Improvements in the configuration of Posix b scheduling

Transcription

1 Improvements n the confguraton of Posx b schedulng Matheu Grener, Ncolas Navet To cte ths verson: Matheu Grener, Ncolas Navet. Improvements n the confguraton of Posx b schedulng. Ncolas Navet and Françose Smonot-Lon and Isabelle Puaut. 15th Internatonal Conference on Real- Tme and Network systems - RTNS 2007, Mar 2007, Nancy, France. pp , 2007, Proceedngs of the 15th conference on Real-Tme and Network Systems - RTNS 07. <nra > HAL Id: nra Submtted on 28 Aug 2007 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 Improvements n the confguraton and analyss of Posx b schedulng Matheu Grener Ncolas Navet LORIA-INRIA Campus Scentfque, BP Vandoeuvre-lès-Nancy- France {grener, nnavet}@lora.fr Abstract Posx b complant systems provde two wellspecfed schedulng polces, namely sched_rr (Round- Robn lke) and sched_ffo (FPP lke). Recently, an optmal prorty and polcy assgnment algorthm for Posx b has been proposed n the case where the quantum value s a system-wde constant. Here we extend ths analyss to the case where quanta can be chosen on a task-per-task bass. The algorthm s shown to be optmal wth regards to the power of the feasblty test (.e. ts ablty to dstngush feasble and non feasble confguratons). Though much less complex than an exhaustve exploraton, the exponental complexty of the algorthm lmts ts applcablty to small or medum-sze problems. In ths context, as shown n the experments, our proposal allows achevng a sgnfcant gan n feasblty over FPP and Posx wth system-wde quanta, and therefore usng the computatonal resources at ther fullest potental. 1. Introducton Context of the paper. Ths study deals wth the schedulng of real-tme systems mplemented on Posx b complant Operatng System (OS). Posx b [7], prevously known as Posx4, defnes realtme extenson to Posx manly concernng sgnals, nterprocess communcatons, memory mapped fles, synchronous and asynchronous IO, tmers and schedulng (a recap of Posx s features related to schedulng s gven n 2.1). Ths standard has become very popular and most of today s OS conform, at least partally, to t. Problem defnton. Posx b complant OSs provde two schedulng polces sched_ffo and sched_rr, whch under some restrctons dscussed n 2.1, are respectvely equvalent to Fxed Preemptve Prorty (FPP) and Round-Robn (RR for short). Thus, under Posx b, each process s assgned both a prorty, a schedulng polcy and, n the case of Round-Robn, a quantum. At each pont n tme, one of the ready processes wth the hghest prorty s executed, accordng to the rules of ts schedulng polcy (e.g. yeldng the CPU after a quantum under RR). The problem addressed here s to assgn prortes, polces and quanta to tasks n such a way as to respect deadlne constrants. For FPP alone, the well-known Audsley algorthm [2] s optmal. A smlar algorthm exsts for both RR and FPP n the case of a system-wde quantum [6]. Here we consder the case where quanta can be chosen on a task-per-task bass. As t wll be seen n 3.2, the complexty of the problem s such that an exhaustve search s usually not feasble even on small sze problems. For nstance, a task set of cardnalty 10 wth quanta chosen among 5 dfferent values requres to analyze the feasblty of more dfferent confguratons (see 3.2). Contrbutons. Tradtonally, the RR polcy s only consdered useful for low prorty processes performng some background computaton tasks when nothng more mportant s runnng. In ths paper, as we dd n [10, 6], we argue that the combned use of RR and FPP allows to successfully schedule a large number of systems that are unschedulable wth FPP alone. The contrbuton of the paper s twofold, frst we propose an algorthm for assgnng prortes, polces and quanta that s optmal n the sense that f there exsts at least a feasble soluton 1, then the algorthm wll return a feasble soluton. The algorthm beng an extenson of the classcal Audsley algorthm [2] and the Audsley-RR- FPP from [6], we name t the Audsley-RR-FPP algorthm. The worst-case complexty of the algorthm s assessed and a set of optmzatons are proposed to reduce the search space. The second contrbuton of the paper s that we gve further evdences that the combned use of both FPP and RR s effectve - especally when quanta can be chosen for each ndvdual task - for fndng feasble schedules even when the workload of the system s hgh. Related work. We dentfy two closely related lnes of research: schedulablty analyses and prorty assgnment. 1 We call here a feasble soluton, a soluton that successfully passes a schedulablty test verfyng property 2 (see 2.5). In the followng, we make use of the response tme bound analyss derved n [9].

3 Audsley n [2, 3] proposes an optmal prorty assgnment algorthm for FPP, that s now well-known n the lterature as the Audsley algorthm. Later on n [5], ths algorthm has been shown to be also optmal for the non-preemptve schedulng wth fxed prortes. The problem of best assgnng prortes and polces under Posx b was frst tackled n [9] but the soluton reles on heurstcs and s not optmal n the general case. Then, n [6], an optmal soluton s proposed for the case where the quantum value s a system-wde constant. As n [6], the problem addressed here s dfferent than n the plan FPP case because the use of RR leads to the occurrence of schedulng anomales, whch are sometmes counter-ntutve. For nstance, as t wll be seen n 2.5, ncreasng the quantum value for a task can leads sometmes to a greater worst-case response tme for ths task. Smlarly, decreasng the set of hgher prorty tasks, can ncrease the response tme (see [6]). Ths prevents us from usng the proposed prorty and assgnment algorthm wth the schedulablty assessed by smulaton, or wth a feasblty test that would not possess some specfc propertes dscussed n 2.5. Indeed there would be cases where the algorthm would dscard schedulable assgnments and thus not be optmal. In ths study, feasblty s assessed by the analyss publshed n [9], whch ensures that the computed response tme bounds decrease when the set of hgher prorty tasks s reduced. Ths property enables us to use an Audsley-lke algorthm for the assgnment that wll be shown to be optmal wth regard to the power of the test, that s ts ablty to dstngush feasble or non feasble confguratons. Organsaton. Secton 2 summarzes the man features of the schedulng under Posx b and ntroduces the model and notatons. In secton 3, we present the optmal prorty, polcy and quantum assgnment Audsley-RR- FPP algorthm. Effcency of the proposal s then assessed n secton Schedulng under Posx b: model and basc propertes In ths secton we present the system model and summarze the man features related to schedulng of Posx b. We then present the assumptons made n ths study and derve some basc propertes of the schedulng under Posx b that wll be used n the subsequent sectons Overvew of Posx b schedulng In the context of OS, we defne a task as a recurrent actvty whch s ether performed by repettvely launchng a process or by a unque process that runs n cycle. Posx b specfes 3 schedulng polces: sched_rr, sched_ffo and sched_other. These polces apply on a process-by-process bass: each process run wth a partcular schedulng polcy and a gven prorty. Each process nherts ts schedulng parameters from ts father but may also change them at run-tme. sched_ffo : fxed preemptve prorty wth Frst-In Frst-Out orderng among same-prorty processes. In the rest of the paper, t wll be assumed that all sched_ffo tasks of an applcaton have dfferent prortes. Wth ths assumpton and wthout change durng run-tme sched_ffo s equvalent to FPP. sched_rr : Round-Robn polcy (RR) whch allows processes of the same prorty to share the CPU. Note that a process wll not get the CPU untl a hgher prorty ready-to-run processes are executed. The quantum value may be a system-wde constant (e.g. QNX OS), process specfc (e.g. VxWorks OS) or fxed for a gven prorty nterval. sched_other s an mplementaton-defned scheduler. It could map onto sched_ffo or sched_rr, or also mplement a classcal Unx tme-sharng polcy. The standard merely mandates ts presence and ts documentaton. Because we cannot rely on the same behavour of sched_other under all Posx complant OSs, t s strongly suggested not to use t f a portablty s a matter of concern. We wll not consder t n our analyss. Assocated wth each polcy s a prorty range. Dependng on the mplementaton, these prorty ranges may or may not overlap but most mplementatons allow overlappng. Note that these prevously explaned schedulng mechansms smlarly apply to Posx threads wth the system contenton scope as standardsed by Posx c standard [7] System model The actvtes of the system are modeled by a set T of n perodc and ndependent tasks T = {τ 1, τ 2,..., τ n }. Each task τ s characterzed by a tuple (C, T, D ) where each request of τ, called an nstance, has an executon tme of C, a relatve deadlne D and a perod equal to T tme unts. One denotes by τ,j the j th release of τ. As usual, the response tme of an nstance s the tme elapsed between ts arrval and ts end of executon. Under Posx b, see 2.1, each task τ possesses both a prorty p and a schedulng polcy sched. In ths study, we choose the conventon the smaller the numercal value, the hgher the prorty. In addton to the prorty, under RR, each task τ s assgned a quantum value ψ. The prorty and schedulng polcy assgnment P s fully defned by a set of n tuples (τ, p, sched P ) (.e. one for each task). A quantum assgnment under P, denoted by Ψ P, defnes the set of quantum values ψ ΨP where ψ ΨP s the quantum of τ. The whole schedulng s fully defned by the tuple (P, Ψ P ) whch s called a confguraton of the system. Under assgnment P, the set of tasks T s parttoned nto separate layers, one layer for each prorty level j

4 where the layer Tj P s the subset of tasks assgned to prorty level j. Under P, Thp(j) P (resp. T lp(j) P ) denotes the set of all tasks possessng a hgher (resp. lower) prorty than j. A layer n whch all tasks are scheduled wth RR (resp. FPP) s called an RR layer (resp. FPP layer). In the followng, P or Ψ P wll be omtted when no confusons are possble. A lst of the notatons s provded n appendx at the end of the paper. In the followng, a task τ s sad schedulable under assgnment (P, Ψ P ) f ts response tme bound, as computed by the exstng Posx b schedulablty analyss [9], s no greater than ts relatve deadlne (.e. maxmum duraton allowed between the arrval of an nstance and ts end of executon). The whole system s sad schedulable f all tasks are schedulable. Note that the test presented n [9] s suffcent but not necessary, there are thus task sets whch won t be classfed as schedulable whle there exst confguratons under whch no deadlnes are mssed Assumptons In ths study, as explaned n 2.1, only sched_ffo and sched_rr are consdered for portablty concern. Due to the complexty of assgnng prortes and schedulng polces, the followng restrctons are made: 1. context swtch latences are neglected, but they could be ncluded n the schedulablty analyss of [9] as classcally done (see, for nstance, [11]), 2. snce a prorty level wthout any tasks has no effect on the schedulng, we mpose the prorty range to be contguous, 3. two tasks havng dfferent schedulng polces have dfferent prortes,.e., j, sched sched j = p p j, 4. all sched_ffo tasks must possess dstnct prortes (sched = sched j = sched_ffo = p p j ). Wth ths assumpton and wthout prorty change at run-tme, sched_ffo s equvalent to fxed-preemptve prorty (FPP). Thus, several tasks havng the same prorty are necessarly scheduled under sched_rr polcy, 5. the quantum value can be chosen on a task-per-task bass n the nterval [Ψ mn, Ψ max ], where Ψ mn and Ψ max are natural numbers whose values are OSspecfc constrants or chosen by the applcaton desgner Schedulablty analyss under Posx: a recap [9] In ths paragraph, we summarze the schedulablty analyss [9] of a confguraton (P, Ψ P ) under Posx. Tasks scheduled under Posx can be descrbed as a superposton of prorty layers [9]. At each pont n tme, one of the ready nstances wth the hghest prorty (let s say p ) s executed as soon as and as long as no nstances n the hgher prorty layers (nstances of tasks n T hp(p)) are pendng. Insde each prorty layer, nstances are scheduled ether accordng to FPP or RR wth the restrctons that all nstances belongng to the same layer have the same polcy. FPP polcy s acheved when a ready nstance τ,j s executed when no hgher prorty nstances s pendng. Under RR, a task τ has repeatedly the opportunty to execute durng a tme slot of maxmal length ψ ΨP. If the task has no pendng nstance or less pendng work than the slot s long, then the rest of the slot s lost and the task has to wat for the next cycle to resume. The tme between two consecutve opportuntes to execute may vary, dependng on the actual demand of the others tasks, but t s bounded by ψ ΨP = τ k T P p ψ ΨP k n any nterval where the consdered task has pendng nstances at any moment. In [9], worst-case response tme bounds for prorty layers have been derved n a way that s ndependent from the schedulng polces used for each layer. Ths analyss s based on the concept of majorzng work arrval functons, whch measure a bound on the processor demand, for each task, over an nterval startng at a generalzed crtcal nstant. The majorzng work arrval functon on an nterval of length t for a perodc task τ s: t s (t) = C. (1) T The worst-case response tme bound can be expressed as max j<j (e,j a,j ), (2) where j = mn{j e,j a,j+1 }, where a,j s the release of the j th nstance of τ after the crtcal nstant and e,j s a bound on the executon end of ths nstance. Snce τ s a perodc task, a,j = (j 1) T (j = 1, 2,...). If τ s n an FPP layer, then e,j = mn{t > 0 s (t) + s,j = t}, (3) where s (t) = τ k T s P k (t) s the demand from hp(p ) hgher prorty tasks (.e. task n Thp(p P ) and s ),j = j =1 C s the demand from prevous nstances and the current nstance of τ. If τ s n an RR layer, then e,j = mn{t > 0 Ψ (t) + s,j = t}, (4) where the demand from hgher prorty tasks and of all other tasks of the RR layer s: ( ) s,j Ψ (t) = mn (ψ ΨP ψ ΨP ψ ΨP ) + s (t), s (t), (5) where ψ ΨP ψ ΨP s the sum of the quanta of all other tasks of the RR layer and s (x) = max (s (u) + s (u + x) + s (u + x) u), u 0 (6)

5 where s (u+x) = τ k T P p \{τ } s k(u+x) s the demand from other tasks than τ n T P p. The algorthm for computng the worst-case response tme bounds can be found n [9]. It s to stress that ths schedulablty analyss s suffcent but not necessary; some task sets may fal the test whle they are perfectly schedulable. Ths wll certanly nduce conservatve results but the approach developed here remans vald wth another - better - schedulablty test as long as t s suffcent and possesses the propertes descrbed n Schedulng under Posx b: basc propertes Under FPP, as well as under RR, any hgher prorty task wll preempt a lower prorty task thus the followng propertes hold for any task τ : 1. all ready nstances, wth hgher prortes than p, wll delay the end-of-executon of the nstances of τ. It s worth notng that ths delay s not dependent on the relatve prorty orderng among these hgher prorty nstances and ther quantum values, 2. lower prorty nstances, whatever ther polcy, wll not nterfer wth the executon of nstances of τ and thus won t delay ther end-of-executon. These two propertes ensure that the followng lemma, whch s well-known n the FPP case, holds. Lemma 1 [3] The worst-case response tme of an nstance of τ only depends on the set of same prorty tasks, the values of ther quantum and the set of hgher prorty tasks. The relatve prorty order among hgher prorty tasks and the values of ther quantum has no nfluence. However, despte lemma 1 holdng, schedulng under RR leads to schedulng anomales. Indeed, schedulng under Posx s often counter-ntutve. For nstance, t has been shown n [4], that early end-of-executons can lead to mssed deadlnes n confguratons that would be feasble wth WCETs. Smlarly, removng a task wth a hgher prorty than τ j may lead to ncreased response tmes for τ (see fgures 1 and 2 n [6]). Here, we hghlght that ncreasng the quantum sze of a task can ncrease ts response tme. Fgures 1 and 2 present the schedulng of task set T = {τ 1, τ 2 } where τ 1 = (C 1 = 2, T 1 = 5) and τ 2 = (4, 10). All the tasks belong to the same layer and the chosen quantum assgnments are Ψ = {ψ 1 = 2, ψ 2 = 2} (fgure 1) and Ψ = {ψ 1 = 2, ψ 2 = 3} (fgure 2). As t can be seen on fgures 1 and 2, surprsngly the response tme of τ 2 s 6 wth a quantum of 2 and 8 wth 3. However, wth the schedulablty analyss used n ths study, property 1 holds and wll be used to restran the search space n secton 3. A proof s gven n appendx A. Property 1 Let τ be a task n a RR layer, ncreasng (resp. reducng) ts quantum value, whle reducng (resp. τ1, ψ 1 = τ 2, ψ 2 = 2 t Fgure 1. Schedulng of task set T = {τ 1, τ 2 } wth Round-Robn and quantum assgnment Ψ = {ψ 1 = 2, ψ 2 = 2}. τ 1, ψ 1 = τ 2, ψ 2 = 3 t Fgure 2. Schedulng of task set T = {τ 1, τ 2 } wth Round-Robn and quantum assgnment Ψ 1 = {ψ 1 = 2, ψ 2 = 3}. ncreasng) the quantum value of the other tasks of ts RR layer, dmnshes (resp. ncreases) the response tme bound of τ computed wth the chosen schedulablty analyss. To be optmal, the Audsley algorthm requres that the schedulablty test fulflls some propertes (see 3.3). In partcular, removng a task wth a hgher prorty must not lead to ncreased response tmes. In the case of Posx b, ths mposes constrants on the schedulablty test whch must fulfll property 2. Property 2 Let τ be a task n RR or FPP layer, reducng ts set of hgher and same prorty tasks, whle keepng the quantum allocaton unchanged wthn ts Round- Robn layer (f τ s scheduled under RR), dmnshes or leaves unchanged the response tme bound of τ computed wth the chosen schedulablty analyss. It has been shown n [6] that the conservatve response tme bound computed wth [9] ensures that property 2 holds. The proof, gven n [6] n the context of a unque sysrem-wde quantum value, s stll vald when dfferent values for the quanta are possble. As t wll be shown n secton 3, a schedulablty test whch ensures that property 2 s verfed, allows to use an extenson of the Audsley algorthm and preserves ts optmalty wth regards to the ablty of the test to dstngush between feasble and nonfeasble solutons (.e., what s called the power of the test n the followng).

6 3. Optmal assgnment algorthm wth taskspecfc quanta We present here an optmal prorty, schedulng polcy and quanta assgnment for Posx b systems when the feasblty s assessed wth schedulablty analyss whch verfes property 2 descrbed n 2.5. Ths algorthm heavly reles on both the Audsley algorthm and the algorthm prevously proposed for system-wde quantum values (called Audsley-RR-FPP n [6]). Here we extend prevous works to the case where quanta can be chosen on a task-per-task bass, the correspondng algorthm s named the Audsley-RR-FPP. Wth the assumpton made n secton 2, the polcy s mpled by the number of tasks havng the same prorty level: should only one task be assgned prorty level then ts polcy s FPP (.e. a RR layer of cardnalty 1 s strctly equvalent to an FPP layer, see 2.1), otherwse the polcy s necessarly RR. The problem s thus reduced to assgnng prortes and quanta to tasks n a RR layer Audsley-RR-FPP algorthm In the same way as the orgnal Audsley algorthm (abrdged by AA n the followng), the dea s to start assgnng the prortes from the lowest prorty n to the hghest prorty 1 (lne 3 n algorthm 1). The dfference wth AA, s that, at each prorty level, the algorthm s not lookng for a sngle task but for a set of tasks (lne 5). For each such set of tasks, our algorthm examnes all possble quantum assgnments untl t fnds one sutable one. Underlyng dea. The underlyng dea of the algorthm s to move, when needed, the maxmum amount of workload to the lower prorty levels and to schedule the tasks under RR. When an nstance τ,j s assgned the same prorty as τ k,h and both are scheduled under RR, τ,j can delay τ k,h less than f τ,j would be scheduled wth a hgher prorty. The same argument holds for the delay nduced by τ k,h to τ,j. Thus, as llustrated wth an example n [10], where a task set that s not feasble under FPP alone, becomes feasble wth RR. Of course, n the general case, combnng the use of both polces s the most effcent and, as t wll be shown, leads to an optmal prorty and polcy assgnment. Step of the algorthm. For each prorty level (lne 3), the Audsley-RR-FPP* algorthm attempts to fnd a schedulable subset T n subset R (lne 5) where R s made of all the tasks whch have not been yet assgned a prorty, a polcy and a quantum. The algorthm tres all possble subsets of R, one by one, and all possble quantum assgnments for each subset untl a schedulable confguraton s obtaned or all confguratons have been consdered. In the latter case, the system s not schedulable (lnes 7-8). Otherwse, we have found a schedulable subset, denoted by T, whch, n the RR case, possesses quantum assgnment {ψ k } τk T (lnes 7 and 8). Precsely, Input: task set T = {τ 1, τ 2..., τ n } Result: schedulable prorty, schedulng polcy and quantum assgnment P k = (P, Ψ k ) Data: : prorty level to assgn R: task-set wth no assgned prorty P: partal prorty and polcy assgnment Ψ P : partal quantum allocaton R = T ; P = ; for = n to 1 do try to assgn prorty : search a schedulable subset of tasks T under quantum allocaton {ψ k } τk T n R f no subset T s schedulable at prorty then falure, return partal assgnement: return (P, Ψ P ); else let T a schedulable subset at prorty wth quantum allocaton {ψ k } τk T ; assgn prorty, polcy and quantum: f #T = 1 then P = P {(τ k,, sched_ffo)} τk T ; else P = P {(τ k,, sched_rr)} τk T ; Ψ P = Ψ P {ψ k } τk T ; end remove T from R: R = R\T ; end f R = then return (P, Ψ P ); end Algorthm 1: Audsley-RR-FPP algorthm wth taskspecfc quantum.

7 > : prorty greater than = : prorty d = = > = b e a f g h = > > = > c depth 0 τ 1 depth 1 τ 2 depth 3 τ 3 depth 4 Fgure 3. Search tree constructed n the search of a feasble subset of R = {τ 1, τ 2, τ 3 } at prorty. For nstance, node b models the partal prorty assgnment where τ 1 s assgned prorty whle node c means that τ 1 s assgned a greater prorty. T s schedulable when all tasks of T are feasble at prorty whle all tasks wthout assgnment (.e., tasks n R\T ) have a prorty greater than. At each step, at least one task s assgned a prorty and a polcy (lnes 11 to 17). Note that, when RR s used at least once, less than n prorty levels are needed (early ext on lne 21). Lookng for the set of schedulable tasks T. There are 2 #R possble subsets T of R that can be assgned prorty level (lne 5). Snce the quantum can take ψ = ψ max ψ mn + 1 dfferent values, there are ψ #T dfferent quantum assgnments for each subset T. Frst, we explan the basc exhaustve tree-search used to set prortes. Then, we explan how we use a smlar search to choose the quantum assgnment for each possble set T. A method that speeds-up the search by prunng away subtrees that cannot contan a soluton s provded n 3.2. A bnary tree structure reflects the prorty choces and the search for the schedulable subset s performed by explorng the tree. In the followng, we call prorty-searchtree the search tree modelng the prorty choces. As an llustraton, fgure 3 shows the prorty-search-tree correspondng to the set R = {τ 1, τ 2, τ 3 }. Each edge s labeled ether wth = (.e., prorty equal to ) or > (.e., prorty greater than ). A label = (resp. > ) on the edge between vertces of depth k and k + 1 means that the (k + 1) th task of R belongs to the layer of prorty (resp. belongs to a layer of prorty greater than ). Thus, a vertex of depth k models the choces performed for the k frst tasks of R. For nstance, on fgure 3, the vertex e mples that tasks τ 1 belongs to layer of prorty whle task τ 2 does not. Each leaf s a complete assgnment for prorty level, for nstance leaf g corresponds to set T = {τ 1, τ 2 }. The search s performed accordng to a depth-frst strategy. The algorthm consders the frst chld of a vertex that appears and goes deeper and deeper untl a leaf s reached,.e., untl the set T s fully defned. When a leaf s reached, the schedulablty of T s assessed. If T s feasble, the algorthm returns, otherwse, t backtracks tll the frst vertex such that not all ts chld vertces have been explored. To assess the schedulablty of T, all possble quantum assgnments are successvely consdered. In the same manner as for the prorty allocaton, a tree -called quantum-search-tree- reflects the choces for quantum values. A depth-frst strategy s used as well to explore the search space. In ths case, a node has ψ chldren where each chld models a dfferent quantum value. Here, we label the edge between vertces of depth k and k + 1 wth the quantum value of the (k + 1) th task of T. Thus, a vertex of depth k models the choces performed for the k frst tasks of T Complexty and mprovements Sze of the search space. Assgnng n tasks to dfferent non-empty layers s lke subdvdng a set of n elements nto non-empty subsets. Let k be the number of layers. The number of possble assgnments s equal, by defnton, to the the Strlng number of the second knd (see [1], page 824): 1 k! k ( k ( 1) (k ) =0 ) n, «k where s the bnomal coeffcent,.e., the number of ways of pckng an unordered subset of elements n a set of k elements. The complexty depends on the number of tasks scheduled under RR snce ther quantum values have to be chosen. When there are k layers, at least n k + 1 tasks are n an RR layer (.e., n k + 1 tasks n a sngle RR layer and one task n each of the remanng k 1 FPP layers) and up to max(n, 2(n k)) (.e., tasks are evenly dstrbuted among RR layers). Snce the quantum can take ψ = ψ max ψ mn + 1 dfferent values, there are between ψ n k+1 and ψ max(n,2(n k)) dfferent quantum assgnments for a confguraton of k layers. In addton, n tasks can be subdvded nto k = 1, 2,..., n many layers and there are k! dfferent possble prorty orderngs among the k prorty layers. Thus, a lower bound for the search space of the problem of assgnng prorty, polcy and quantum for a set of n tasks s n k=1 =0 k ( k ( 1) (k ) ) n ψ n k+1. In a smlar way, we derve an upper bound by replacng ψ n k+1 wth ψ max(n,2(n k)). For nstance, as can be seen on fgure 4, the sze of the search space comprses about schedulng confguratons for a set of 10 tasks. The search space grows

8 1e+90 1e+80 1e+70 1e+60 1e+50 1e+40 1e+30 1e+20 1e+10 1 Sze of the problem (lower bound) Audsley-RR-FPP task-specfc quanta Audsley-RR-FPP system-wde quantum Number of tasks (n) Fgure 4. Complexty of the problem for a number of tasks varyng from 5 to 50 when the quantum value can be chosen n the nterval [1, 5]. more than exponentally, thus an exhaustve search s not possble n practce n a wde range of real-tme problems. Audsley-RR-FPP. Our algorthm looks at each prorty level for a subset T n R whch s schedulable at prorty (lne 5). Snce at least one task s assgned to each prorty level, the number of tasks belongng to R when dealng wth prorty level s lower than or equal to. In addton, we know that there are ψ k dfferent quantum assgnments for a subset of k tasks. Thus, at each prorty level, the algorthm examnes ( ) j=1 ψ j = j ( ψ + 1) 1 assgnments n the worst-case. Thus, for prorty level from 1 to n, the algorthm consders n the worst-case a number of assgnments gven by: n ( ψ + 1) 1 = =1 1 ( ψ + 1)n+1 1 ( ψ + 1) (n + 1) Ths complexty for a varyng number of tasks s shown on fgure 4, for nstance, for a set of 10 tasks wth ψ mn = 1 and ψ max = 5 t s approxmately equal to Fgure 4 shows also the sze of the search space and, for comparson, the worst-case complexty of the soluton proposed n [6] n the case where the quantum sze s a system-wde constant. Although we acheve a great complexty reducton wth regards to an exhaustve search, the complexty remans exponental n the number of tasks. Thus, n practce, our proposal s not suted for large-sze task sets that would, for nstance, be better handled by heurstcs gudng the search towards promsng parts of the search space. Ths s left as future work. Complexty reducton. As seen before, the Audsley- RR-FPP performs an exhaustve search for each prorty level. To a certan extent, t s possble to reduce the number of sets that are to be consdered. Indeed, the property 3 gven n ths paragraph shows that t s possble to dentfy prorty and polcy assgnments that are not schedulable whatever the quantum allocaton. Thanks to property 2 and property 3, one can dentfy and prune away branches of the prorty-search-tree whch necessarly lead to subsets T that are not schedulable whatever the quantum assgnments. Furthermore, wth property 3, one can reduce n a smlar manner the number of quantum assgnments to consder for a partcular subset T n a quantum-searchtree. Wth the basc algorthm explans n 3.1, feasblty of a prorty allocaton s assessed at the leafs when all tasks have been gven a prorty by testng all quantum assgnments. The dea s here to evaluate feasblty at ntermedate vertces as well, by assgnng a prorty lower than to the tasks for whch no prorty choce has been made yet. Under that confguraton, f a task τ whch s assgned the prorty s not schedulable whatever the quantum assgnment, there s no need to consder the chldren of ths vertex. Indeed, from property 2, snce the prorty assgnment of the chldren of ths node wll ncrease the set of same or hgher prorty tasks, the response tme of τ cannot decrease. Thus, all chld vertces corresponds to prorty assgnments that are not schedulable. Now, t remans to dentfy prorty and polcy assgnments that are not schedulable whatever the quantum allocaton. The followng property, proven n appendx A.3, can be stated. Property 3 Let S be a schedulablty test for whch property 2 holds. Let T be a task set and P be a global prorty and polcy assgnment. Let τ be a task wth the maxmum quantum value ψ max n an RR layer. Let the quantum values of all other tasks n the RR layer be set to the mnmum ψ mn. If the response tme bound of τ, computed wth S, s greater than ts relatve deadlne, then, whatever the quantum assgnment under P, τ wll reman unschedulable wth S. Thus, at each vertex of the prorty search tree, a prorty assgnment P s not feasble whatever the quantum assgnment, f a task τ k whch has a prorty s not feasble wth the quantum allocaton gven n property 3. Smlarly, we can cut branches when explorng the quantum-search-tree of a set T. The dea s agan to evaluate feasblty at ntermedate vertces. Snce an ntermedate vertex models a partal quantum assgnment for a set T, we assgn the lowest quantum value to each task n T whch has no quantum assgned yet. In that case, f a task τ k for whch the quantum has already been set at ths vertex s not schedulable, then there s no need to consder the chldren of ths vertex. Indeed, gven property 1, the response tme of τ k can only ncrease when the the chldren of ths vertex are consdered. The fndng of ths paragraph allows a very sgnfcant decrease n the average number of confguratons tested by the Audsley-RR-FPP algorthm. For nstance, for task sets consttuted of 10 tasks, the algorthm examnes on average only about 4000 confguratons before comng up

9 wth a feasble soluton or concludng that the task set s unfeasble whle t would requre about tests otherwse Proof of optmalty Here we show that the Audsley-RR-FPP algorthm s optmal n the sense that f there s a prorty, polcy and quantum assgnment that can be dentfed as feasble by the schedulablty analyss, t wll be found by the algorthm. Let us frst remnd the followng theorem whch has been proven n [2, 3, 5, 9] for varous contexts of fxed prorty schedulng. Theorem 1 [3] Let (P, Ψ P ) be a schedulable confguraton up to prorty,.e. tasks that have been assgned the prortes from n to are schedulable. If there exsts a schedulable confguraton (A, Ψ A ), then there s at least one schedulable confguraton (Q, Ψ Q ) havng an dentcal confguraton as (P, Ψ P ) for prortes n to. From theorem 1, we can prove the optmalty of Audsley-RR-FPP. Indeed, f Audsley-RR-FPP happens to fal at level, the prorty, schedulng polcy and quantum assgnment (P, Ψ P ) provded by Audsley-RR-FPP leads to a schedulable soluton up to level + 1. Snce Audsley-RR-FPP performs an exhaustve search to assgn level, there cannot be any schedulable assgnment (Q, Ψ Q ) possessng the same assgnment as (P, Ψ P ) for prorty + 1 to n. Thus, from theorem 1, there s no schedulable assgnment. We gve here an ntutve proof of theorem 1, whch bascally s vald under Posx thanks to lemma 1 and property 2. It should be ponted out that theorem 1, and thus the optmalty result of Audsley-RR-FPP, does not hold where property 2 s not verfed by the schedulablty test. Theorem 1 holds f a schedulable confguraton (A, Ψ A ) can be transformed nto a schedulable confguraton (Q, Ψ Q ) for whch the confguraton s the same as (P, Ψ P ) for prorty to n. Ths transformaton can be done teratvely by changng the confguraton of certan tasks n (A, Ψ A ) to the confguraton they have n (P, Ψ P ). The procedure s the followng: for prorty level k from n to, assgn n (A, Ψ A ) the prorty k+n to the tasks of prorty k n (P, Ψ P ) (.e., the set T P k ) and set ther quantum value to ther values ψ P n Ψ P ( τ j Tk P, pa j = p P j + n, sched A j = sched P j and ψ ΨA j = ψ ΨP j ). Snce at each step, tasks n T P k have the same quantum assgnment, the same set of hgher and equal prorty tasks under the current confguraton (A, Ψ A ) as under (P, Ψ P ), they reman schedulable under (A, Ψ A ) by lemma 1. From property 2, the other tasks (T \ Tk P ) meet ther deadlne too snce the quantum assgnment and the set of hgher and same prorty task s reduced or stay unchanged under current confguraton compared to the ntal confguraton (A, Ψ A ). Note that n the proof the prorty range has been artfcally extended by addng n lower prorty levels n order to avod the case where a hgher prorty tasks s moved to a non-empty layer snce property 2 does not cover ths stuaton. 4. Expermental results Here our am s to quantfy the extent to whch usng task-specfc quanta enables us to mprove the schedulablty of the system by comparson 1) wth FPP and 2) wth system-wde quanta Expermental setup. In the followng experments, we only consder task sets that are unschedulable wth FPP alone. Snce we choose to consder perodc tasks wth deadlnes equal to perods (D = T ), we use the Rate Monotonc prorty assgnment, whch s optmal n that context. The global load U (.e., n C =1 T ) has to be necessarly greater than n (2 1/n 1) (from [8]) n order to be able to exhbt nonfeasble task sets. In the followng, we choose a quantum value of 1 for the system-wde quantum or, when taskspecfc quanta s consdered, a quantum value whch can be chosen n the nterval [1, 5]. The actual parameters of an experment are defned by the tuple (n, U). The utlzaton rate ( C T ) of each task τ s unformly dstrbuted n the nterval [ U n 0.9, U n 1.1] where n s the number of tasks. The computaton tme C s randomly chosen wth an unform law n the nterval [1, 30] and the perod T s upper bounded by 500. The results shown on fgure 5 have been obtaned wth 200 task sets randomly generated wth the aforementoned parameters Schedulablty mprovement over FPP and systemwde quanta Fgure 5 shows the percentage of task sets that are not schedulable wth FPP alone and become schedulable when usng the Audsley-RR-FPP (task-specfc quanta) and Audsley-RR-FPP (system-wde quanta - see [6]) algorthms that are both optmal n ther context. One observes that the mprovement wth task-specfc quanta s very mportant, at least 3 tmes better than wth a system-wde quantum. When the load s lower than 84%, a soluton s found n almost all cases, the percentage of successes remanng greater than 50% up to a load equal to 88%. As t was to be expected, when the load gets hgher, feasble schedulng soluton tends to rarefy. Our experments show that the combned used of RR and FPP wth process-specfc quanta allows to schedule a large number of task sets whch are nether schedulable wth FPP nor wth a system-wde quantum. It s worth notng that context swtch latences were neglected whle RR nduces more context swtches than FPP. Ths fact weakens to a certan extent our conclusons. A future work s to fnd the feasble quantum allocaton that mnmzes the global number of context swtches.

10 Audsley-RR-FPP task-specfc quanta Audsley-RR-FPP system-wde quantum [3] N.C. Audsley. On prorty assgnment n fxed prorty schedulng. Inf. Process. Lett., 79(1):39 44, % of schedulable task sets Load Fgure 5. Percentage of task sets unschedulable wth DM whch become schedulable under Posx usng the Audsley-RR-FPP (task-specfc quanta) and Audsley-RR-FPP (system-wde quanta - see [6]) algorthms. The CPU load ranges from 0.82 to The number of tasks s equal to Concluson In ths paper, we propose a prorty, polcy and quantum assgnment algorthm for Posx b complant OS that we named the Audsley-RR-FPP. We have shown ths algorthm to be optmal n the sense that f there s a feasble schedule usng FPP and RR that can be dentfed as such by the schedulablty test, t wll be found by the algorthm. A result yelds by the experments s that the combned used of FPP and RR wth process-specfc quanta enables to sgnfcantly mprove schedulablty by comparson wth FPP alone and wth system-wde quanta. Ths s partcularly nterestng n the context of embedded systems where the cost pressure s hgh, whch lead us to explot the computatonal resources at ther fullest. In terms of worst-case complexty, the algorthm greatly mproves upon an exhaustve exploraton of the search space but s stll exponental n the number of tasks n the worst-case. Therefore, t s not suted to large task sets and future work s needed to develop technques able to handle such systems. A future work s to take nto account context swtches and come up wth a way of assgnng quantum values n such a manner as to mnmze the context-swtch overhead. References [1] M. Abramowtz and I.A. Stegun. Handbook of Mathematcal Functons. Dover Publcatons (ISBN ), [2] N.C. Audsley. Optmal prorty assgnment and feasblty of statc prorty tasks wth arbtrary start tmes. Report YO1 5DD, Dept. of Computer Scence, Unversty of York, England, [4] R. Brto and N. Navet. Low-power round-robn schedulng. In Proc. of the 12th nternatonal conference on real-tme systems (RTS 2004), [5] L. George, N. Rverre, and M. Spur. Preemptve and non-preemptve real-tme unprocessor schedulng. Techncal Report RR-2966, INRIA, Avalable at rrrt/rr-2966.html. [6] M. Grener and N. Navet. Schedulng confguraton on Posx b systems. Techncal report, INRIA, to appear, [7] (ISO/IEC) :2004 and IEEE Std , 2004 Edton. Informaton technology portable operatng system nterface (POSIX R ) part 1: Base defntons. IEEE Standards Press, [8] C.L. Lu and J.W Layland. Schedulng algorthms for multprogrammng n hard-real tme envronnement. Journal of the ACM, 20(1):40 61, [9] J. Mgge, A. Jean-Mare, and N. Navet. Tmng analyss of compound schedulng polces : Applcaton to Posx1003.1b. Journal of Schedulng, 6(5): , [10] N. Navet and J. Mgge. Fne tunng the schedulng of tasks through a genetc algorthm: Applcaton to Posx1003.1b complant OS. Proc. of IEEE Proceedngs Software, 150(1):13 24, [11] K. Tndell. An extendble approach for analyzng fxed prorty hard real-tme tasks. Techncal Report YCS , Department of Computer Scence, Unversty of York, A. Proof of propertes 1 and 2 In ths appendx, we prove that the schedulablty analyss [9] ensures that propertes 1 and 3 hold. The frst paragraph s devoted to the study of the executon end e,j of τ,j computed wth [9] under two confguratons (P, Ψ P ) and (P, Ψ P ) that only dffer by ther quantum assgnment. Ths result s used n subsequent proofs. A.1. Executon end bound: basc propertes We compare bounds on the executon end of τ under the same prorty and polcy assgnment P wth two dfferent quantum allocatons. Let e,j and e,j be respectvely the executon end bound of τ under (P, Ψ P ) and under (P, Ψ P ). Snce τ s n an RR layer, e,j s computed wth equaton 4 of 2.4: e,j = mn{t > 0 Ψ (t) + s,j = t},

11 where (equaton 5 of 2.4) ı Ψ Ψ P s,j (t) = mn (ψ Ψ P ψ Ψ P «ψ Ψ P) + es (t), s (x), where ψ ΨP ψ ΨP s the sum of the quanta of all other tasks of the RR layer. Snce s,j, s (t) and s (x) are ndependent of the quantum assgnment (see 2.4), t s enough to compare the frst term of the mn() to decde whch task wll have the smallest response tme bound. Two cases arse: 1. s,j ψ Ψ P (ψ ΨP ψ ΨP s ) >,j (ψ Ψ P ψ Ψ ψ Ψ P) P then we conclude e,j e,j, 2. otherwse: s,j ψ ΨP (ψ ΨP and e,j e,j. ψ ΨP ) s,j ψ Ψ P (ψ Ψ P ψ Ψ P ), When s (x) s the mnmum, we have e,j = e,j. From ths fndng we can deduce that for any other assgnment Ψ P, f the two followng requrements are met: requrement 1: the quantum ψ Ψ P of τ n Ψ P s lower than or equal to ts quantum ψ ΨP under Ψ P, requrement 2: the sum of the quanta of all other tasks of the RR layer T P under Ψ P s greater than or equal to the one under Ψ P,.e., ψ Ψ P where ψ ΨP = τ k T ψ ΨP τ k ψ Ψ P ψ ΨP ψ ΨP s the sum of the quantum of all tasks of the RR layer T P allocaton Ψ P, then we have: s,j ψ Ψ P (ψ Ψ P s,j ψ Ψ P) ψ ΨP under quantum (ψ ΨP ψ ΨP ), (7) and thus τ,j, e,j e,j whch mples that the response tme bound of τ under (P, Ψ P ) s greater than or equal to the response tme bound under (P, Ψ P ). A.2. Proof of property 1 Snce the prerequstes of property 3 are exactly requrements 1 and 2 of A.1, the response tme bound of τ n property 3, s no less under (P, Ψ P ) than under (P, Ψ P ). Snce τ s not schedulable under (P, Ψ P ), t cannot be schedulable under (P, Ψ P ). A.3. Proof of property 3 We show that the bound on the executon end e,j for a task n an RR layer under P, s mnmum under P when the quantum of τ s equal to ψ max whle the quanta of the other tasks n the layer are set to ψ mn. Let Ψ P be the correspondng quantum assgnment where s,j ψ ΨP (ψ ΨP ψ ΨP ) = s,j ψ max ( τ k T P p \{τ } ψ mn ) and one notes that whatever a dfferent quantum assgnment Ψ P : s,j s,j ψ mn (ψ Ψ P ψ Ψ P) ψ max τ k T P p \{τ } ψ Ψ P snce, by defnton, ψ max ψ Ψ P and ψ mn ψ Ψ P k. From equaton 7, the executon end bound e,j of τ,j s thus mnmum wth Ψ P among the set of all possble quantum assgnments. Notatons T = {τ 1,..., τ n }: a set of n perodc tasks P: prorty and polcy assgnment Ψ P : a specfc quantum allocaton under assgnment P (P, Ψ P ) : a prorty, polcy and a quantum assgnment T P : subset of tasks assgned to prorty level under P Thp() P : subset of tasks assgned to a hgher prorty than under P Tlp() P : subset of tasks assgned to a lower prorty than under P ψ ΨP : Round-Robn quantum for task τ under Ψ P ψ ΨP : sum of the quanta of all tasks n layer T under Ψ P t s (t) = C T : majorzng work arrval functon on an nterval of length t for a perodc task τ s (t) = τ k T s P k (t): the demand from hgher hp(p ) prorty tasks under P s,j = j =1 C : the demand from prevous nstances plus demand of current nstance τ,j of τ s (x) = τ k T P p \{τ } s k(x) s the demand from all other tasks than τ at prorty level under assgnment P.