Improving the Quality of Control of Periodic Tasks Scheduled by FP with an Asynchronous Approach

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1 Improvng the Qualty of Control of Perodc Tasks Scheduled by FP wth an Asynchronous Approach P. Meumeu Yoms, L. George, Y. Sorel, D. de Rauglaudre AOSTE Project-team INRIA Pars-Rocquencourt Le Chesnay, France {patrck.meumeu, laurent.george, yves.sorel, danel.de Abstract The am of ths paper s to address the problem of correctly dmensonng real-tme embedded systems scheduled wth Fxed Prorty (FP) schedulng. 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 affectng a task, thereby 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 worst case response tme of the tasks, hence reducng the jtter, when all the 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 n terms of mnmum deadlne reducton factor preservng the schedulablty of tasks set n both cases. Keywords: Real-tme systems, Fxed-prorty schedulng algorthms, Senstvty analyss, Robust control. 1. Introducton In ths paper we consder the problem of correctly dmensonng real-tme embedded systems ([1], [2], [3], [4]). 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) ([5], [6], [7]) 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. : 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. A recent research area called senstvty analyss ams at provdng nterestng nformaton on the valdty of feasblty condtons by consderng possble devatons of task WCETs ([2]), task perods ([2]), or task deadlnes ([3]). 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 a change n archtecture on the feasblty of a task set. 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. We are nterested n the senstvty of deadlnes. Computer controllng systems are very much affected by delays and jtter occurrng n the control loop. A deadlne reducton has been consdered by ([8]) as one soluton to reducng the jtter affectng a task and therefore obtanng better loop stablty n the control loop. The jtter of a task depends on the mnmum and on the worst case response tmes. Reducng the deadlne of a task can be a way to reduce the worst case response tme of a task and thus can reduce the jtter of the task. However, ths deadlne reducton should be performed n such a way that t does not cause any task to fal at run-tme. Ths supposes a schedulng drven by deadlnes. Ths paper proposes a soluton to reduce as much as possble the worst case response tme of each task when tasks are scheduled wth fxed prortes, accordng to Deadlne Monotonc Algorthm, by usng a specfc asynchronous frst release tmes scenaro. We show the benefts of our asynchronous scenaro by comparng the mnmum deadlne reducton factor appled preservng the schedulablty of the tasks n the synchronous and n the our asynchronous scenaro. Wth Deadlne Monotonc Algorthm, 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 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

2 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. Reducng the deadlne of a task makes t possble to reduce the jtter resultng from the executon of a task. In ths paper we show that the maxmum deadlne reducton s mnmzed for the synchronous scenaro where all the tasks are frst released at the same tme. We then propose a partcular asynchronous frst release tmes scenaro that allows us to obtan better feasblty condtons and a better deadlne reducton factor than the one obtaned wth the synchronous scenaro, thus reducng the jtter of the tasks for a better control. The feasblty problem of asynchronous task sets s known to be more complex than for synchronous task sets. We ntroduce a new formalsm to compute the worst case response tme of a task for asynchronous task sets. We apply ths approach to the case where the perods of the tasks are harmonc. We then show that n ths case, the worst case response tme s always obtaned for the second nstance of a task, whch represents a sgnfcant reducton n complexty. The rest of the paper s organzed as follows. In secton 2, we gve a state of the art regardng senstvty analyss of deadlnes consderng dynamc and fxed prorty drven schedulngs. We then focus on asynchronous task sets and recall exstng feasblty condtons. In secton 3, we ntroduce the concepts and notatons and establsh mportant propertes for the partcular asynchronous scenaro that we have chosen. 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 1. In secton 4, we ntroduce a new schedulng representaton whch s more compact than the classcal lnear representaton / Gantt Chart for a schedule. In secton 5, we ntroduce the concept of Mesod whch s used to compute the worst case response tme of an asynchronous task set. In secton 6, 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 7 based on extensve smulatons comparng the deadlne reducton factor for several load confguratons n both the synchronous case and n our asynchronous scenaro. Fnally, we conclude n secton Throughout the paper all subscrpts refer to tasks whereas all superscrpts refer to nstances. 2. State of the art Senstvty analyss for deadlnes has been consdered for Earlest Deadlne Frst (EDF) schedulng algorthm by ([3]) showng how to compute the mnmum feasble deadlnes such that the deadlne of any task τ equals αd, where α s reducton factor 0 < α 1. In ([8]), 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 algorthms. Few results have been proposed to deal wth the deadlne assgnment problem. As far as the authors are aware, no results are avalable for Fxed Prorty (FP) schedulng. Baruah & al., n [9] propose modfyng the deadlnes of a task set to mnmze the output, seen as a secondary crtera. In Cervn & al. ([10]), the deadlnes are modfed to guarantee close-loop stablty of a real-tme control system. Marnca & al. ([11]) focus on the deadlne assgnment problem n the dstrbuted case for multmeda flows. The deadlne assgnment problem s formalzed n terms of a lnear programmng problem. The schedulng consdered on every node s non-preemptve EDF or FIFO, wth a jtter cancellaton appled on every node. A performance evaluaton of several deadlne assgnment schemes s proposed. A recent paper proposed by Balvastre & al. ([3]) proposes an optmal deadlne assgnment for perodc tasks scheduled wth preemptve EDF n the case of deadlnes less than or equal to perods. The goal s to fnd the mnmum deadlne reducton factor preservng all the deadlnes of the tasks. They frst focus on the case of a sngle task deadlne reducton and show how to compute D mn, the mnmum deadlne of task τ such that any deadlne smaller than D mn for task τ wll lead to a non-feasble task set. They also show n [3] that when consderng the reducton of a sngle task τ, D mn s the worst case response tme of task τ for EDF schedulng. The maxmum deadlne reducton factor α for task τ s then: α = 1 Dmn D. In the case of a deadlne reducton appled to n tasks, the goal s to mnmze all tasks deadlnes assumng the same reducton factor for all the tasks (wth no preference regardng whch task requres the greatest deadlne reducton). Balbastre & al. n [3] show how to compute the maxmum deadlne reducton factor α appled to all the deadlnes usng an teratve algorthm. The prncple s to compute the mnmum slack t h(t) for any tme t [0, L) to determne the deadlne reducton factor appled to all the tasks, where h(t) = n t D =1 max(0, 1 + )C and L s the length of the frst synchronous busy perod, soluton of the equaton n t t = C. =1

3 τ = {τ 1,...,τ n } : task set; L compute-l(τ) : nteger; α 1 : real slack = mn t [0,L) (t h(t)) : real; Whle (slack 0) do α = mn =1...n (1 slack D ); For ( = 1; < n; + +) do D = αd ; end For slack = mn t [0,L) (t h(t)); done Return α; Algorthm 1: Computaton of α for EDF schedulng 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 ([12], [6]). 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 where all the tasks are frst 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. It can be shown that consderng only non-concrete tasks can lead to a pessmstc dmensonng [13]. The complexty of ths approach depends on the worst case response tme computaton complexty. In the case of deadlnes less than or equal 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 ([12]) 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. 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 than τ wthn the nterval [0, D ]. Ths feasblty has been mproved by ([14]), where the authors show how to reduce the tme nstants of S. For any task τ, they show how to sgnfcantly reduce the number of checkng nstants durng the nterval [0, D ] to at most 2 1 tmes rather than 1+ D τ j hp() T j. When deadlnes and perods are ndependent, ([6]) shows that the worstcase response tmes of a sporadc task τ are not necessarly obtaned for the frst actvaton request of τ at tme 0. The number of actvatons to consder s 1 + L, where L t s the length of the worst-case level-τ busy perod defned n ([15]) 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: Mn τ j hp() τ 1 C j τ j hp() τ C j C j, P ([7]). T j τ j hp() τ T j where P = LCM(T 1,..., T n ) s the least common multple (LCM) of the perods of all tasks and t leads to a pseudo-polynomal 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 system. ([13]) 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, ([16]) 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 ([16]). Ths feasblty result has been sgnfcantly mproved by ([17]) 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, ([16]) proves the non optmalty of Deadlne Monotonc schedulng algorthm for asynchronous systems when task deadlnes are less than or equal to perods. An optmal prorty assgnment can be obtaned n O(n 2 ) usng the Audsley procedure ([18]). A partcular case denoted offset free systems corresponds to the case where offsets can be chosen arbtrarly. An optmal offset assgnment s gven n ([19]). An offset assgnment s optmal f t can fnd a schedulable offset whenever a feasble assgnment exsts. The complexty of

4 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 ([19]) 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 j } such that the dstance belongs to [0, gcd{, 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] )+log(n 2 ))). Other offset assgnment strateges have been consdered by ([13]) usng the Audsley procedure to determne the subset of 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, 1 (.e. there exsts k Z such that = k 1 ) wth partcular offsets. In the case of non-concrete harmonc tasks, when tasks are scheduled wth Rate Monotonc Algorthm (the shorter the perod, the hgher the prorty) and n the case where deadlnes are equal to perods, a necessary and suffcent condton for the feasblty of such a system s gven by U = =1...n 1 (see [20]). Ths potentally proves the benefts of consderng harmonc tasks n order to get 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 ([4]) 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. C More recently, for control systems, [21] has proposed to nclude the control delay resultng from the response tme of a task as a cost functon for the controllers. They show how to solve the optmal perod assgnment problem analytcally. 3. Propertes of the asynchronous harmonc task set 3.1. 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 [22], 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 s commonly called 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 ([23]). 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 [23] 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, ) used to compute the worst case response tme of a task: τ k : The kth nstance of τ

5 r 1 : Release tme of the frst nstance of τ r k = r1 + (k 1): Release tme of τ k R k : Response tme of τk released at tme r k R : Worst-case response tme of τ 3.2. The specfc asynchronous scenaro Here we gve some nterestng propertes whch are satsfed by the specfc asynchronous scenaro we propose and 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 =. Ths assumpton wll be weakened n secton 6. 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. +1, {1,, n 1}) preemptve tasks ordered by decreasng prortes ( +1, {1,, n 1}). If there exst two tasks τ, τ j Γ n, ( < j) such that r 1 j r1 mod[] 2, then k, l 0 such that r k j = rl. Proof: (by contradcton) Let us assume that there exst two tasks τ, τ j Γ n, ( < j) such that rj 1 r1 mod[], and k, l 0 such that rj k = r l ṙ j k = rl rj 1 + (k 1)T j = r 1 + (l 1) rj 1 = r1 + (l 1) (k 1)T j rj 1 = r1 mod[] as 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. Theorem 1: (nspred by theorem 2.48 n [24]) Let Γ n = {τ 1, τ 2,, τ n } be a system of n asynchronous ndependent perodc preemptve tasks ordered by decreasng prortes ( +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 )+ {2,, n} (1) Then, f Γ n s schedulable up to s n + H n, wth H n = 2. Gven a, b, c Z: a = b mod[c] means that there exsts d Zsuch that a = b + cd. 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. 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,, ). 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, ) = 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 5 by usng corollary 2, and then a pseudo polynomal FC detaled n secton 6.1. 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 ( +1, {1,, n 1}) such that +1 and r 1 +1 = r1 C +1, the schedule s perodc from the second nstance wth perod H =. 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 +1 and r+1 1 = r 1 C +1, 1. Thanks to the prevous theorem, t s suffcent to prove that s r 1 =, 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 T 2 T 2 = r second release snce s 2 = r2 1 + (s1 r2) 1 + T 2 C2 T 2 = r2 1 +T 2 s the frst release tme of task τ 2 after (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 )+ = r 1 + (T 1 + r 1 1 r1 )+ by nducton hypothess. Thus, s = r 1 + (T 1 + C ) + snce r 1 1 = r1 +C. Now, as 0 < 1 + C < due to the scenaro mposed

6 to the frst nstance of each task and the fact that 1, then we obtan s = r 1 +. Corollary 2: The worst response tme R of each task τ s obtaned n the second nstance and s equal to that n all 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. Fgure 2. Release tmes of each task n the classcal lnear representaton or Gantt Chart 4. A new schedulng representaton A drect consequence of corollary 2 leads us to the concluson that n the case of a vald schedule,.e. when all deadlnes are met for all tasks, the schedule obtaned at level (the resultng schedule of the tasks wth the hghest prorty) s perodc wth the perod = LCM{T j j = 1,, } from the second nstance. As such, from the pont of vew of each task, the nterval preceedng the second nstance necessarly contans the transent phase, correspondng to the ntal part of the schedule at level, and the nterval startng at date r 2 wth the length s somorphc to the permanent phase of the schedule at level, correspondng to the perodc part of the schedule. The transent phase s always fnte due to the exstence of the permanent phase and the permanent phase repeats ndefntely. For a system of n perodc harmonc tasks for whch there exsts a vald schedule, snce the permanent phase repeats ndefntely, we ntroduce a new schedulng representaton. Ths schedulng representaton s obtaned by graphcally usng an orented crcular dsk called Damed wth a reference tme nstant t 0 = 0 correspondng to the tme reference n the classcal lnear representaton or Gantt Chart. The postve drecton n Damed s the trgonometrcal one,.e. opposte to that of the hands of a watch. The crcumference of Damed at level n corresponds to H n = LCM{ = 1,, n} where means the perod of the th task and n denotes the number of tasks n the system. In Damed, the dfferent release tmes for each task are unambguously determned by the value of ther frst release tme relatvely to that of other tasks wth respect to the reference date t 0 = 0, and the rato H n for task τ. As an example, fgure 2 llustrates the release tmes of each task for a system consstng of 4 perodc harmonc tasks. In ths fgure, the frst release tme of task t 1 s 2, whle that of task t 3 s 0. Fgure 3 clarfes our dea for the constructon of Damed for a gven set of harmonc perodc tasks. Ths fgure llustrates, for the same system wth 4 tasks (see Fgure 2), the correspondence of the release tmes of each perodc task n Damed relatve to the reference date t 0 = 0. The man ntuton behnd ths new representaton s to reduce the nterval of analyss for a system harmonc perodc tasks whatever ther frst release tmes are. Fgure 3. Release tmes of each task n Damed Now, n addton to the release tmes of each task, let us add the WCETs and explan how Damed can represent schedules. Durng the schedulng process from the hghest prorty task to the lowest prorty task, some of the avalable tme unts at a gven level,.e. those whch are not executed after the schedule of the frst 1 hghest prorty tasks, are executed by tme unts correspondng to the WCET C of the current task τ. Ths s done n order to obtan the next result for the schedulng analyss of the next task τ +1 wth respect to the prortes. As the consdered schedulng polcy (DM) determnes the total order n whch to perform the schedulng analyss, t follows that the crcular representaton,.e. Damed, of crcumference correspondng to the LCM of perods of all tasks that we have ntroduced allows us to buld drectly the permanent phase of the system f t s schedulable. Indeed, Damed can be constructed completely ndependently from the lnear representaton. In ths representaton, the WCETs of the tasks correspond to angular sectors, where the angular unt s gven by 1 H n and

7 H n = LCM{T 1, T 2,, T n }. Fgure 6 shows an example of the Damed for the system of whch the schedule and the curve of response tme as a functon of tme for each task are llustrated n fgure 4 and fgure 5. For ths system, whose characterstcs are summarzed n table 1, we assume that task t 1 has a hgher prorty than task t 2,.e. tasks are scheduled by usng DM. In fgure 4, the permanent phase s llustrated by the hghlghted zone (blue zone). The curve of the response tme of each task accordng to tme (see fgure 5) shows that from the tme t = 15, the response tme of each task s constant. We fnd ths result by constructng the Damed. Indeed, the LCM of the perods of both tasks t 1 and t 2 s gven by H 2 = LCM(5, 15) = 15. The release tmes of task t 1 n Damed wth respect to ts frst release tme are gven by r1 1 = 4, r1 2 = 9 and r1 3 = 14. For task t 2, we have a sngle release tme equal to r2 1 = 0 because ts perod T 2 = H 2 = 15. Snce task t 1 has a hgher prorty than task t 2, then at each release tme of t 1,.e. at the dates r1 1, r2 1 and r3 1, a sector correspondng to ts worst case executon tme (C 1 = 2 tme unts) s executed. As task t 2 has a lower prorty than task t 1, the fllng of the sectors of crcumference correspondng to ts worst case executon tme (C 2 = 4 tme unts) can only be done between the tme nstants 1 and 4, then tme nstants 6 and 7. Damed bulds the permanent phase of the system drectly: n fgure 4, task t 2 has two dstnct response tmes, 4 tme unts for the frst actvaton and 7 tme unts afterwards, whle n the crcular representaton through Damed, t has a sngle response tme, 7 tme unts, whch corresponds to ts response tme n the permanent phase. Fgure 5. Response tme of each task as a functon of tme. Tche r 1 C D t t Table 1. Characterstcs of the tasks Fgure 6. Crcular representaton of the schedule by usng Damed. Fgure 4. schedule. Lnear representaton / Gantt Chart of the Ths new representaton of the schedule s more nterestng than the lnear representaton / Gantt Chart because t s more compact and puts greater emphass on the avalable tme unts n the resultng schedule. In hs thess ([24]), Joel Goossens suggested that the permanent phase s suffcent to guaranteeng the schedulablty of a gven perodc task set when the cost of preempton s neglected and ths permanent phase s drectly bult by usng Damed. We now suppose the asynchronous task set defned n corollary 1 and present the Mesod approach used to compute the worst case response tme of each perodc task. 5. 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 ([25]), the worst response tme of each task, or the processor

8 demand ([26]). 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 to fll 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 [4] to a system where the tasks are not all released smultaneously and where the cost of a preempton s assumed to be zero. Ths method, unlke those proposed n ([27], [7], [28]), 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 ( 1, {2,, n}) such that 1 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 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 ([4]) 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 ) = as 1 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 that 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 +, r 1 +2] 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, and hence we obtan the next current schedule. Consequently, we represent the prevous schedule of every nstance τ k of the current task τ = (C, ) by an ordered set of 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 - mesod. More detals on the defnton of a -mesod are gven n [4]. For the current task τ = (C, ), there are as many -mesods as nstances. We call M b,2 the -mesod correspondng to the second nstance of task τ before beng scheduled n the current schedule. The process used to buld M b,2 for task τ wll be detaled later n ths subsecton. Stll, from the pont of vew of task τ, we defne for the mesod M b,2 the correspondng unverse X 2 to be the ordered set, compatble wth that of the mesod, whch conssts of all the avalabltes of M b,2 that s to say, all the possble values that C can take n M b,2. 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 -mesods correspondng to the second nstance of task τ after τ has been scheduled. s a functon of M b,2 whch tself s a functon of 1, both detaled as follows. Let f be the functon such that M b,2 = f( 1 ) whch transforms the 1 -mesod after task τ 1 has been scheduled at level 1 nto the -mesod before task τ s scheduled at level. As mentoned n [4], 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 M b,2 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 C, then ψ starts from the tme unt rght after γ = C mod[ 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 -mesod M b,2. Ths - mesod s obtaned when ψ =. Indeed, the prevous schedule at level (the schedule obtaned at level 1) conssts of H 1 = 1 tme unts whereas the schedule of the current task τ s computed upon H = tme unts. Thus, that amounts to extendng the prevous schedule from 1 to 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 [4] whch started from the frst tme unt, starts from the tme unt rght after γ = C mod [ 1 ] tme unts n the mesod 1. Snce τ 1 s the task wth the shortest

9 perod, then sp(τ 1 ) = {τ 1 }. Because τ 1 s never preempted, we have M b,2 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 = g(m b,2 ) whch transforms the -mesod M b,2 before task τ has been scheduled at level nto the -mesod after task τ has been scheduled at level Worst case response tme wth a Mesod For the -mesod M b,2, 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 -mesod before the avalablty correspondng to C [4]. 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(m b,2 ): functon g transforms a tme unt already executed n M b,2 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 M b,2 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 -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 : M b,2 : -mesod before τ s scheduled at level : -mesod after τ s scheduled at level. 6. Deadlne reducton factor 6.1. 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 n addton to 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 M b,2 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 M b,2 1 = {1, 2,, T 1 } and X1 2 = {1, 2,, T 1}. Moreover, ts response tme s also equal to C 1. Consequently, the correspondng T 1 -mesods assocated to task τ 1 are gven by M b,2 1 = {1, 2,, T 1 } τ 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 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 -mesod M b,2 = 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 = C mod[ 1 ] tme unts rather than the frst tme unt as n [4]. 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 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 R2 of task τ n ts k th nstance (wth k 2),.e. n the k th -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 (3) If for task τ expresson (3) holds, then = g(m b,2 ) 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 = C mod [ 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 C. The schedulng algorthm has the followng nne steps. Snce the task wth the shortest

10 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 = r1 1 C and compute γ = C mod [ 1 ] of the second nstance of τ w.r.t. τ 1. 3: Buld the -mesod M b,2 = f( 1 ) of task τ before t s scheduled. Ths constructon s based on a modulo arthmetc usng ndex ψ on 1 wthout forgettng to start from the tme unt rght after γ = C mod[ 1 ] tme unts rather than the frst tme unt as n [4]. Ths s due to the partcular release of tasks. 4: For the -mesod M b,2 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 M b,2. M b,2 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 -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(m b,2 ), 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, )} 1 n s not schedulable. 9: end for Thanks to the above algorthm, a system of n tasks {τ = (C, )} 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) 6.2. Computaton of α The value of α s gven by: α = max 1 n ( R Ths value of α guarantees that no task fals at run-tme. We recall that for the synchronous scenaro, the worst case response tme of task τ s gven by: R = C + Example j hp() R T j C j 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 2. Table 2. Characterstcs of the tasks C τ τ τ τ The shorter the perod of a task s, the hgher ts level s. Thus, as depcted n table 2, 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 M b,2 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 M b,2 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 τ 3 : ). M b,2 3 = {(1), 1, 2, 3, (8), 4, 5, (2), 6, 7, 8, (8), 9, 10, (1)} R 3 = = 14 3 = {(16), 1, 2, 3, (8), 4, 5, (1)} 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 = r3 1 C 4 = 9 7 = 16, we have τ 4 : M b,2 4 = {(4), 1, 2, (17), 3, 4, 5, (8), 6, 7, (17), 8, 9, 10, (4)} R 4 = = 36 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

11 Fgure 7. Executon of a set of harmonc tasks wth r 1 = r 1 1 C, {2,, 4} Fgure 10. Crcular representaton of the schedule for a set of harmonc tasks wth r 1 = 0 {1,, 4} Fgure 8. Crcular representaton of the schedule for a set of harmonc tasks wth r 1 = r1 1 C, {2,, 4} s depcted n fgure 7 and the crcular representaton of the schedule by usng Damed s depcted n fgure 8. The schedule of the same set of tasks released smultaneously s depcted n fgure 9 and the crcular representaton of the schedule by usng Damed s depcted n fgure s 36 tme unts whereas t s 55 tme unts n fgure 9 and fgure 10. 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 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% 7. Expermental results Fgure 9. 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. In fact, the worst case response tme of task τ 4 n fgure 7 and fgure In ths secton we present some expermental results found by usng the approach we have developed above. To acheve these expermental results, we proceed n two steps. Frst, we compare the mnmum deadlne reducton factor α obtaned n the synchronous scenaro wth that obtaned n our specfc asynchronous scenaro. Second, we extend ths comparson concernng the value α to the value of α obtaned for an arbtrarly generated scenaro of the frst release tmes for all tasks. Ths extenson s performed by usng more extensve experments n order to get more accurate conclusons wth

12 regard to the contrbutons of the proposed approach. As n ([1]), we consder a set of harmonc tasks scheduled wth the Deadlne Monotonc algorthm. The frst step n our process of comparng the value of α for gven scenar of frst release for all tasks conssts n performng experments for each graph, where 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 3, compared to the synchronous one. We set α = D, 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. The resultng graphc s dsplayed n fgure 11. 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 α both n the synchronous and n the specfc asynchronous cases. Concernng the second step n our process of comparng the value of α for gven scenar of frst release tmes for all tasks, we perform twce as many experments than for the frst step. That s to say, we perform experments for each graph, and every task set stll conssts of n = 10 harmonc tasks. Agan, the total utlzaton factor of the processor s randomly chosen between 0.7 and 1 for each task set. As such, we can evaluate the gan of α obtaned n our specfc asynchronous scenaro, compared to that obtaned n the synchronous scenaro on the one hand, and to the mean value obtaned for a set of arbtrarly generated scenar on the other hand. As for the frst step, we set α = D, and we gradually and unformly decrease the value of the relatve deadlnes D by the same factor for all tasks n each set. For the synchronous, and the asynchronous scenar, we plot the curves correspondng to the smallest value of α. For the set of arbtrarly generated scenar, we plot the curves correspondng to the mean value of α. Ths s performed n each case as a functon of the total utlzaton factor of the processor, for the task set to reman schedulable. The curves obtaned are dsplayed n fgure alpha load Fgure 11. Value of α wth our asynchronous scenaro and wth the synchronous scenaro In fgure 11, the sold curve represents the result obtaned for α n our specfc asynchronous case whereas the dotted curve represents the result obtaned n the synchronous case. In both cases, we start wth a schedulable task set τ, D =. From [20], U 1 s a necessary and suffcent condton for the schedulablty of a harmonc task set as tasks are scheduled wth DM, equvalent to RM when τ, D =. We can see that for a small load, we obtan almost the same Fgure 12. Value of α wth our asynchronous scenaro, then wth the synchronous scenaro and the mean of a set of arbtrarly generated scenar In fgure 12, the curve n red represents the result obtaned for α by usng our specfc asynchronous scenaro. The curve n green represents the result obtaned for the synchronous case and the curve n blue represents the mean value obtaned for a set of arbtrarly generated scenar. In all the cases, we start wth a schedulable task set τ, D = and U 1 remans a necessary and suffcent condton for the schedulablty of a harmonc task set as tasks are scheduled

13 wth DM. It s worth notcng that DM s equvalent to RM when τ, D =. We can see that we always obtan almost the same value for α both n the synchronous case and for the mean value obtaned for a set of arbtrarly generated scenar. For a small load, the value of α vares very slghtly whatever the scenaro of frst release for all tasks s. In both steps, ths s due to the fact that wth a small load the worst case response tmes of the tasks are less nfluenced by the frst release tmes of other tasks. When the load ncreases, the gan also ncreases, reaches and remans at 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 spacfc asynchronous scenaro becomes less sgnfcant. 8. Concluson In ths paper we have proposed a new approach for a better control of perodc tasks scheduled wth Deadlne Monotonc schedulng algorthm. We have consdered a specfc asynchronous task set and harmonc tasks that enables us to sgnfcantly reduce the worst case response tme of each task thus reducng the jtter of each task for a better control. The asynchronous scenaro we consdered 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 characterzng the beneft n terms of worst case response tme reducton. 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 and to an arbtrarly generated scenaro. Ths makes t possble to better control the jtter of the tasks when consderng control loops. Future work wll compare the deadlne reducton factor obtaned wth EDF wth the one we have obtaned wth our specfc asynchronous scenaro. References [1] P. Meumeu Yoms, L. George, Y. Sorel, and D. De Rauglaudre. Improvng the Senstvty of Deadlnes wth a Specfc Asynchronous Scenaro for Harmonc Perodc Tasks scheduled by FP. The Fourth Internatonal Conference on Systems (ICONS 09), Cancun, Mexco, March [2] 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, [3] 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, [4] P. Meumeu Yoms and Sorel Y. Extendng Rate Monotonc Analyss wth Exact Cost of Preemptons for Hard Real-Tme Systems. Proceedngs of 19th Euromcro Conference on Real- Tme Systems, ECRTS 07, Psa, Italy, Jul [5] 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 , [6] K. Tndell, A. Burns, and A. J. Wellngs. Analyss of hard real-tme communcatons. Real-Tme Systems, Vol. 9, pp , [7] L. George, N. Rverre, and M. Spur. Preemptve and non-preemptve schedulng real-tme unprocessor schedulng. INRIA Research Report, No. 2966, September [8] 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 [9] S. Baruah, G. Buttazo, S. Gornsky, and G. Lpar. Schedulng perodc task systems to mnmze output jtter. In 6 th Conference on Real-Tme Computng Systems and Applcatons, pp , [10] A. Cervn, B. Lncoln, J. Eker, K. Arzen, and Buttazzo G. The jtter margn and ts applcaton n the desgn of real-tme control systems. In proceedngs of the IEEE Conference on Real-Tme and Embedded Computng Systems and Applcatons, [11] D. Marnca, P. Mnet, and L. George. Analyss of deadlne assgnment methods n dstrbuted real-tme systems. Computer Communcatons, Elsever, To appear, [12] M. Joseph and P. Pandya. Fndng response tmes n a realtme system. BCS Comp. Jour., 29(5), pp ,, [13] 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, [14] Gorgo Buttazzo Enrco Bn. Schedulablty Analyss of Perodc Fxed Prorty Systems. IEEE Transactons On Computers, Vol. 53, No. 11, Nov [15] 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, [16] 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