Scheduling within Temporal Partitions: Response-time Analysis and Server Design

Transcription

1 chedulng wthn Temporal Parttons: Response-tme Analyss and erver Desgn Lus Almeda LE-IEETA/DET Unversdade de Avero Avero, Portugal Paulo Pedreras LE-IEETA/DET Unversdade de Avero Avero, Portugal ABTRACT As the bandwdth of CPUs and networks contnues to grow, t becomes more attractve, for effcency reasons, to share such resources among several applcatons wth the mnmum level of nterference. Ths can be acheved usng temporal parttons, wth each applcaton assgned to ts own partton and executng as f t was executng alone on a resource wth lower bandwdth. The parttons are assocated to servers that execute the applcaton tasks accordng to a gven applcaton-level scheduler. On the other hand, the set of servers s scheduled by a system-level scheduler. Ths paper addresses the partcular case of fxed prortes-based applcaton-level schedulers together wth a perodc server model at the system level. It starts wth an adequate response tme analyss based on the noton of server avalablty for a known server. Then t addresses the nverse problem of desgnng a server wth mnmum system-level resource requrements to fulfll the applcaton tme constrants. In ths context, the paper shows that response tme based schedulablty tests wth lnear tme bounds do not need to consder all tasks but ust a small subset, whch may lead to substantal speed-ups. The proposed method goes a step further wth respect to other recent works n the lterature by consderng a more complete task model, effectvely computng the server parameters and establshng a better trade-off concernng complexty and tghtness. Categores and ubect Descrptors D..7 [Operatng ystems]: Organzaton and Desgn Real-tme systems and embedded systems. General Terms Algorthms, Desgn, Theory. Keywords Real-tme systems, real-tme schedulng, herarchcal schedulng, response-tme analyss. Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. EMOFT, eptember 7--9,, Psa, Italy. Copyrght ACM //9...$5... INTRODUCTION Herarchcal schedulng has been generatng a consderable nterest, recently, due to ts ablty to separate the concerns of schedulng at the system and applcaton levels. It s a fundamental brck n the current trend towards hgher ntegraton and flexblty n embedded systems [8], whch opens the way to hgher effcency and lower costs by means of resource sharng, as well as to hgher reslence to hardware falures by means of dynamc reallocaton of computng or communcaton enttes [7]. Herarchcal schedulng s ntmately connected wth resource temporal parttonng accordng to whch a shared resource, e.g. CPU or network, s used by several complex applcatons each of whch s composed of a set of enttes, e.g. tasks or streams. These enttes must be scheduled nternally to the applcaton nsde one specfc resource partton to whch they were allocated. At a hgher level, all resource parttons are scheduled usng a gven system-level polcy. The concept of server s well adapted to ths level, supportng temporal solaton among parttons (Fgure ). ystem scheduler Applcaton n ystem level Applcaton Applcaton level Task Task Task m Applcaton ervers (capacty Cs, perod Ts ) Applcaton scheduler Fxed prortes Fgure. Herarchcal schedulng framework. However, two problems arse: how to derve real-tme guarantees for the applcatons runnng wthn each server and, conversely, how to desgn the server for a gven applcaton so that t fulfls the applcaton requrements wth the least resource utlzaton. Ths paper addresses both problems. It relates drectly to recent work avalable n the lterature and consttutes a further contrbuton to the analyss and desgn of temporally parttoned 95

2 systems. A prelmnary work-n-progress verson was presented n []. The related work s dscussed n secton, whch also hghlghts the contrbutons of ths paper, secton presents the task model, secton descrbes the response-tme analyss and secton 5 shows the server desgn approach, wth an example n secton 6. Fnally, secton 7 extends the prevous results ncludng ntra-server blockng and secton 8 concludes the paper.. RELATED WORK The problem of herarchcal schedulng bears many resemblances wth other schedulng problems that have been tackled n the past, such as those regardng exclusons [5] and nserted dle-tme [9][]. In fact, lookng to the system from the perspectve of one applcaton executng wthn a server,.e. a temporal partton, the perods of tme n whch the respectve server cannot execute, e.g. because another server has been scheduled at the system level, can be seen as excluson perods or perods of nserted dle-tme. Moreover, wthn specfc scopes, such as real-tme communcaton over shared meda, some forms of herarchcal schedulng/ temporal parttonng have long been used. Ths s the case wth TDMA (as n TTP/C), n whch each node has one or more dedcated slots n the TDMA round to transmt ts traffc, as well as wth the mult-phased cyclc framework (as n WorldFIP, FlexRay or FTT-CAN []), wth several phases used n sequence wthn a mcro-cycle, each for a gven type of traffc (Fgure ). In both cases, ether slots or phases can be taken as perodc servers wthn whch several message streams must be scheduled. TDMA round wth nodes Bus tme Mcro-cycle wth phases sync async sync async sync Bus tme Fgure. Common parttonng schemes for shared buses used n real-tme communcaton. However, recent work has brought to lght new results that are more general and abstract away the specfc applcaton scope. In [7] the authors use the concepts of vrtual resource, vrtual tme, bounded-delay resource parttonng and server supply functon to deduce real-tme guarantees for herarchcal partton schedulng. In [6] the focus s on herarchcal resource parttonng usng fxed prortes local schedulers and both deferrable and sporadc servers. It presents a response-tme analyss as well as utlzaton bounds. In [], the authors deduce a tghter schedulablty test for parttons usng fxed prortes local schedulers and address the ssue of server desgn n order to meet the applcaton tme constrants. In [], the authors present a response tme analyss for a composed model n whch EDF local schedulng s executed wthn a system-level fxed prortes framework. The task model consders several practcal ssues such as release tter and ntertask synchronzaton blockng. Fnally, n [] the authors present an analyss for herarchcal partton schedulng, consderng both fxed prortes and EDF local schedulng, together wth a generc perodc resource model. A general schedulng nterface s also presented that facltates the composton of parttons and dervaton of real-tme guarantees. The paper also addresses the nverse problem of defnng a partton n order to meet a gven set of real-tme requrements. Ths paper relates closely to the works referred above. It s nterestng to note that these works are very recent and some of them were submtted n parallel, leadng to overlappng of short parts. Our motvaton was to extend the work started n [] and later mproved n [], n whch we developed a response-tme analyss for the asynchronous traffc wthn a double phase cyclc framework as depcted n Fgure. In ths paper we use the same reasonng but wth a more general model that fts well n task schedulng. We wll consder a fxed prortes local scheduler together wth a perodc server model to manage a resource partton. Then, we derve upper bounds to the worst-case response tme of tasks executng wthn a gven server. We wll also use the concept of server supply functon as n [7], or resource supply n [], but we wll refer to t as server avalablty functon for coherence wth our prevous work. The response-tme analyss we propose s smlar to the one for fxed prortes presented n [] but we extend t to cover a more realstc task model that ncludes deadlnes shorter than or equal to perods, release tter and synchronzaton blockng. Our response-tme analyss s also equvalent, despte dfferent, to the one n [6] but we beleve ours s more flexble snce t can be easly adapted to rregular parttons as n [] and [], by descrbng them analytcally n the respectve server avalablty functon (see further on). The server desgn problem s addressed n both [] and []. When compared to the former one, our method s smpler but less tght and, as shown later, may represent a more favorable compromse between tghtness and complexty n certan crcumstances. On the other hand, [] follows the same reasonng as we dd n [] of matchng the applcaton demand wth the server supply to derve the worst-case response tme. The correspondng result presented n [] s equvalent to the one we presented n []. In ths paper we extend that work by generatng the server perod that mnmzes the server bandwdth together wth the overhead caused by context swtchng at the system level, usng the cost functon proposed n []. Fnally, we consder release tter as well as ntra-server blockng n our task model, whch none of the other approaches does. In summary, ths paper bulds upon the work n [],[], [] and [] and proposes the followng: extenson of the task model to a more realstc one; alternatve way of deducng the soluton space for the server parameters; new method to search the server soluton space based on the new concept of applcaton external ponts; full computaton of the server parameters that mnmze a gven cost functon.. TAK MODEL In ths work, we consder that a gven actve resource, say a CPU, s used to execute a set of ndependent applcatons. Each applcaton Ω s composed by a set Γ Ω of N Ω tasks, Γ Ω {τ (C,T,D,J,P ), =..N Ω }, n whch each task, at ths pont, wll be consdered ndependent and fully preemptve. Each task τ 96

3 s characterzed by a perod T, a worst-case executon tme C, a relatve deadlne D that s shorter than or equal to the respectve perod, a maxmum release tter of J and a fxed prorty P that may possbly be derved from the perod, the deadlne, or any suboptmal crteron. Also, the set of tasks Γ Ω executes by prorty order wthn a perodc server Ω, whch s characterzed, at a system level, by a perod T and a capacty C. The server can be of any type as long as, under contnuous demand, t behaves lke a perodc task executng for C tme unts wthn every T tme unts nterval. Ths s also the server model consdered n [] and [] ([6] consders deferrable and sporadc models, only). The system-level scheduler that schedules parttons, and the current system load, determne when the server s avalable to execute applcaton tasks. We thus defne the server avalablty functon, A (t), whch returns for each nstant t the cumulatve CPU tme avalable for the applcaton to execute snce an arbtrary tme orgn. For statc system level schedulng such as TDMA (Fgure ), A (t) may be exactly characterzed a pror. However, f an on-lne system-level schedulng polcy s used, the specfc pattern of A (t) may be dffcult to predct. Therefore, for the sake of ndependence wth respect to the system-level schedulng polcy and load, t s helpful to use a lower bound A_ (t) that assures that the cumulatve CPU tme avalable for an applcaton s never smaller than a gven value. In order to determne a lower bound to the avalablty functon, we can use the same reasonng as n [] or []. Bascally, t consders the worst-case server avalablty pattern wth respect to the arbtrary tme orgn n whch the server suffers maxmum latency ( ) n the begnnng and then follows a perodc pattern wth ts capacty avalable at the end of each perodc nstance (Fgure ). The value of depends drectly on the maxmum fnshng tter of the server executon, as determned by the system scheduler. In certan cases, e.g. n regular TDMA schedules, such tter s elmnated because the server executes n a strct perodc fashon, leadng to = T -C. Thus, wll vary between ths best-case value and the worst-case depcted n Fgure n whch = *(T -C ). For the sake of generalty we wll consder such ntal latency as gven by Equaton. Cs T s possble avalablty pattern avalablty lower bound A s(t) A_ s(t) tme T s -C s C s Fgure. erver avalablty functons A (t) and A_ (t). Fnshng tter refers to the absolute tter that affects the nstant n whch the server exhausts ts capacty wthn each perod. = (+β )*(T -C ) () wth β = (R -C ) / (T -C ) Notce that β ( β ) s a normalzed measure of the maxmum server fnshng tter, whle R (C R T ) s the maxmum relatve fnshng nstant of all server nstances. To mantan ndependency from the system schedulng polcy β = should be consdered (worst-case). For smplcty, n the remander of the paper we wll use the same expresson avalablty functon for the lower bound functon, and refer to t as A_ (t) (Equaton ). In [] ths functon s called server characterstc functon, n [7] server supply functon and n [] resource supply bound functon. A _, ( t) = t < t ( + k * ( T C ), + k * T t < + k ( k + ) * C, + k * T + C t < + ( k + ) k = ( t ) / T * T + C * T. REPONE-TIME ANALYI A relatvely smple but effectve way of upper boundng the response-tme for each task τ wthn a gven server s to use the same reasonng explaned n [] and []. The fact that we are now usng a fully preemptve task model together wth a perodc server further smplfes the analyss theren presented, whch was based on non-preempton wth nserted dle-tme together wth a background server. Therefore, we compute for each task τ and for each nstant t the maxmum load submtted to the server by the task tself after ts release together wth all hgher prorty tasks. We call ths the level submtted load functon, H (t) (Equaton ). It can be determned by the usual methods n fxed-prortes response tme analyss [] snce the crtcal nstant for each task s not changed by the presence of the server [6]. Equaton consders Γ Ω sorted by decreasng prortes,, < P >P. ( t + J ) / T = () H ( t) = * C () The worst-case response tme for task τ, referred to as R, can thus be obtaned as expressed n Lemma. Lemma. Gven the task set Γ Ω executed wthn a server wth avalablty functon A_ (t), the worst-case response tme R for task τ Γ Ω s obtaned by determnng the earlest nstant n whch the maxmum level submtted load H (t) matches the least server avalablty A_ (t). Proof: Lemma can be proved by consderng the defntons of both A_ (t) and H (t). In fact, A_ (t) stands for the mnmum executon tme that the server can delver to the applcaton counted from t=. On the other hand, H (t) stands for the maxmum executon tme requred to execute τ to completon, when released at t= and consderng the maxmum nterference t may suffer by hgher prorty tasks wthn the applcaton. Thus, the worst-case response tme s gven by the nstant when the least avalablty s ust enough to cover the longest requested 97

4 executon tme (Fgure ). If the server, n one or more nstances, executes before than consdered n A_ (t) the response tme can only be shorter. level subm tted load 8 6 A_ (t) s H (t) 5 5 tm e Fgure. Worst-case response tme of task τ. Wth Lemma, we can perform a trval schedulablty test as stated n theorem. Theorem. The task set Γ Ω executed wthn a server wth an avalablty functon A_ (t) s schedulable f (and only f) τ Γ Ω R D Proof: The f s trvally proved but the only f requres R to be an accurate value. Ths depends on the accuracy of the avalablty lower bound A_ (t). In practce, ths lower bound wll be pessmstc because A (t) wll not suffer maxmum delay n all nstances after startup, thus there wll always be a gven load for whch some exact worst-case response tmes are lower than the computed R. In ths case, the test n Theorem wll be suffcent, only. However, n partcular stuatons, such as regular TDMA schedules, A_ (t) = A (t) and thus the test wll be necessary (ths s why we kept the only f wthn parenthess). The value of R can be determned usng Equaton or, more effcently, the equvalent Equaton 5, whch makes use of the nverse of the avalablty functon A_ (t), referred to as A nv (t). Ths s formalzed n Equaton 6 (t s referred to as servce tme bound functon n []). R = earlest t: A_ (t) = H (t) () R = earlest t: t = A nv (H (t)) (5) Formally, nvertng A_ (t) requres specfyng the value of the nverse correspondng to the flat segments of the orgnal functon. We consder the lowest values of such ntervals, as ndcated n Fgure, whch result n the lowest avalablty. A nv (u) = + m * T + (u m*c ), m = u/c A nv (u) = (β + u/c ) * (T C ) + u (6) Equaton 5 can be solved teratvely wth R R =H () and R n+ = A nv (H (R n )) that ether converges to R = R n+ = R n or grows beyond the deadlne D, wthn fnte teratons. Ths can be easly demonstrated by the fact that, from teraton to teraton, the ncrement n H (t) s lower bounded by mn =.. (C ). If the whole CPU s avalable to execute the applcaton, then A_ (t) = A nv (t) = t and Equatons and 5 reduce to the usual response tme analyss for fxed prortes schedulng for a smlar task model []. A smpler but less tght upper bound R for the response tme of each task can be obtaned consderng a lnear lower bound to the avalablty functon, also proposed n [] and [], heren referred to as lnear avalablty functon A_ (t) (Equaton 7). Ths functon s depcted n Fgure 5, wth an ntal latency of, such as A_ (t), and then grows lnearly wth slope α = C / T,.e. the server bandwdth. A_ (t) = (t ) * α, for t > and otherwse (7) The response tme upper bounds can be obtaned rewrtng Equaton 5 usng the nverse of A_ (t) (Equaton 8). R = earlest t: t = + H (t)/α (8) Ths allows us to state Theorem, whch wll be partcularly helpful n the followng secton. Theorem. The task set Γ Ω, executed wthn a server wth ntal latency and bandwdth α, s schedulable f τ Γ Ω R D. Proof: Ths can be proved notcng that, for every task τ the ntersecton between H (t) and A_ (t) (.e. R ), s always later than or concdent wth that between H (t) and A_ (t) (.e. R ). Thus, τ Γ Ω R R, provng the theorem. 5. ERVER DEIGN In ths secton we tackle the opposte problem of the prevous one,.e. gven an applcaton Ω, whch parameters (C,T ), or equvalently (α, ), should the server Ω have so that t requres the least system resources and stll meets the applcaton tme constrants? In order to address ths problem we wll start by notcng that such a server should be as tght as possble concernng fulfllng the applcaton tme constrants, otherwse t would over consume system resources. Therefore, usng the approach presented n the prevous secton for Theorem, we wll consder that, for each task τ the respectve R s ust on the deadlne D. Ths means that the avalablty functon A_ (t) should be such that ntersects H (t) exactly at t=d. To acheve ths, we start by defnng the set of deadlne ponts DP Ω {DP (D,H (D )), =..N Ω } (Fgure 5), whch represent the lowest avalablty requred for meetng all applcaton deadlnes. Our purpose, then, s to defne A_ (t) so that t s hgher than but as close as possble to the set of such ponts. In [] (theorem 5), gven a fxed server perod T, the authors suggest performng an extensve search for the mnmum server capacty C that stll allows meetng the tasks deadlnes. However, the search method s not specfed. We suggest usng Bnary earch n the nterval [, T ], whch s relatvely effcent. For example, f a resoluton of /56 of T s enough, only 8 teratons are needed. Correspondngly, we can use the approach suggested n Theorem 98

5 to fnd the lowest lnear bound A_ (t) that passes above or through all deadlne ponts DP. Ths s also proposed n [] (theorem 6), where C s determned as a functon of the perod T so that A_ (t) touches at least one deadlne pont. Notce that all task deadlnes,.e. all deadlne ponts, are tested aganst the lnear lower bound. 8 deadlne ponts external ponts A_ (t) A_' (t) N Ω and N Ω. Also, n general, the number of external ponts N E s substantally smaller than the number of deadlne ponts N Ω (.e. the number of tasks), whch contrbutes to decrease the complexty of the remanng part of the process. Fgure 6 plots the number of external ponts aganst the number of tasks for 5 randomly generated sets, each constraned to a total bandwdth smaller than but close to 5% and wth up to tasks. It s surprsng to see how small the number of external ponts normally s, between and 5, and that there s stll a tendency for reducton as the number of tasks grows (for large task sets, the most frequent number of external ponts s!). H (D ) 6 slope α 5 5 D Fgure 5. Determnng the avalablty functon A_ (t). However, we are not nterested n determnng ust C as a functon of T but rather n analyzng the space of possble solutons (C, T ) to choose the one that mnmzes the use of system resources. For ths purpose we determne the subset of external ponts, E Ω {E (x,y ), =..N E } DP Ω (Fgure 5), that s the subset of deadlne ponts through whch a lnear bound (α, ) can be drawn that fulflls Theorem. Necessarly, such lnear bound must also respect ts boundary constrants namely α (the server cannot use more than the total CPU bandwdth) and, for each E, α y /x (otherwse, < and, snce α >, t would lead to T < and C <). Ths drectly leads to Theorem. Theorem. The task set Γ Ω, executed wthn a server wth ntal latency and bandwdth α, s schedulable f τ : DP E Ω, R D. Proof: Informally, Theorem says that t s suffcent to test the schedulablty of the external ponts to guarantee that the task set s schedulable (Fgure 5). It can be proved ust by observng the defnton of external ponts. If these ponts are below a gven lnear avalablty functon, then all other deadlne ponts are and thus the whole set s schedulable. The determnaton of the E Ω subset s carred out usng a smple algorthm that goes through all deadlne ponts n ascendng deadlne order, startng wth the assumpton that the frst deadlne pont s an external pont. Notce that the slopes of the segments that on every two consecutve external ponts must be monotoncally decreasng from the maxmum α = to the mnmum α = y NE /x NE. Therefore, the algorthm removes from the set of deadlne ponts all those that would volate ths decreasng slope pattern and the referred lmts. At the end, the ponts that reman are the external ponts. The tme complexty of the algorthm vares wth the characterstcs of the task set between Fgure 6. Number of external ponts n random sets. Hence, the use of external ponts may accelerate substantally the executon of repettve response tme based schedulablty tests, for example n optmzaton processes, n whch the task set remans unchanged and several server parameters are tested. Ths s depcted n Fgure 7 for several mplementatons of the test correspondng to theorem 6 n [], to determne C for dfferent values of T, and usng random task sets of dfferent szes (each pont corresponds to the average of sets wth the same sze). We used a drect mplementaton n Matlab ([](th.6)) and an mproved verson ([] (th.6)_mp), as well as both cases usng external ponts, (wth ext.pt.) and (wth ext.pt._mp), respectvely. The benefts of usng external ponts are clear for task sets wth more than 8 tasks. The set of external ponts defnes a soluton space for the server desgn problem. In fact, all lnear avalabltes A_ (t) that touch at least one external pont wth a vald slope are possble solutons. Vald slopes are those n between the slopes of the segments that on each pont wth the prevous and wth the next, consderng the absolute maxmum and mnmum for the ponts n the extremes. Ths can be represented by the unon of the followng N E subntervals: α {[ α(e,e )] [α(e,e ) α(e,e )] [α(e N E-,E N E) y N E /x N E]} 99

6 Executon tm e n seconds wth ext.pt._m p [](th.6)_mp wth ext.pt. [](th.6) 6 8 Number of task per taskset Fgure 7. peed up by usng external ponts. Each of these subntervals corresponds to A_ (t) lnes that pass through one external pont E (x,y ), =..N E, whch can be characterzed by the straght-lne Equaton 9 (left). y=α ( x- x ) + y and = x - y /α (9) Equaton 9(rght) shows a hyperbolc relatonshp between α and around each pont E and also allows determnng the respectve server perod T usng Equaton. Ts = /((+β)(-α )) () The complete soluton space n terms of the server parameters (α, ) can be obtaned by calculatng the followng ntersecton for all N E external ponts: =..NE x - y /α and α () Ths soluton space can equvalently be expressed n terms of the server parameters (C,T ) by nsertng from Equaton 9 (rght) nto and replacng α wth Cs /Ts. The result s the followng ntersecton for all N E external ponts: Cs =..NE Cs Ts and ( x ( + β ) Ts ) + x ( + β ) Ts ( + β ) ( ) + ( + β ) y Ts () Condton s equvalent to the result presented n [] (theorem 6). Recall that x = D and y =H (D ). Fnally, n order to fnd one specfc soluton, we use the cost functon proposed n [], whch consders both the bandwdth drectly requested by the server, α=c /T, as well as the overhead bandwdth mplctly used by the server n context swtchng between applcatons at the system level. Ths latter factor can be roughly computed as C O /T where C O s a system parameter representng the context swtchng tme. The cost functon can thus be expressed as F=α+C O /T. Mnmzng F corresponds to fndng the balance between mnmzng α and maxmzng T. Agan nsertng from Equaton 9 (rght) nto Equaton we obtan Ts =Ts (α ), for each α subnterval,.e. each external pont. Then, usng Ts (α ) wthn the cost functon results n F=F(α ). The α of mnmum cost for each subnterval (referred to as α,mn ) can be easly determned wth a closed formula (Equaton ) obtaned by dfferentatng F(α ) wth respect to α and calculatng the respectve root. Whenever α,mn les outsde the respectve subnterval, the closest extreme s consdered for mnmum. After havng determned α,mn for all subntervals (=..N E ) t s then ust a matter of selectng the one that generates the absolute mnmum cost (α mn ). It s also necessary to dentfy to whch subnterval the α mn value belongs to, n order to determne (Equaton 9 rght) and then T (Equaton ). ( y ( + β)* CO ) /( x ( + β )* CO ) α = + y / x * (),mn y / x The server parameters generated ths way are not optmal due to several factors such as the pessmsm ncluded n the server ntal latency, the successve approxmatons of the effectve server avalablty functon A (t) and the use of the deadlne ponts that may lead to worse than necessary bandwdth requrements. However, concernng the A_ (t) approxmaton by A_ (t),.e. the lnear bound, t s possble to carry out a smple fnal mprovement step. The fact that A_ (t) touches at least one deadlne (external) pont accordng to the desgn method presented above, t does not mply that the correspondng A_ (t) functon also touches one, because t A_ (t) A_ (t). Hence, we may allow A_ (t) to actually enter the area bounded by A_ (t) untl t also touches one deadlne pont (ths assures that Theorem s stll met). Ths can be done decreasng C, ncreasng T or a combnaton of both. For smplcty and effcency accordng to tests wth random sets, we propose ncreasng T, accordng to Equaton, whch corresponds to mantanng the heght of the steps n A_ (t) whle expandng the functon to the rght. Ths also leads to a further reducton n α. The amount of beneft, however, depends on the task set. Usng random task sets wth unformly dstrbuted perods, we acheved a best mprovement of 8% ncrease n T and 7.% reducton n α. However, on average, the benefts were lower, wth.8% ncrease n T and.7% reducton n α. nv D A ( H ( D )) T = + mp T mn = N + + (),.. Ω ( D ( β ) * C ) / T Ths mprovement can be carred out over any non-optmal result obtaned by any perodc server based method. 6. EXAMPLE To llustrate the desgn methodology presented above we make use of the example suggested n []. Ths example conssts of an applcaton Ω, wth the task set Γ Ω ={(C,T )= (,), (,), (,5) and (D =T, J =, P =/)}. Frstly, we obtan the sets of deadlne ponts DP Ω ={(,), (,), (5,)} and external ponts E Ω ={(,), (5,)}. Then, usng the external ponts we can derve the α subntervals and usng expressons and we can establsh the (α, ) and (C,T ) soluton spaces, respectvely (Fgure 8). In the (α, ) fgure we can see that our soluton space s contaned n the one derved n [] and thus t s worse, although for a small dfference (less than % n the low values of α). On the other

7 hand, the worst-case tme complexty of the method n [] may reach NΩ -, contrastng wth the smplcty of our method wth a worst-case tme complexty of N Ω. Executng the fnal mprovement step wth Equaton consderng C O =.6 generates an operatng pont (Fgure 8) that s slghtly better than both soluton spaces, wth (α, ) = (.5,.8) or equvalently (C,T ) = (.,.9). Just for comparson, the corner n the sold lne of Fgure 8-a) corresponds to (α, ) = (.55,.8). Fgure 9 shows the obtaned avalablty functons (top) and varaton of the cost functon F wth respect to T (bottom). 5 5 A_s(t) A_'s(t) o deadlne ponts + external ponts.5 Our soluton space oluton space n [] 5 t 5 5 a) Avalablty functons plus deadlne and external ponts delta.5.5 Operatng pont wth mproved Ts alfa.8.6 F.. alfa Co overhead Total bandwdth Cs 5 a) (α, ) soluton space Cs Cs Ts max(cs) (Cs,Ts) soluton space Operatonal pont wth mproved Ts 5 Ts b) (C s,t s) soluton space Fgure 8. oluton spaces for the gven example 7. YNCHRONIZATION AMONG TAK In many practcal applcatons there s the need for synchronzaton n the access to shared resources that consttute crtcal sectons. There are several protocols specfcally developed for real-tme systems that allow boundng prorty nversons and blockng ntervals as well as preventng deadlocks. Our ssue here s to dscuss whether and how can such protocols be used wthn the temporally parttoned framework consdered n ths paper, wth fxed prortes local schedulers. 6 8 Ts b) cost functon wth respect to Ts Fgure 9. Avalablty and cost functons for the example In ths paper, we consder ntra-server synchronzaton, only,.e. synchronzaton among tasks of the same applcaton (Fgure ), whch execute wthn the same partton and requre access to resources that are exclusve to that applcaton (applcaton resources). These tasks eventually cause blockng to one another, the duraton of whch depends on the specfc synchronzaton protocol used. uch blockng (applcaton level blockng - B a ) conssts on the executon of crtcal sectons of lower prorty tasks,.e. the blockng tasks, whle there are hgher prorty ones pendng,.e. the blocked tasks. Therefore, the applcaton level blockng B a causes nterference to the blocked tasks, delayng them, and consequently must be consdered n the respectve level submtted load H (t), whch must be updated accordngly (Equaton 5). server- τ τ ( t + J ) / T a H ( t) = B + * C (5) shared resource = blockng of τ server suspenson Fgure. Intra-server blockng. server- tme

8 Moreover, we can state Lemma wth respect to the propertes of the synchronzaton protocols. Lemma. All propertes of any synchronzaton protocol appled to shared applcaton resources wthn a server, reman the same as establshed n a non-parttoned processng system, namely the number and duraton of prorty nversons and deadlock avodance. Proof: To prove Lemma notce that semaphore lockng and unlockng and the related blockng take place wthn the applcaton, only, durng the executon of the respectve server. When the server s suspended untl the next perodc nstance, the status of all semaphores stays unaltered. Therefore, from the pont of vew of the synchronzaton protocol, whchever t s, these perods of tme have no mpact on the protocol propertes, except on the nflaton of the blockng that, obvously, can now extend across consecutve server nstances. However, ths extra delay due to server suspenson s the same that the task would suffer ust because of executng wthn a partton and that s already accounted for n the avalablty functon of the server and should not be consdered blockng. Thus, excludng ths suspenson delay, the applcaton level blockng (B a ) that a task can suffer s the same as t would be f the respectve applcaton was executed n a dedcated system,.e. non-parttoned. Lemma plus the updated load functon lead to Theorem : Theorem. The response tme analyss and the server desgn method proposed prevously n ths paper apply equally when the task model consders applcaton level blockng as long as: the updated load functon s used (Equaton 5); the applcaton tasks can always be suspended at any pont of ther executon when the server capacty s exhausted or the server s preempted as determned by the system scheduler. Proof: The proof of ths theorem can be easly establshed by notng that, from a response tme analyss pont of vew, and gven Lemma, the blockng ust represents extra load that s accounted for n the updated load functon. Thus, all the reasonng behnd the prevously shown analyss that s based on the matchng between server avalablty and submtted load stll apples. The second requrement guarantees that there s no couplng between the applcaton level synchronzaton protocol and the system level schedulng. Ths assures that a server does not execute more than ts capacty n any nstance, not causng/sufferng extra nterference on/from other servers runnng n the system, thus behavng accordng to our perodc model. The second requrement of Theorem may conflct wth synchronzaton protocols based on non-preempton. Thus, ether preemptve protocols are used, e.g. Prorty Inhertance (PIP), Prorty Celng (PCP) and tack Resource Protocols (RP), or the executon platform must be able to separate preempton wthn the applcaton from preempton at the system level and assurng that the latter s always possble, even when the server s executng a local non-preemptve applcaton secton. 8. CONCLUION Ths paper consdered the case n whch an applcaton composed by several tasks executes wthn a perodc server wth a fxed prortes local schedulng polcy. Two man results are presented, the response tme analyss for such tasks and the desgn of the server to allow fulfllng the applcaton tme constrants usng the least system resources. The former contrbuton s a generalzaton of the well-known worst-case response tme analyss for fxed-prorty systems [] that copes wth the lmted processor avalablty delvered by a server and t s based on the analyss prevously developed for traffc schedulng wthn the asynchronous messagng system of FTT-CAN []. It s also equvalent to the one n [] but ncludes a more complete task model wth release tter, deadlnes earler than perods and synchronzaton blockng. The latter contrbuton goes n the same drecton as that of [] but presents a dfferent method that leads to a more favorable compromse between tghtness of the soluton (slghtly lower) and complexty of the process (substantally lower). Agan, the presented method also bears smlartes wth the one n [] but, not only t ncludes a more complete task model as referred above, as t also presents an heurstc to deduce the server parameters that mnmze resource utlzaton takng nto account the context swtch overhead at the system level. Moreover, we present a fnal optmzaton step that s applcable to our method as well as to [] and [], whch reduces the requred server utlzaton when a lnear bound to the server servce s used. A spn-off result that seems to have potental for more generalzed use s the utlzaton of external ponts n the response tme analyss. Further work wll address the case of nter-server synchronzaton blockng as well as non-preempton at the applcaton level and ts mpact at the system level. 9. ACKNOWLEDGMENT The authors would lke to thank E. Bn and G. Lpar for the frutful dscussons concernng the server desgn problem that helped n mprovng the respectve part of ths paper.. REFERENCE [] Almeda L., P. Pedreras, J. A. Fonseca, The FTT-CAN Protocol: Why and How, IEEE Transactons on Industral Electroncs, 9(6), December. [] Almeda L., J. Fonseca. Analyss of a mple Model for Non- Preemptve Blockng-Free chedulng. Proc. of ECRT (EUROMICRO Conf. on Real-Tme ystems). Delft, Holland. June. [] Audsley, N., A. Burns, M. 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