϕ (next work nstant); next work nstant = ρ (next dle nstant); f( next work nstant > t max ) return work nstants; work nstants = work nstants {next work nstant}; }} functon ϕ (work nstant){ //returns the next dle nstant total dle tme = work nstant w (work nstant); terator = work nstant + w (work nstant) w (work nstant); whle(true) { last terator = terator; terator = w (terator)+total dle tme; f( terator = last terator ) return terator; }} Fgure 8. Pseudo-code to determne all the canddates of the asynchronous crtcal nstant n the nterval ]t mn,t max ] Fgure 10. Analyss of the canddates to the worst sporadc release Fgure 9. Examples of the ϕ (t) teratons the canddates to the worst release nstant are n the nterval ] l 1 r, l r ]. If there are no crtcal nstant canddates, then the task s not schedulable snce durng that nterval the system never completed ts workload and hence the prevous job dd not fullfll ts deadlne. The nterval ] l 1 r, l r ] must not be nsde the ntal transent phase. So the hyperperod must start T after the ntal phase has ended. Hence the need to make S = max(r 1,...,r )+T n Equaton (9). Ths change n the hyperperod does not affect sgnfcantly the analyss tme snce the LCM s stll by far the man factor. Snce the dle perods from the worst release nstant, κ, to l r are now occuped processng the hgher prorty tasks, these have to be subtracted when calculatng the response nstant. The total dle tme from κ to l r s gven by [L 1 ( l r ) w 1 (L 1 ( l r ))] [L 1(κ ) w 1(L 1(κ ))] }{{}}{{} total dle tme untl l r total dle tme untl κ Snce at κ the hgher prorty perodc tasks are dle, then L 1 (κ )) = κ. The response nstant of an asynchronous perodc task s found by addng to equaton (8) the workload from the hgher prorty sporadc tasks and subtractng the total dle tme l R = κ + c + w 1 ( l R ) w 1 (κ )+ω K ( l R κ ) (16) where the smallest soluton (larger than l r ) corresponds to the response nstant. Ths equaton can be solved usng the fxed-pont method wth l R (0) = l r + c. Ths equaton has to be solved for every κ ] l 1 r, l r ] and the worst response nstant found s used n expresson (10) to determne f Γ s schedulable. Note that f a partcular κ s not the worst nstant and t does not completely remove the dle perods, then the correspondng response nstant calculated wll be smaller than t should be (because we are subtractng all the dle perods), but snce ths s not the worst case t wll not be consdered n the fnal schedulablty test. If there are no hgher prorty sporadc tasks then the only κ canddate that does not have any dle perods between κ and l r s L 1 ( l r ) (κ (4) n Fgure 10), so equaton (16) becomes equvalent to equaton (8) Response Tme Analyss of Sporadc Tasks To determne the worst possble response tme of a sporadc task we have to consder all asynchronous crtcal n-

10 stant canddates. Consder Γ to be the lowest prorty perodc task wth hgher prorty than τ m. The worst response nstant of a sporadc job s found when all hgher prorty sporadc tasks are released at the same asynchronous crtcal canddate. Hence, t s the smallest soluton (larger than κ )of R m = κ + e m + w (R m ) w (κ )+ω m 1 (R m κ ) (17) Lkewse, ths equaton can be solved usng the fxedpont method, wth R (0) m = e m + κ. The schedulablty analyss can be summarzed n the condton { κ Λ R m κ H m, τ m s schedulable κ Λ R m κ > H m, τ m s unschedulable (18) As an example take the nterference gven by Γ 1,...,Γ 3 (Table 1). The canddates for the asynchronous crtcal nstant are (frst 10): 37; 45; 57; 60; 67; 75; 77; 87; 89; 97. In total, there are 55 canddates n Λ 3. The response tmes of a sporadc task wth e 1 = 1 and released at the canddates nstants are respectvely: 3; 9; 3; 2; 8; 2; 3; 9; 7; 3. For a e 1 = 10 the response tmes are: 20; 21; 23; 21; 20; 21; 20; 23; 21; 20. Note that f the task s released at t = 57, for a e 1 = 1 the response tme s very low compared to other nstants, but when e 2 = 10 t has the hghest response tme. Therefore all crtcal nstants canddates must be consdered, even f for lower WCET they gve very good response tmes. Ths s because for a task wth a very low WCET, then the worst canddate corresponds to the sngle longest computaton perod. But for a task wth a large WCET t can be the longest chan of computng perods wth lttle dle perods n between. 7. Improvng the Response Tme Calculaton To speedup the schedulablty test of a partcular task t s possble to determne the response tme assumng a crtcal nstant, R. If the task s schedulable under ths condton then t s always schedulable, whatever the release tme [1] Explotng Last Idle Instant Knowledge As descrbed earler, the complexty of determnng the last dle nstant for all jobs of a gven task Γ s O(E LCM(T 1,...,T ) 2 ). The quadratc dependency comes from the rentalzaton of the teratve method to determne L (t) for each job. But by keepng track of the last known dle nstant, the algorthm can start from ths pont nstead of dong the same analyss snce the begnnng. Ths reduces the overall complexty to O(E LCM(T 1,...,T )). Snce the functon L (last known dle,t){ f( = 0)return t; // there are no hgher prorty tasks last dle nstant = last known dle; total dle tme = last known dle w (last known dle); terator = last known dle; whle(true) { last terator = terator; terator = w (terator)+total dle tme; f(terator > t) return last dle nstant; f(last terator = terator) { // arrved at an dle nstant f(terator t ρ (terator)) return t; last dle nstant = ρ (terator); total dle tme += ρ (terator) terator; terator = ρ (terator)+w (ρ (terator)) w (ρ (terator)); }}} Fgure 11. Improved pseudo-code to determne the L (t) startng from the last known dle nstant LCM s obvously the lmtng factor, ths mprovement s very sgnfcant. The Fgure 11 llustrates the mproved pseudo-code that determnes the last dle nstant startng from the last known dle nstant. Note the changes made only n the ntalzaton phase. Fgure 12 descrbes the mproved pseudo-code that determnes κ usng the new functon L (last,t). Fnally, Fgure 13 llustrates the mproved pseudocode to determne the response nstants for the jobs l ntal,...,l fnal. Note that the last dle nstant s updated after the maxmum known crtcal nstant canddate snce, as prevously stated, the crtcal nstant canddates are themselves dle nstants. If the response tme s larger than the task perod, then ths analyss cannot determne the response nstants of the followng jobs (snce the next job wll have some addtonal nterference) and the method halts. As an example of the mprovement made, the determnaton of the response tmes for the Γ 7 jobs took approxmately one hour wth the standard method 4. Wth the mproved method t took 5 seconds - Table 1. A smlar approach was also taken n [1] Handlng Large Hyperperods The schedulablty analyss s rgdly dependent on the hyperperod defned by the perodc tasks. For a hgh number of tasks, ths analyss can be deemed unfeasble, specally when the perods are co-prme. Ths paper ntroduces an nnovatve dea to handle large hyperperods by allowng some degree of pessmsm. Instead of consderng that all tasks are released at a crtcal 4 Usng MatLab under Wndows XP on a PIII processor at 1300 MHz.

11 functon κ (t mn,t max,last dle nstant){ next work nstant = L (last dle nstant,t mn ); work nstants = {}; whle(true) { next dle nstant = ϕ (next work nstant); next work nstant = ρ (next dle nstant); f( next work nstant > t max ) return work nstants; work nstants = work nstants {next work nstant}; }} Fgure 12. Improved pseudo-code to determne κ n the nterval ]t mn,t max ] usng the functon L (last,t) functon R (l ntal,l fnal ) { dle = 0; // last known dle nstant for ( l = l ntal ; l l fnal ; l++ ) { l R = l r + c ; // smallest possble value of l R κ = κ ( l 1 r, l r,dle); dle = max(κ ); // update last known dle nstant foreach(κ (x) κ ){ R = l r + c ; // ntal teraton do { last R = R; R = κ (x) + c + w 1 (R) w 1 (κ (x) )+ω K (R κ (x) ); f(r > l+1 r ) return cannot determne response nstant ; } whle(last R R); f(r > l R ) // update l R wth largest response nstant l R = R; }} return l mnr,..., l maxr ; } Fgure 13. Improved pseudo-code to determne l ntalr,..., l fnal R usng the functon κ (t mn,t max,last) nstant, t s possble to allow that only a small porton s released at pessmstc nstants. Hence t s possble to compromse between the pessmsm nduced and the number of teratons requred. By knowng the asynchronous crtcal nstants mposed by the perodc tasks, the analyss can encompass a reduced set of hgh prorty perodc tasks and determne the worst nstants n whch lower prorty tasks (perodc and sporadc) can be released. For example, suppose a task set of 40 perodc and 40 sporadc tasks. If the hyperperod defned by the 40 perodc tasks s too large, the system desgner determnes the asynchronous crtcal nstants of only the frst 20 perodc tasks. A suffcent test can thus be found by assumng that all lower prorty tasks, perodc and sporadc, are released at these nstants. Note that even though the number of canddates can be very large, t s fxed, makng the analyss of the lower prorty tasks take relatvely the same amount of tme. As a concrete example take the schedulablty analyss of Γ 8 n Example 1 (Table 1). The determnaton of all canddates of the asynchronous crtcal nstant of the tasks Γ 1,...,Γ 7 n Λ 7 took 4.28s and the determnaton of the response tmes of Γ 8, assumng t was released at these canddates, took 9.2s (not shown n Table 1). The worst response tme s found when the task s released at t = 925;49435;97945 and equals to 110. Snce the deadlne s 120, ths analyss s suffcent to say that Γ 8 s schedulable. If the worst response tme s larger than the deadlne, then nothng can be concluded. 8. Conclusons Ths paper presents an algorthm to analyze the schedulablty of asynchronous perodc and sporadc tasks scheduled by a fxed-prorty preemptve algorthm. The results presented n ths paper can be used to Dmnsh the pessmsm nduced by the crtcal nstant Allow offset relatonshps Determne the worst response tme of each job analyze response jtter account for mssed deadlnes Calculate the asynchronous crtcal nstant ntegrate sporadc tasks ncrease analyss speed By assumng that the release tmes are not equal to zero and hence not consderng the crtcal nstant scenaro, more than just one response tme calculaton s requred. The determnaton of the response tmes of all jobs under a hyperperod s necessary. Ths makes the overall computaton tme strongly dependent on the LCM of the tasks perods. Therefore, the proposed methods do not scale well aganst the number of perodc tasks, especally task sets usng co-prme perods. Hence, t s only expected to be used as an offlne test. However, there are a number of applcatons where the schedulablty analyss presented s manageable and useful, such as the automotve, satellte and other aerospace ndustres, where the typcal task frequency s around Hz. The ncluson of the sporadc tasks also enables a new faster schedulablty method by manpulatng the degree of pessmsm ntroduced. In essence, the lower prorty tasks

12 are consdered to be sporadc tasks released at a set of asynchronous crtcal nstants (worst possble nstants). Ths method strongly reduces the hyperperod under analyss. The ncorporaton of major model extensons found for synchronous systems, such as blockng factors, context swtch overhead, non-preemptve schedulers, dynamc prorty schedulers, arbtrary deadlnes, etc, s very appealng as future work. The ntroducton of release jtter s also of great nterest. Furthermore, a study s beng performed to allow the determnaton of L (t) wthout havng to analyze the whole nterval [0,t[. Ths enables an effcent dstrbuton of the overall method through several ndependent machnes, each one analysng a small porton of the hyperperod wth a very small overhead. Acknowledgments Ths work was motvated by our research actvtes under the scope of Projects DARIO (Dstrbuted Agency for Relable Input/Output), AIR (ARINC 653 Interface n RTEMS) [16] and RTEMS CENTRE [19]. The authors are n debt to researchers and techncal offcers at ESTEC (ESA - European Space Agency, Noordwjk) for some nformal dscussons on task tmng characterstcs of typcal spacecraft on-board software applcatons. References [1] N. Audsley. Optmal prorty assgnment and feasblty of statc prorty tasks wth arbtrary start tmes. Techncal report, Department of Computer Scence, Unversty of York, [2] N. Audsley. On Prorty Assgnment n Fxed Prorty Schedulng. Informaton Processng Letters, 79(1):39 44, [3] N. Audsley, K. Tndell, and A. Burns. The end of the lne for statc cyclc schedulng? In Ffth Euromcro Workshop on Real-tme Systems, pages 36 41, Oulu, Fnland, IEEE Computer Socety Press. [4] I. Bate and A. Burns. Schedulablty analyss of fxed prorty real-tme systems wth offsets. In Proc. of 9th Euromcro Workshop on Real-Tme Systems, pages , Toledo, Span, June IEEE Computer Socety. [5] G. Bernat. Response tme analyss of asynchronous realtme systems. Real-Tme Syst., 25(2-3): , [6] R. Devllers and J. Goossens. General response tme computaton for the deadlne drven schedulng of perodc tasks. Fundamenta Informatcae, 40(2 3): , November- December [7] J. Goossens. Schedulng of Hard Real-Tme Perodc Systems wth Varous Knds of Deadlne and Offset Constrans. PhD thess, Faculté des Scences, December [8] J. Goossens and R. Devllers. Feasblty ntervals for the deadlne drven scheduler wth arbtrary deadlnes. In I. C. Socety, edtor, The 6th Internatonal Conference on Real- Tme Computng Systems and Applcatons (RTCSA 99), pages 54 61, December [9] M. Grener, J. Goossens, and N. Navet. Near-optmal fxed prorty preemptve schedulng of offset free systems. In Proceedngs of the 14th Internatonal Conference on Network and Systems (RTNS 2006), Poters, France, May [10] K. Jeffay, D. Stanat, and C. Martel. On non-preemptve schedulng of perodc and sporadc tasks. In I. C. S. Press, edtor, Proceeedngs of the 12th IEEE Real-Tme Systems Symposum, pages , December [11] M. Joseph and P. Pandaya. Fndng response tmes n a realtme system. The Computer Journal, 29: , [12] J. Lehoczky, L. Sha, and Y. Dng. The rate monotonc schedulng algorthm: Exact characterzaton and average case behavor. In Proceedngs IEEE Real-Tme Systems Symposum, pages , Santa Monca, USA, [13] J. Leung and J. Whtehead. On the complexty of fxedprorty schedulng of perodc real-tme tasks. Performance Evaluaton, 2: , [14] J. Mäk-Turja and M. Noln. Fast and tght response-tmes for tasks wth offsets. In 17th EUROMICRO Conference on Real-Tme Systems, page 10, Palma de Mallorca Span, July IEEE. [15] R. Pellzzon and G. Lpar. Feasblty analyss of real-tme perodc tasks wth offsets. Real-Tme Systems, 30(1-2): , [16] J. Rufno, S. Flpe, M. Coutnho, S. Santos, and J. Wndsor. ARINC 653 nterface n RTEMS. In Proceedngs of the DASIA 2007 DAta Systems In Aerospace Conference, Naples, Italy, June EUROSPACE. [17] K. Shn and X. Chu. Computng tme delay and ts effects on real-tme control systems. IEEE Transactons on Control Systems Technology, 3(2): , June [18] H. Slva, A. Constantno, D. Fretas, M. Coutnho, S. Faustno, and M. Zulanello. RTEMS CENTRE - support and mantenance CENTRE to RTEMS operatng system. In Proceedngs of the DASIA 2008 DAta Systems In Aerospace Conference, Palma de Majorca, Span, May EUROSPACE. [19] H. Slva, A. Constantno, M. Mota, D. Fretas, and M. Zulanello. RTEMS CENTRE - support and mantenance to RTEMS operatng system. In Proceedngs of the DASIA 2007 DAta Systems In Aerospace Conference, Naples, Italy, June EUROSPACE. [20] K. Tndell. Addng tme-offsets to schedulablty analyss. Techncal report, Unversty of York, 1994.