Improving the Sensitivity of Deadlines with a Specific Asynchronous Scenario for Harmonic Periodic Tasks scheduled by FP

Transcription

1 Improvng the Senstvty of Deadlnes wth a Specfc Asynchronous Scenaro for Harmonc Perodc Tasks scheduled by FP P. Meumeu Yoms, Y. Sorel, D. de Rauglaudre AOSTE Project-team INRIA Pars-Rocquencourt Le Chesnay, France {patrck.meumeu, yves.sorel, danel.de rauglaudre}@nra.fr L. George LACSC ECE Pars, France lgeorge@ece.fr Abstract The am of ths paper s to address the problem of correctly dmensonng real-tme embedded systems. It s well known that computers whch control systems are greatly affected by delays and jtter occurrng n the control loop. In the lterature, a deadlne reducton approach has been consdered as one soluton to reducng the jtter experenced by a task and therefore obtanng better loop stablty n the control loop. Here, n order to mprove the senstvty of the deadlnes, we propose another soluton for reducng the deadlne of a task when all tasks are scheduled wth the Deadlne Monotonc Algorthm. Ths s performed for a specfc asynchronous scenaro for harmonc perodc tasks. We compare the results to those for the synchronous scenaro. 1. Introducton In ths paper, we consder the problem of correctly dmensonng real-tme embedded systems. The correct dmensonng of a real-tme system strongly depends on the determnaton of the tasks Worst-Case Executon Tmes (WCETs). Based on the WCETs, Feasblty Condtons (FCs) ([1], [2], [3]) can be establshed to ensure that the tmelness constrants of all the tasks are always met when tasks are scheduled by a fxed or a dynamc prorty drven schedulng algorthm. We consder an applcaton composed of a perodc task set Γ n = {τ 1,..., τ n } of n perodc tasks, scheduled wth Fxed Prorty (FP) preemptve schedulng. The classcal defnton of a perodc task τ, s: C : the Worst Case Executon Tme (WCET) of τ. T : the perod of τ. D : the relatve deadlne of τ (a task requested at tme t must be termnated by ts absolute deadlne t + D ), where D T. A recent research area called senstvty analyss ams at provdng nterestng nformaton on the valdty of feasblty condtons consderng possble devatons of task WCETs ([4]), task perods ([4]), or task deadlnes ([5]). Ths makes t possble, for example, to fnd a feasble task set, f the current one s not feasble, by modfyng the task parameters or determnng the mpact of an archtecture change on the feasblty of a task set. We are nterested n the senstvty of deadlnes when tasks are scheduled wth fxed prorty Deadlne Monotonc schedulng. Senstvty analyss for deadlnes has been consdered for Earlest Deadlne Frst (EDF) schedulng by ([5]) showng how to compute the mnmum feasble deadlnes such that deadlne of any task τ s equal αd, where α s reducton factor 0 < α 1. In ([6]), the space of feasble deadlnes (D-space), a space of n dmensons has been consdered. Any task set havng deadlnes n the D-space s consdered to be schedulable. To the knowledge of the authors, no work has been done on the senstvty of deadlnes for fxed prorty schedulng. Computer controllng systems are very much affected by delays and jtter occurrng n the control loop. A deadlne reducton has been consdered by ([6]) as one soluton to reducng the jtter experenced by a task and therefore obtanng better loop stablty n the control loop. Ths paper proposes a soluton reducng as much as possble the deadlne of a task when tasks are scheduled wth Deadlne Monotonc. Wth Deadlne Monotonc, tasks are scheduled accordng to ther relatve deadlnes. The smaller the relatve deadlne, the hgher the prorty. Startng from a schedulable task set, we want to characterze the mnmum deadlne reducton factor 0 < α 1 such that any task τ, = 1...n havng a deadlne D = α T s schedulable. α s such that any smaller reducton factor would lead to a non schedulable task set. We compare the value of α obtaned n the worst case synchronous scenaro (all the tasks are frst released at the same tme) to that obtaned wth a partcular asynchronous scenaro that we propose, and whch has some nterestng propertes. We show that the mnmum reducton factor obtaned n our asynchronous scenaro s always less than or equal to the mnmum reducton factor obtaned n the synchronous scenaro.

2 For Fxed Prorty (FP) schedulng, necessary and suffcent FCs have been proposed n the case of non-concrete tasks where the frst release tmes of the tasks can be arbtrary. A classcal approach s based on the computaton of the tasks worst-case response tmes ([7], [2]). The worst-case response tme, defned as the worst case tme between the request tme of a task and ts latest completon tme, s obtaned n the worst case synchronous scenaro correspondng to the case where all the tasks are released at the same tme, and s computed by successve teratons. Ths worst case response tme provdes a bound on the response tme vald for any other task frst release tmes. A task set s declared feasble f for any task n the synchronous scenaro, ts worst case response tme s less than or equal to ts deadlne. It can be shown that consderng only non-concrete tasks can lead to a pessmstc dmensonng [8]. The complexty of ths approach depends on the worst case response tme computaton complexty. In the case of deadlnes less than or equal and to perods for all tasks, the worst-case response tme R of a task τ s obtaned n the synchronous scenaro for the frst release of τ at tme 0 and s the soluton of the equaton ([7]) R = W (R ), where W (t) = C + τ j hp() T j C and hp() denotes the set of tasks wth a prorty hgher than or equal to that of τ except τ tself. The value of R s computed by successve teratons and the number of teratons s bounded by 1+ τ j hp() D T j. The FC has been revsted by ([9]), whch the authors showed that a necessary and suffcent feasblty condton for a task set s: t S, such that W (t)/t 1, where S = τj hp(){kt j, k N} [0, D ]. For any task τ, the checkng nstants correspond to the arrval tmes of the tasks wth a hgher prorty wthn the nterval [0, D ]. Ths feasblty has been mproved by ([9]), where the authors showed how to reduce the tme nstants of S. For any task τ, they show how to sgnfcantly reduce the number of checkng nstants durng nterval [0, D ] to at most 2 1 tmes rather than 1+ D τ j hp() T j. When deadlnes and perods are ndependent, ([2]) shows that the worstcase response tmes of a sporadc task τ s not necessarly obtaned for the frst actvaton request of τ at tme 0. The t L T, where L number of actvatons to consder s 1 + s the length of the worst-case level-τ busy perod defned n ([10]) as the longest perod of processor actvty runnng tasks of prorty hgher than or equal to τ n the synchronous scenaro. It can be shown that L = L τ j hp() τ T j C j. From ts defnton, L s bounded by: C j, C j τ j hp() τ 1 Mn T j τ j hp() τ ([3]). C j P T j τ j hp() τ where P = LCM(T 1,...,T n ) and t leads to a pseudopolynomal tme complexty for the feasblty condtons. Ths s an nterestng approach as t provdes a pseudopolynomal tme complexty but t may lead to a pessmstc dmensonng as the synchronous scenaro mght not be lkely to occur. In order to mprove the schedulablty of the systems, offset strateges on the frst release tmes of the tasks have been consdered. A system where offsets are mposed s called an asynchronous systems. ([8]) shows sgnfcant feasblty mprovements consderng offsets. Smulatons show that the number of feasble schedulable systems wth offsets (whle unfeasble n the synchronous case) ncreases wth the number of tasks for a processor load of 0.8 and ranges from 40.5% to 97% for dfferent offset assgnment strateges. Ths percentage strongly decreases when the load s hgh (tends to 1). Wth asynchronous tasks, ([11]) shows that for a gven offset assgnment, the schedulablty of the tasks must be checked n the nterval [0, max =1...n (O )+2P) where P s the least common multple of the tasks and O s the offset of task τ, leadng to an exponental tme complexty. To provde less pessmstc FCs, t s furthermore mandatory to prove that the offsets wll not result later n a synchronous scenaro. Ths problem s referred to as the K-smultaneous congruence problem n the state of the art ([11]). Ths feasblty result has been sgnfcantly mproved by ([12]) showng that the nterval to check the feasblty of a perodc task set wth offsets can be reduced to [0, max =1...n (O ) + P). Furthermore, ([11]) proves the non optmalty of Deadlne Monotonc schedulng for asynchronous systems when tasks deadlnes are less than or equal to perods. An optmal prorty assgnment can be obtaned n O(n 2 ) usng the Audsley procedure ([13]). A partcular case denoted offset free systems corresponds to the case where offsets can be chosen arbtrarly. An optmal offset assgnment s gven n ([14]). An offset assgnment s optmal f t can fnd a schedulable offset whenever a feasble assgnment exsts. The complexty of the offset assgnment algorthm s exponental and s n O((max 2 j n T j ) n 1 ). The offset of task τ 1 s set to 0. Dfferent offset strateges heurstcs have been consdered n the lterature. Among them, we can cte the dssmlar offset assgnment proposed by ([14]) that conssts n shftng (computng a dstance between the offsets) the offset of the tasks to be as far as possble from the synchronous scenaro. The algorthm sorts the couple of tasks (τ, τ j ) by decreasng values of gcd{t, T j } such that the dstance belongs to [0, gcd{t, T j }). The dssmlar offset assgnment sgnfcantly reduces the number of offsets to consder, leadng to a complexty n O(n 2.(log(max [1,n] T )+log(n 2 ))). Other offset assgnment strateges have been consdered by ([8]) usng the Audsley procedure to determne the subset of

3 tasks of τ that can be feasbly scheduled n the synchronous scenaro (settng ther offset to 0). The offsets are only computed for the subset of tasks that are unfeasble wth the Audsley procedure n the synchronous case. The authors consder dfferent crtera to assgn the offsets, based on the crtera used to sort the couple of tasks (τ, τ j ). The complexty s the same as that of the dssmlar offset assgnment. In ths paper we consder a partcular asynchronous harmonc concrete task set where 2 n, T 1 T (.e. there exsts k Z such that T = kt 1 ) wth partcular offsets. In the case of non-concrete harmonc tasks, when tasks are scheduled wth Rate Monotonc (the shorter the perod, the hgher the prorty) and n the case deadlnes are equal to perods, a necessary and suffcent condton for the feasblty of such a system s gven by C T U = =1...n 1 (see [15]). Ths potentally proves the benefts of consderng harmonc tasks gettng better feasblty condtons. Ths property does not hold when deadlnes can be shorter than perods. In ths case we show how to determne n O(n) the offset of the tasks to obtan a pseudo-polynomal tme feasblty condton nstead of an exponental one. In the case of asynchronous tasks, the worst case response tme cannot be computed wth a recursve equaton as for the synchronous tasks. Ths s due to the fact that wth offsets, there s not necessarly a contnuous busy perod from tme 0 to the release tme of a task. In ths paper we nvestgate a new approach to compute the worst case response tme of a task based on the Mesod approach. Ths approach was frst ntroduced by ([16]) n the context of real-tme schedulng wth preempton cost. Ths approach does not requre a contnuous busy perod to compute the worst case response tmes of the tasks. We propose a partcular offset assgnment, such that the worst case response tme of any task s obtaned for ts second request tme, provdng an exponental tme mprovement n the complexty of the FCs. The rest of the paper s organzed as follows. In secton 2, we ntroduce the concepts and notatons and establsh mportant propertes for the partcular asynchronous scenaro that we consder. We consder harmonc perods. We show that, usng ths partcular scenaro, the worst case response tme of every task s obtaned for ts second nstance. In secton 3, we ntroduce the concept of Mesod whch s used to compute the worst case response tme of an asynchronous task set. In secton 4, we gve an algorthm for the computaton of the worst case response tme of any task n our asynchronous scenaro, then we show how to compute the mnmum deadlne reducton factor. An example s gven n order to compare the deadlne reducton factor obtaned wth our asynchronous scenaro to that n the synchronous scenaro. We provde expermental results n secton 5 based on extensve smulatons comparng the deadlne reducton factor for several load confguratons n both the synchronous and n our asynchronous scenaro. Fnally we conclude the paper n secton Propertes of the asynchronous harmonc task set 2.1. Concepts and notatons We recall classcal results n the unprocessor context for real-tme schedulng. Tme s assumed to be dscrete (task arrvals occur and task executons begn and termnate at clock tcks; the parameters used are expressed as multples of the clock tck); n [17], t s shown that there s no loss of generalty wth respect to feasblty results by restrctng the schedules to be dscrete, once the task parameters are assumed to be ntegers (multples of the clock tck).e. a dscrete schedule exsts, f and only f a contnuous schedule exsts. A task set s sad to be vald wth a gven schedulng polcy f and only f no task occurrence ever msses ts absolute deadlne wth ths schedulng polcy. U = n C =1 T s the processor utlzaton factor assocated to the task set Γ n,.e., the fracton of processor tme spent n the executon of the task set ([18]). If U > 1, then no schedulng algorthm can meet the tasks deadlnes. The synchronous scenaro corresponds to the scenaro where all the tasks are released at the same tme (at tme 0). The model depcted n fgure 1 s Lu & Layland s poneerng model [18] for systems executed on a sngle processor. Fgure 1. Model Throughout the paper, we assume that all tmng characterstcs are non-negatve ntegers,.e. they are multples of some elementary tme nterval (for example the CPU tck, the smallest ndvsble CPU tme unt): We ntroduce several notatons for a perodc task τ = (C, D, T ) used to compute the worst case response tme of a task: τ k : The kth nstance of τ

4 r 1 : Release tme of the frst nstance of τ r k = r1 + (k 1)T : Release tme of τ k R k : Response tme of τk released at tme r k R : Worst-case response tme of τ 2.2. The specfc asynchronous scenaro Now, we gve some nterestng propertes whch are satsfed by the specfc asynchronous scenaro we propose whch lead to the concluson that the worst case response tme of a task n our asynchronous scenaro s obtaned for any task for ts second release. In ths secton we assume that the relatve deadlne for each task equals ts perod,.e. D = T. Ths assumpton wll be weakened n secton 4. We frst show n lemma 1 that wth harmonc asynchronous tasks, two nstances belongng to any two tasks can never be released at the same tme f ther release tmes are not equal modulo ther perods. Lemma 1: Let Γ n = {τ 1, τ 2,,τ n } be a system of n ndependent harmonc (.e. T T +1, {1,,n 1}) preemptve tasks ordered by decreasng prortes (T T +1, {1,,n 1}). If there exst two tasks τ, τ j Γ n, ( < j) such that rj 1 r1 mod[t ] 1, then k, l 0 such that rj k = rl. Proof: (by contradcton) Let us assume that there exst two tasks τ, τ j Γ n, ( < j) such that rj 1 r1 mod[t ], and k, l 0 such that rj k = rl. rj k = rl rj 1 + (k 1)T j = r 1 + (l 1)T rj 1 = r1 + (l 1)T (k 1)T j rj 1 = r1 mod[t ] as T T j. Contradcts the hypothess and thus, ends the proof. We now show n theorem 1 that from the pont of vew of any task n the system, the schedule repeats dentcally from the second nstance. (nspred by theorem 2.48 n [19]) Let Γ n = {τ 1, τ 2,,τ n } be a system of n asynchronous ndependent perodc preemptve tasks ordered by decreasng prortes (T T +1, {1,, n 1}). Let r1 1, r1 2,, r1 n be respectvely the release tme of ther frst nstances. Let (s ) 1 n be the sequence nductvely defned by s 1 = r 1 1 s = r 1 + (s 1 r 1 )+ T T {2,, n} (1) Then, f Γ n s schedulable up to s n + H n, wth H n = LCM(T 1, T 2,, T n ) and x + = max{x, 0}, then Γ n s schedulable and perodc from s n wth perod H n. 1. Gven a, b, c Z : a = b mod[c] means that there exsts d Z such that a = b + cd. Proof: (By nducton on the number of tasks n) The property s straghtforward for the smple case where n = 1: ndeed, the schedule for task τ 1 s perodc of perod T 1 from ts frst release (s 1 = r1) 1 snce C 1 T 1, otherwse the deadlne of the frst nstance s mssed. Let us now assume that the property s true up to n = 1 and Γ = {τ 1, τ 2,,τ } s schedulable up to s + H, wth H = LCM(T 1, T 2,, T ). Notce that s s the frst release tme of task τ after (or at) s 1. We have s + H s 1 + H 1 and by nducton hypothess, the subset Γ 1 = {τ 1, τ 2,,τ 1 } s schedulable and perodc from s 1 of perod H 1. As tasks are ordered by prorty, the nstances of the frst ones are not changed by the requests of task τ and the schedule repeats at tme s + lcm(h 1, T ) = s + H. Consequently, Γ = {τ 1, τ 2,, τ } s schedulable and ts schedule repeats from s wth perod H. We now characterze the asynchronous scenaro we consder n ths paper n corollary 1. Ths leads to provdng a smple method for computng the worst response tme of each task n secton 3 by usng corollary 2, and then a pseudo polynomal FC detaled n secton 4.1. Corollary 1: From the pont of vew of any task τ of a schedulable system Γ n = {τ 1, τ 2,, τ n } ordered by decreasng prortes (T T +1, {1,,n 1}) such that T T +1 and r+1 1 = r1 C +1, the schedule s perodc from the second nstance wth perod H = T. Proof: (By nducton on the ndex of the task) Let us consder a task τ of a schedulable system Γ n = {τ 1, τ 2,, τ n }, we assume that T T +1 and r+1 1 = r 1 C +1, 1. Thanks to the prevous theorem, t s suffcent to prove that s r 1 = T, 2. Ths s done by nducton on. The property s straghtforward for the smple case where = 2: ndeed, as C 2 T 2 and H 2 = LCM(T 1, T 2 ) = T 2, the schedule for task τ 2 s perodc of perod T 2 from ts second release snce s 2 = r2 1 (s1 r )+ T 2 = r2 1 + T 2 C2 T 2 = r2 1 +T 2 s the frst release tme of task τ 2 after T 2 (or at) s 1 = r1 1. Let us now assume that the property s true up to ndex 1 and Γ = {τ 1, τ 2,,τ } s schedulable. Thanks to the prevous theorem, we have s = r 1 + (s 1 r 1 T )+ = r 1 T + (T 1 + r 1 1 r1 )+ T T by nducton hypothess. Thus, s = r 1 + (T 1 + C ) + T snce r 1 1 = r1 +C. T Now, as 0 < T 1 + C < T due to the scenaro mposed to the frst nstance of each task and the fact that T 1 T, then we obtan s = r 1 + T. Corollary 2: The worst response tme R of each task τ s obtaned n the second nstance and s equal to that n all

5 nstances greater than 2. Proof: Immedately follows from corollary 1 and the fact that R 1 = C by constructon, R k C k 1, and we consder harmonc tasks. We now suppose n the followng the asynchronous task set defned n corollary 1. We now present the Mesod approach used to compute the worst case response tme of perodc tasks n an asynchronous task set. 3. Worst case response tme: the Mesod approach In ths secton, we provde the method for computng the worst response tme of each task n order to check ts schedulablty. Actually, three classcal methods may be used to do so: the utlsaton factor of the processor ([20]), the worst response tme of each task, or the processor demand ([21]). In ths paper, we have chosen to use the second approach as t provdes a schedulablty condton for each task ndvdually. The man dea behnd the Mesod approach s fllng some avalable tme unts left by the schedule of hgher prorty tasks wth executed tme unts correspondng to the executon tme of the current task. Snce the worst response tme s obtaned n the second nstance w.r.t. corollary 2, we wll acheve ths goal by applyng the method descrbed n [16] to a system where all tasks are not released smultaneously and where the cost of a preempton s assumed to be zero. Ths method, contrary to those proposed n ([22], [3], [23]), s of lesser complexty snce t s not necessary to determne the releases of every task w.r.t. those of hgher prorty tasks. As we are n a fxed prorty context the proposed method checks for the schedulablty of each task by computng ts worst response tme, from the task wth the hghest prorty to that wth the lowest prorty. Hence, from the pont of vew of any task τ of a system Γ n = {τ 1, τ 2,,τ n } ordered by decreasng prortes (T 1 T, {2,,n}) such that T 1 T and r 1 = r1 1 C, the elapsed duraton between the release of the second nstance and the frst release r 1 1 of task τ 1 s gven by T C. Before provdng the computaton method of the worst case response tme, we provde some necessary defntons below Defntons All the defntons and termnologes used n ths secton are drectly nspred by ([16]) and are appled here to the case of a model where the cost of preempton s assumed to be zero. From the pont of vew of any task τ, the hyperperod at level, H, s gven by H = LCM{T j } τj sp(τ ) = T as T 1 T for every {2,,n}, and sp(τ ) s the set of tasks wth a perod shorter than that of task τ. Wthout any loss of generalty we assume that the frst task τ 1 starts ts executon at tme t = 0 and all tasks have dfferent perods. Snce at each level the schedule repeats ndefntely from the second nstance thanks to corollary 1, t s suffcent to perform the schedulng analyss n the nterval [r 1 +T, r 1 +2T ] for task τ as ts response tme n ts frst nstance equals ts WCET. We proceed the schedule from the task wth the shortest perod towards the task wth the longest perod. Thus, at each level n the schedulng process the goal s to fll avalable tme unts n the prevous schedule, obtaned up to now, wth slces of the WCET of the current task, hence we obtan the next current schedule. Consequently, we represent the prevous schedule of every nstance τ k of the current task τ = (C, T ) by an ordered set of T tme unts where some have already been executed because of the executon of tasks wth shorter perods, and the others are stll avalable for the executon of task τ n that nstance. We call ths ordered set whch descrbes the state of each nstance τ k the M k T -mesod. More detals on the defnton of a T -mesod are gven n [16]. For the current task τ = (C, T ), there are as many T -mesods as nstances. We call the T -mesod correspondng to the second nstance of task τ before beng scheduled n the current schedule. The process used to buld for task τ wll be detaled later n ths subsecton. Stll, from the pont of vew of task τ, we defne for the mesod the correspondng unverse X 2 to be the ordered set, compatble wth that of the mesod, whch conssts of all the avalabltes of. That s to say, all the possble values that C can take n. Task τ wll be sad to be potentally schedulable f and only f C X 2 {1,,n} (2) Ths equaton verfes that C belongs to the unverse at level. If t does not, then the system s clearly not schedulable. When equaton (2) holds for a gven task τ, we call the T -mesods correspondng to the second nstance of task τ after τ has been scheduled. s a functon of whch tself s a functon of 1, both detaled as follows. Let f be the functon such that = f( 1 ) whch transforms the T 1 -mesod after task τ 1 has been scheduled at level 1 nto the T -mesod before task τ s scheduled at level. As mentoned n [16], a mesod conssts only of tme unts already executed denoted by e and tme unts stll avalable denoted by a. Moreover, the cardnal of a mesod s equal to the perod of the task under consderaton whatever the level s. As such, the functon f transforms a tme unt already executed (resp. stll avalable) n 1 nto a tme unt already executed (resp. stll avalable) n by followng an ndex ψ whch enumerates, accordng to naturals, the tme unts (already executed or stll avalable) n 1 of task τ 1 after τ 1 has been scheduled. As the elapsed duraton between the release of the second nstance of task τ and the release of the frst nstance of τ 1 s T C, then ψ

6 starts from the tme unt rght after γ = T C mod[t 1 ] tme unts n the mesod 1 towards the last tme unt, and then crcles around to the begnnng of the mesod 1 agan, untl we get the T -mesod. Ths T - mesod s obtaned when ψ = T. Indeed, the prevous schedule at level (the schedule obtaned at level 1) conssts of H 1 = T 1 tme unts whereas the schedule of the current task τ s computed upon H = T tme unts. Thus, that amounts to extendng the prevous schedule from T 1 to T tme unts by dentcally repeatng the prevous schedule as often as necessary to obtan H tme unts. Due to the partcular releases of the frst nstance of each task,.e. r+1 1 = r1 C +1 {1,, n 1}, notce that ndex ψ n contrast to ndex ζ used n [16] whch started from the frst tme unt, starts from the tme unt rght after γ = T C mod [T 1 ] tme unts n the mesod 1. Snce τ 1 s the task wth the shortest perod, then sp(τ 1 ) = {τ 1 }. Because τ 1 s never preempted, we have 1 = {1, 2,, T 1 } and therefore we obtan 1 = {(C 1 ), 1, 2,, T 1 C 1 }. Let g be the functon such that ) whch transforms the T -mesod scheduled at level nto the T -mesod has been scheduled at level. = g( before task τ has been after task τ 3.2. Worst case response tme wth a Mesod For the T -mesod, we wll compute the response tme R 2 of task τ n the second nstance by addng to the WCET C all the consumptons appearng n that T -mesod before the avalablty correspondng to C [16]. Ths yelds the worst-case response tme R of task τ snce at each level the schedule becomes perodc from the second nstance, that s to say R k = R2 k 2, and R 1 = C 1. Now we can buld = g( ): functon g transforms a tme unt already executed n nto a tme unt already executed n, and transforms a tme unt stll avalable nto ether a tme unt stll avalable or a tme unt already executed w.r.t. the followng condton. We use an ndex whch enumerates accordng to numerals the tme unts n from the frst to the last one, at each step n the ncremental process, f the current value of the ndex s less than or equal to R 2, functon g transforms the tme unt stll avalable nto a tme unt already executed due to the executon of nstance τ 2, otherwse g transforms t nto a tme unt stll avalable. Indeed, functon g flls avalable tme unts n the current schedule wth slces of the WCET n each T -mesod, leadng to the prevous schedule for the next task at level + 1 w.r.t. prortes. To summarze, for every task τ, we have τ : : T -mesod before τ s scheduled at level : T -mesod after τ s scheduled at level. 4. Deadlne reducton factor 4.1. Worst case response tme computaton The approach proposed here leads to a new schedulablty condton for harmonc hard real-tme systems. Ths condton s new n the sense that besdes provdng a necessary and suffcent schedulablty condton, t also reduces the feasblty nterval for a gven harmonc asynchronous system. In the schedulng process, at each level, the basc dea conssts n fllng avalabltes n the mesod before task τ s scheduled, wth slces of ts WCET. Ths s why t s fundamental to calculate the correspondng response tme. Ths yelds the worst case response tme and allows us to conclude on the schedulablty of task τ w.r.t. prortes. In the case where τ s schedulable, we buld, after τ has been scheduled, n order to check the schedulablty of the next task, and so on, otherwse the system s not schedulable. Thanks to everythng we have presented up to now, τ 1 s scheduled frst and r1 1 = 0. The latter statement mples that before τ 1 s scheduled, ts WCET can potentally take any value from 1 up to the value of ts perod T 1. Snce task τ 1 s never preempted, then 1 = {1, 2,,T 1 } and X1 2 = {1, 2,, T 1 }. In addton, ts response tme s also equal to C 1. Consequently, the correspondng T 1 -mesods assocated to task τ 1 are gven by τ 1 : 1 = {1, 2,, T 1 } 1 = {(C 1 ), 1, 2,,T 1 C 1 } We assume that the frst 1 tasks wth 2 n have already been scheduled,.e. the T 1 -mesod 1 of task τ 1 s known, and that we are about to schedule task τ. As explaned n the prevous secton, the T -mesod = f( 1 ) of task τ s bult thanks to ndex ψ on 1 of task τ 1 wthout forgettng to start from the tme unt rght after γ = T C mod [T 1 ] tme unts rather than the frst tme unt as n [16]. Agan ths s due to the partcular release of the frst nstances of tasks: r 1 = r1 1 C. We can therefore determne the unverse X 2 when the T 1 -mesod 1 s known. Unless the system s not schedulable,.e. C X 2, we assume that task τ s potentally schedulable,.e. C X 2. The response tme R 2 of task τ n ts k th nstance (wth k 2),.e. n the k th T -mesod wll be obtaned by summng C wth all consumptons pror to C n the correspondng mesod. The worst-case response tme R of task τ wll then be gven by R = R 2 Ths equaton leads us to say that task τ s schedulable f and only f R T (3)

7 If for task τ expresson (3) holds, then = g( ) wll be deduced as explaned n the prevous secton. For the sake of clarty, whenever there are two consecutve consumptons n a mesod, ths amounts to consderng only one consumpton whch s the sum of the prevous consumptons. That s to say that after determnng the response tme of task τ n ts k th mesod, f M a,k = {(c 1 ), (c 2 ), 1, 2, }, then ths s equvalent to M a,k = {(c 1 + c 2 ), 1, 2, }. Below, we present our schedulng algorthm whch, for a gven task, on the one hand frst determnes the value of γ = T C mod [T 1 ] relatve to prortes, then, on the other hand the schedulablty condton. Recall that the elapsed duraton between the release of the second nstance and the frst release s T C. The schedulng algorthm has the followng nne steps. Snce the task wth the shortest perod, namely task τ 1, s never preempted, the loop starts from the ndex of the task wth the second shortest perod, namely task τ 2 as the schedule proceeds towards tasks wth longer perods. 1: for = 2 to n do 2: Determne the release tme of the frst nstance of task τ : r 1 = r 1 1 C and compute γ = T C mod [T 1 ] of the second nstance of τ w.r.t. τ 1. 3: Buld the T -mesod = f( 1 ) of task τ before t s scheduled. Ths constructon s based on a modulo T arthmetc usng ndex ψ on 1 wthout forgettng to start from the tme unt rght after γ = T C mod[t 1 ] tme unts rather than the frst tme unt as n [16]. Ths s due to the partcular release of tasks. 4: For the T -mesod resultng from the prevous step, buld the correspondng unverse X 2 whch conssts of the ordered set of all avalabltes of. Notce that ths set corresponds to the set of all possble values that the WCET C of task τ can take n. 5: Snce τ s potentally schedulable,.e. ts WCET C X 2, we must verfy that t s actually schedulable. Clearly, f C X 2, then task τ s not schedulable because the deadlne of the task s exceeded. 6: Determne the response tme R k of task τ n ts k th nstance,.e. n the k th T -mesod. Ths s obtaned by summng C wth all the consumptons pror to C n the correspondng mesod. Deduce the worst-case response tme R of task τ. R = R 2 It s worth notcng that task τ s schedulable f and only f R D. 7: If R D, then buld = g( ), ncrement, and go back to step 2 as long as there reman potentally schedulable tasks n the system. 8: If R > D, then the system {τ = (C, T )} 1 n s not schedulable. 9: end for Thanks to the above algorthm, a system of n tasks {τ = (C, T )} 1 n, wth harmonc perods and frst released such that r 1 = r 1 1 C, s schedulable f and only f R = R 2 2 D {1, 2,, n} (4) 4.2. Computaton of α ( ) R The value of α s gven by: α = max 1 n We recall that for the synchronous scenaro, the worst case response tme of task τ s gven by: R = C + R C j Example j hp() Let us consder {τ 1, τ 2, τ 3, τ 4 } to be a system of four tasks wth harmonc perods and frst released such that r 1 = r 1 1 C. The characterstcs are defned n table 1. T j Table 1. Characterstcs of the tasks C T τ τ τ τ The shorter the perod of a task s, the hgher ts level s. Thus, as depcted n table 1, τ 1 has the hghest level and task τ 4 the lowest level. Thanks to our schedulng algorthm, for task τ 1 whose frst release tme s r1 1 = 0, we have 1 = {1, 2, 3, 4, 5} τ 1 : R 1 = 2 1 = {(2), 1, 2, 3} γ 2 = T 2 C 2 mod [T 1 ] = 15 4 mod [5] = 1, thus for task τ 2 whose frst release tme s r2 1 = r1 1 C 2 = 4, we have 2 = {(1), 1, 2, 3, (2), 4, 5, 6, (2), 7, 8, 9, (1)} τ 2 : R 2 = = 7 2 = {(7), 1, 2, (2), 3, 4, 5, (1)} γ 3 = T 3 C 3 mod[t 2 ] = 30 5mod[15] = 10, thus for task τ 3 whose frst release tme s r3 1 = r1 2 C 3 = 4 5 = 9, we have 3 = {(1), 1, 2, 3, (8), 4, 5, (2), 6, 7, 8, (8), 9, 10, (1)} τ 3 : R 3 = = 14 3 = {(16), 1, 2, 3, (8), 4, 5, (1)} T

8 γ 4 = T 4 C 4 mod[t 3 ] = 60 7mod[30] = 23, thus for task τ 4 whose frst release tme s r4 1 = r1 3 C 4 = 9 7 = 16, we have τ 4 : 4 = {(4), 1, 2, (17), 3, 4, 5, (8), 6, 7, (17), 8, 9, 10, (4)} R 4 = = Expermental results 4 = {(53), 1, 2, 3, (4)} Consequently, the set of tasks {τ 1, τ 2, τ 3, τ 4 } wth harmonc perods and frst released such that r 1 = r1 1 C s schedulable. The schedule wth the above characterstcs s depcted n fgure 2, Fgure 2. Executon of a set of harmonc tasks wth r 1 = r 1 1 C, {2,,4} whereas the schedule of the same set of tasks released smultaneously s depcted n fgure 4. α asynchrnous = max(2/5, 7/15, 14/30, 36/60) = 0.60, whch means the mprovement performed n ths case s of 34.54% In ths secton we present some expermental results comparng the mnmum deadlne reducton factor α obtaned n the synchronous scenaro and n our partcular asynchronous scenaro. We consder harmonc tasks scheduled wth the Deadlne Monotonc algorthm. We perform experments for each graph, and every task set conssts of n = 10 harmonc tasks. The total utlzaton factor of the processor s randomly chosen between 0.7 and 1 for each task set. Hence, we can evaluate the gan of our specfc asynchronous scenaro defned n corollary 1 n secton 2, compared to the synchronous one. We set α = D T, and we gradually and unformly decrease the value of the relatve deadlnes D by the same factor for all tasks n each set. In both the synchronous and the asynchronous scenaro, we plot the curves correspondng to the smallest value of α, as a functon of the total utlzaton factor of the processor, for the task set to reman schedulable. If the value of α s denoted α synchronous n the synchronous scenaro and α asynchronous n our asynchronous scenaro, the gan can be computed as follows:. gan = αsynchronous α asynchronous α synchronous 100 Fgure 3. Executon of a set of harmonc tasks wth r 1 = 0 {1,,4} It s worth notcng here the large varaton between the two scenaros n terms of the tasks response tmes. As a matter of fact, the worst case response tme of task τ 4 n fgure 2 s 36 whereas t s 55 n fgure 4. Ths phenomenon s even more apparent n the next secton wth the expermental results where we gradually and unformly decrease the value of the relatve deadlnes for all tasks by the same factor to hghlght the advantage of our approach. Tasks τ τ τ τ R synchronous R asynchronous Ths leads us to obtan α synchrnous = max(2/5, 8/15, 15/30, 55/60) = 0.91 whereas Fgure 4. Value of α wth our asynchronous scenaro and wth the synchronous scenaro In fgure 4, the curve n plan style represents the result obtaned for α n our asynchronous case whereas the curve n dot style represents the result obtaned n the synchronous case. In both cases, we start wth a schedulable task set τ, D = T. From [15], U 1 s a necessary and suffcent feasblty condton for the schedulablty of an harmonc task set as tasks are scheduled wth DM equvalent to RM when τ, D = T. We can see that for a small load, we obtan almost the same α both n the synchronous and n the asynchronous

9 cases. Ths s due to the fact that at small load the worst case response tmes of the tasks are less nfluenced by the release tmes of other tasks. As the load ncreases, the gan ncreases and reaches a maxmum of 14, 3% for U = Over the load U = 0.95, the gan steadly decreases when U tends to 1 and α tends to 1. At hgh loads, the worst case response tme of a task tends to ts perod and thus α tends to 1. In ths latter case, the mprovement obtaned wth our asynchronous scenaro becomes less sgnfcant. 6. Concluson In ths paper, we have studed the senstvty of the deadlnes for perodc tasks when the tasks are scheduled wth the Deadlne Monotonc schedulng. We have consdered a specfc asynchronous task set and harmonc tasks. The asynchronous scenaro we consder makes t possble to sgnfcantly reduce the complexty of the worst case response tme computaton. We have then consdered the Mesod approach to compute the worst case response tme of a task n an asynchronous scenaro. We have used the Mesod approach to compute the mnmum deadlne reducton factor. We have proved by extensve smulatons that the gan n terms of deadlne reducton can reach 14, 3% wth our partcular asynchronous scenaro compared to the synchronous scenaro. Ths makes t possble to better control the jtter of the tasks when consderng control loops. References [1] S. Baruah, R. Howell, and L. Roser. Algorthms and complexty concernng the preemptve schedulng of perodc real-tme tasks on one processor. Real-Tme Systems, Vol. 2, pp , [2] K. Tndell, A. Burns, and A. J. Wellngs. Analyss of hard real-tme communcatons. Real-Tme Systems, Vol. 9, pp , [3] L. George, N. Rverre, and M. Spur. Preemptve and non-preemptve schedulng real-tme unprocessor schedulng. INRIA Research Report, No. 2966, September [4] Gorgo Buttazzo Enrco Bn, Marco D Natale. Senstvty Analyss for Fxed-Prorty Real-Tme Systems. Proceedngs of the 18th Euromcro Conference on Real-Tme Systems (ECRTS 06), Dresden, Germany July 5-7, [5] Ismael Rpoll Patrca Balbastre and Alfons Crespo. Optmal deadlne assgnment for perodc real-tme tasks n dynamc prorty systems. Proceedngs of the 18th Euromcro Conference on Real-Tme Systems (ECRTS 06), Dresden, Germany July 5-7, [6] E. Bn and G. Buttazzo. The Space of EDF Feasble Deadlnes. Proceedngs of the 19th Euromcro Conference on Real-Tme Systems (ECRTS 07), Psa, Italy July [7] M. Joseph and P. Pandya. Fndng response tmes n a realtme system. BCS Comp. Jour., 29(5), pp ,, [8] M. Grener, J. Goossens, and N. Navet. Near-optmal fxed prorty preemptve schedulng of offset free systems. Proc. of the 14th Internatonal Conference on Network and Systems (RTNS 2006), Poters, France, May 30-31, [9] Gorgo Buttazzo Enrco Bn. Schedulablty Analyss of Perodc Fxed Prorty Systems. IEEE Transactons On Computers, Vol. 53, No. 11, Nov [10] J.P. Lehoczky. Fxed prorty schedulng of perodc task sets wth arbtrary deadlnes. Proceedngs 11th IEEE Real-Tme Systems Symposum, pp , Dec. Lake Buena Vsta, FL, USA, [11] J. Y. T. Leung and M.L. Merrl. A note on premptve schedulng of perodc, Real Tme Tasks. Informaton Processng Letters, Vol 11, num 3, Nov [12] Anne Choquet-Genet and Emmanuel Grolleau. Mnmal schedulablty nterval for real-tme systems of perodc tasks wth offsets. Theor. Comput. Sc., 310(1-3): , [13] N. C. Audsley. Optmal prorty assgnment and feasblty of statc prorty tasks wth arbtrary start tmes. Dept. Comp. Scence Report YCS 164, Unversty of York, [14] J. Goossens. Schedulng of offset free systems. Real-Tme Systems, 24(2): , March [15] G. C. Buttazzo. Rate Monotonc vs. EDF: Judgment Day. Real-Tme Systems, 29, 5-26, [16] P. Meumeu and Y. Sorel. Extendng rate monotonc analyss wth exact cost of preemptons for hard real-tme systems. In Proceedngs of 19th Euromcro Conference on Real-Tme Systems, ECRTS 07, Psa, Italy, July [17] S. Baruah, A. K. Mok, and L. Roser. Preemptvely schedulng hard real-tme sporadc tasks on one processor. Proceedngs of the 11th Real-Tme Systems Symposum, pp , [18] L. C. Lu and W. Layland. Schedulng algorthms for multprogrammng n a hard real tme envronment. Journal of ACM, Vol. 20, No 1, pp , January [19] J. Goossens. Schedulng of Hard Real-Tme Perodc Systems wth Varous Knds of Deadlne and Offset Constrants. PhD thess, Unversté Lbre de Bruxelles, [20] C.L. Lu and J.W. Layland. Schedulng algorthms for multprogrammng n a hard-real-tme envronment. Journal of the ACM, [21] A.K. Mok S.K. Baruah and L.E. Roser. Preemptvely schedulng hard realtme sporadc tasks on one processor. In proc. 11th IEEE Real-Tme Systems Symposum, [22] Joseph Y.-T. Leung and M. L. Merrll. A note on preemptve schedulng of perodc, real-tme tasks. Informaton Processng Letters, [23] J. Leung and Whtehead J. On the complexty of fxedprorty schedulng of perodc real-tme tasks. Performance Evaluaton(4), 1982.