Mixed-Criticality Scheduling on Multiprocessors using Task Grouping

Transcription

1 Mxed-Crtcalty Schedulng on Multprocessors usng Task Groupng Jankang Ren Lnh Th Xuan Phan School of Software Technology, Dalan Unversty of Technology, Chna Computer and Informaton Scence Department, Unversty of Pennsylvana, U.S.A. Abstract Real-tme systems are ncreasngly runnng a mx of tasks wth dfferent crtcalty levels: for nstance, unmanned aeral vehcle has multple software functons wth dfferent safety crtcalty levels, but runs them on a sngle, shared computatonal platform. In addton, these systems are ncreasngly deployed on multprocessor platforms because ths can help to reduce ther cost, space, weght, and power consumpton. To assure the safety of such systems, several mxed-crtcalty schedulng algorthms have been developed that can provde mxed-crtcalty tmng guarantees. However, most exstng algorthms have two mportant lmtatons: they do not guarantee strong solaton among the hgh-crtcalty tasks, and they offer poor real-tme performance for the low-crtcalty tasks. In ths paper, we present a parttoned schedulng scheme for mxed-crtcalty tasks on multprocessor platforms that addresses both ssues. Our schedulng scheme conssts of () a task-toprocessor packng algorthm that takes nto account the demands of tasks wth respect to ther crtcalty levels, and () a mxedcrtcalty unprocessor schedulng strategy that s based on task groupng. Our strategy assocates each hgh-crtcalty task wth a subset of the low-crtcalty tasks and encapsulates them n a task group, whch s scheduled wth the other task groups under the Earlest Deadlne Frst (EDF) polcy. Wthn each task group, the low-crtcalty task and the hgh-crtcalty tasks are scheduled usng a server-based strategy, so as to enable more of the former to meet ther deadlnes wthout affectng the latter. We present a schedulablty analyss for our schedulng strategy, and we show how tasks can be grouped usng Mxed Integer Nonlnear Programmng. Our evaluaton shows that our proposed scheme sgnfcantly outperforms exstng parttoned mxed-crtcalty schedulng algorthms, n terms of both the fracton of schedulable task sets and ts ablty to schedule low-crtcalty tasks. I. INTRODUCTION Due to recent advances n chp technology, multprocessor platforms are ncreasngly appled n safety-crtcal domans (such as avoncs, automotve, and ndustral control) to accommodate the ncreasng demand for computatonallyntensve workloads. Many systems n these domans are mxed-crtcalty (MC) real-tme systems that s, they ntegrate multple functonaltes (tasks) wth dfferent safety crtcalty levels, such as flght-crtcal and msson-crtcal tasks, on a sngle, shared hardware platform. Ths s typcally done to meet strngent requrements on space, weght, and power consumpton. Therefore, t s mportant to provde strong solaton and tmng guarantees for the mxed-crtcalty tasks wth respect to ther crtcaltes. Snce MC real-tme systems run a mx of tasks wth dfferent degrees of crtcalty, these systems are more complex and unpredctable than tradtonal real-tme systems. In order to certfy the correctness of a MC system, t s common to make certan assumptons about the worst-case behavor of the system for nstance, some task parameters are allowed to depend on the crtcalty levels of the tasks. A typcal approach s to assume more pessmstc worst-case executon tmes (WCETs) for hgher crtcalty levels for each task; n systems wth two levels of crtcalty, a task s assumed to be n the hgh-crtcalty mode f ts executon tme exceeds the low-crtcalty WCET. A maor challenge n ths approach s to guarantee mxed-crtcalty schedulablty (.e., hgh-crtcalty tasks always meet ther deadlnes and low-crtcalty tasks meet ther deadlnes f the hgh-crtcalty tasks are always n the low-crtcalty mode) of the tasks whle achevng effcent resource use. There s a rch lterature on schedulng mxed-crtcalty real-tme systems on both unprocessor and multprocessor platforms (e.g., [] []); however, most exstng solutons have at least one of the followng three key lmtatons. Frst, they assume that, whenever a ob of a hgh-crtcalty task exhbts hgh-crtcalty behavor that s, ts executon tme exceeds the task s WCET as estmated at the low-crtcalty level all current and future obs of every hgh-crtcalty task n the system wll also exhbt hgh-crtcalty behavor, whch does not always happen n general. Second, exstng technques cannot fully capture the dynamc behavor of the system: each task can dynamcally transton back and forth between the hgh-crtcalty and the low-crtcalty mode. Thrd, exstng solutons do not offer good performance for the low-crtcalty tasks: snce these are klled as soon as a hgh-crtcalty ob n the system enters ts hgh-crtcalty mode, t s possble that no low-crtcalty ob can ever meet ts deadlne for nstance, when the crtcalty change happens shortly after the system begns ts executon. Ths s undesrable because low-crtcalty tasks requre a certan tmng performance as well (although wth weaker guarantees than the hgh-crtcalty tasks) []. In ths paper, we propose a compostonal approach to mxed-crtcalty task schedulng that can solve these problems. Our approach, called TG-PEDF, provdes strong solaton among the hgh-crtcalty tasks, and t enables more lowcrtcalty tasks to be scheduled whle stll ensurng the mxedcrtcalty schedulablty guarantee of the entre system. TG- PEDF s based on task groupng and compostonal schedulng: each hgh-crtcalty task s encapsulated n a separate task group (to enable solaton) together wth a subset of the lowcrtcalty tasks (to enable overbookng of resources). Wthn each task group, the tasks are scheduled wth a server-based mxed-crtcalty schedulng strategy, and the dfferent task groups are scheduled on the platform as conventonal tasks under the EDF schedulng polcy. Ths ensures that the hghcrtcalty behavor of a hgh-crtcalty task cannot affect other hgh-crtcalty tasks, and t also enables more low-crtcalty obs n the system to meet ther deadlnes. Ths s because () the nterference of a hgh-crtcalty task on low-crtcalty tasks beyond ts local task group s sgnfcantly reduced, and because () low-crtcalty tasks are not always dscarded, but rather scheduled accordng to best-effort servce when the hgh-crtcalty task n the same task group s n the hghcrtcalty mode. Furthermore, our schedulng algorthm can dynamcally adapt to the crtcalty mode of each ob (by

2 enablng tasks to return to the low-crtcalty mode, dependng on the executon tme of ther current obs), and can thus save resources at runtme. Contrbutons. Ths paper makes the followng contrbutons: A parttoned mxed-crtcalty schedulng approach based on task groupng for mplct-deadlne perodc MC tasks on multprocessor platforms. In contrast to exstng solutons, our approach provdes strong solaton between hgh-crtcalty tasks and better tmng performance for low-crtcalty tasks; A schedulablty analyss for a set of mxed-crtcalty tasks that have been assgned to a sngle processor under our task group based schedulng algorthm; A Mxed Integer Nonlnear Programmng (MINLP) formulaton for the task groupng, based on the schedulablty constrants, that can mprove the schedulablty of the system; and A task packng algorthm that can save resources by takng nto account the demand of both low-crtcalty tasks and hgh-crtcalty tasks durng deployment. Our evaluaton on randomly generated task systems shows that the schedulablty of our proposed scheme outperforms the state-of-the-art MC schedulng algorthms. The rest of the paper s organzed as follows. In Secton II, we summarze the related work. Secton III presents the system model. The detaled descrpton of the task group based MC schedulng on a unprocessor s gven n Secton IV. Secton V presents the task packng algorthm. We dscuss our evaluaton n Secton VI and conclude the paper n Secton VII. II. RELATED WORK Snce Vestal s ntal work on mxed-crtcalty systems [], a rch lterature on mxed-crtcalty schedulng of real-tme systems has developed; see [3] for a survey. Tradtonally, work n ths doman has focused on unprocessor platforms [] [7], [4] [], but recently efforts have shfted towards multprocessor platforms. Several multprocessor mxed-crtcalty schedulng algorthms have already been developed [8] [], [] [], ncludng both global and parttoned approaches. For nstance, Pathan [] derved a suffcent schedulablty condton of a global fxed-prorty schedulng algorthm on preemptve multprocessors based on response tme analyss. Baruah et al. [] proposed global and parttoned schedulng algorthms for mxed-crtcalty mplct-deadlne sporadc task systems by combnng fpedf [3] wth EDF-VD, and they showed that parttoned schedulng gves better system schedulablty than global schedulng. For harmonc task systems, Mollson et al. [8] proposed an archtecture to use approprate schedulng technques for tasks wth dfferent crtcaltes on dfferent processors, whch provdes tmng solaton between these levels through a bandwdth reservaton server. Kelly et al. [9] studed the schedulablty of varous parttonng technques nspred by bn packng, as well as tradtonal parttonng heurstcs under fxed-prorty schedulng for MC systems. To address the effects of the physcal envronment on the system, Nz et al. [] presented a parttoned multprocessor schedulng scheme based on a mult-mode extenson of the zero slack rate-monotonc schedulng. Gu et al. [4] extended the demand-based sngle-processor mxed-crtcalty schedulng [] (called EY n ths paper) to multprocessor platforms and proposed two enhancements based on heavy low-crtcalty task awareness and balance factor to mprove the system schedulablty. However, these exstng solutons assume that, whenever a ob of a hgh-crtcalty task exhbts hgh-crtcalty behavor, all current and future obs of every hgh-crtcalty task n the system wll exhbt hgh-crtcalty behavor and thus all low-crtcalty tasks wll be dropped. Our work removes ths assumpton by consderng the specfc mode of each hgh-crtcalty tasks n the schedulng. There has also been work on offerng degraded servce to low-crtcalty tasks when the system s n the hgh-crtcalty mode. For nstance, Su et al. [7], [] studed an elastc mxed-crtcalty task model that allows low-crtcalty task to have dfferent perods n dfferent system modes. To maxmze the executon rate of low-crtcalty tasks, Jan et al. [8] ntroduced stretchng factors and proposed an assocated onlne decson algorthm based on the exstng elastc task model. Santy et al. [] proposed a technque that ams to avod or delay the rse n crtcalty level n order to mnmze the number of tasks that wll mss ther deadlnes. From the mplementaton perspectve, Burns et al. [9] ponted out the lmtatons of the exstng standard mxed-crtcalty models, and they proposed a more practcal model that consders the servce adapton of low-crtcalty tasks and the system mode swathng problem from the low-crtcalty mode to the hgh-crtcalty mode. Huang et al. [], [7] studed the servce reconfguraton and resettng for low-crtcalty tasks by quantfyng the degraded servce of low-crtcalty tasks. Gu et al. [] proposed a herarchcal executon model that can solate the mpact of hgh-crtcalty tasks from low-crtcalty tasks, and that can allow low-crtcalty executons n the hgh-crtcalty behavor by usng component boundares. Our work dffers from these approaches n that t can reduce the propagaton of hghcrtcalty ob behavor for mxed-crtcalty systems through the solaton of dfferent task groups. Further, our approach enables the system to swtch back to low-crtcalty mode from hgh-crtcalty mode once all hgh-crtcalty tasks exhbt lowcrtcalty ob behavor. To the best of our knowledge, the propagaton reducton of hgh-crtcalty ob behavor has not been nvestgated n exstng MC schedulng solutons. III. MIXED-CRITICALITY SYSTEM MODEL The system conssts of a set τ of mplct-deadlne perodc mxed-crtcalty tasks that need to be scheduled on a multprocessor platform wth p dentcal, unt-capacty processors. We follow the same task model as n []: Each task τ s characterzed by a tuple τ = (ζ, C τ (), C τ (), T τ ), where ζ {, } denotes the crtcalty level of τ ; C τ () and C τ () specfy the WCET estmates of τ at the low and hgh crtcalty levels, respectvely, where C τ () = C τ () f ζ =, and C τ () C τ () f ζ = (that s, the WCET at the hgh crtcalty level s more pessmstc than the WCET at the low crtcalty level); and T τ denotes the perod (relatve deadlne) of τ. We refer to C τ () and C τ () as the -WCET and - WCET of τ, respectvely. We assume that all tasks are ntally released smultaneously (at tme t = ).

3 Mxed-crtcalty schedulablty. A task set τ s consdered mxed-crtcalty schedulable under a schedulng polcy f the followng condtons are met: Low-crtcalty guarantee: If all obs run for at most ther -WCETs, all hgh-crtcalty and lowcrtcalty obs must complete before ther deadlnes. Hgh-crtcalty guarantee: If all hgh-crtcalty obs run for at most ther -WCETs and at least one runs for more than ts -WCET, all hgh-crtcalty obs must complete before ther deadlnes (whereas lowcrtcalty obs may be dropped). We note that although our system model and algorthms consder only two levels of crtcalty, t should be possble to extend them to more than two crtcalty levels. One potentally effcent approach s to perform task groupng n a herarchcal fashon accordng to the crtcalty levels of tasks, whch we plan to explore n future work. In the next secton, we present the algorthm for schedulng mxed-crtcalty tasks on a (un)processor. The algorthm for assgnng tasks to processors s dscussed n Secton V. Due to space constrants, we omt the proofs of lemmas and theorems here; they can be found n our techncal report [8]. IV. TASK-GROUP-BASED MIXED-CRITICALITY SCHEDULING ON UNIPROCESSORS To provde solaton between hgh-crtcalty tasks and to reduce ther nterference wth low-crtcalty tasks, we assocate each hgh-crtcalty task wth a subset of the low-crtcalty tasks and encapsulate both n a separate task group. Tasks wthn each task group are scheduled usng a server-based strategy to maxmze mxed-crtcalty schedulablty, and the task groups themselves are scheduled on the processor under the Earlest Deadlne Frst (EDF) polcy. (In a specal case where all tasks assgned to a processor have the same crtcalty level, they are scheduled as conventonal tasks usng EDF.) A. Task groups A task group conssts of one hgh-crtcalty task and several low-crtcalty tasks. It s specfed as T G = { τ, τ,..., τn, τ }, where the τ ( n) are the n low-crtcalty tasks, and τ s the sngle hgh-crtcalty task. The perod of a task group T G s a common dvsor of the perods of all tasks n the task group,.e., T T G = CD ( T τ, ) T τ,..., T τ n, T τ, where Tτ ( n) s the perod of the low-crtcalty task τ and T τ s the perod of the hghcrtcalty task τ. (Snce t s common for real-tme systems to have harmonc perods n practce, we make ths choce to smplfy the analyss; however, extensons to other perod values are possble.) All task groups are smultaneously actvated at tme t =. By defnton, the number of task group budget replenshments durng a perod of a low-crtcalty task τ s gven by l = T τ /T T G ( n), and the number of task group budget replenshments durng a perod of the hgh-crtcalty task τ s gven by h = T τ /T T G. We denote by B the mnmum budget (to be computed) that a task group T G must receve n each of ts perods to ensure all of ts tasks meet the mxed-crtcalty schedulablty. In our schedulng strategy, each task group T G wll be scheduled wth other task groups n the system as a conventonal perodc task wth perod (relatve deadlne) T T G and executon tme B. As usual, we defne the utlzaton of a task group to be ts budget dvded by ts perod. For ease of presentaton, we refer to a perod of a task group as a TG-perod. B. Three-phase Schedulng Strategy wthn Task Groups Basc deas. The schedulng wthn each task group T G s done n rounds; each round corresponds to a perod of the hgh-crtcalty task τ. Wthn each round, there can be three dfferent schedulng behavors: one s used whle the actual executon tme of τ s current ob has not yet reached ts -WCET; another s used when τ s ob mght reach ts -WCET durng the current perod of the task group; and a thrd s used when τ s ob has ether fnshed or exceeded ts -WCET,.e., swtched to hgh-crtcalty ob mode. Ths corresponds to three dstnct phases, whch are llustrated n Fgure : t May swtch to hgh-crtcalty ob mode Phase I Phase II Phase III Perod of task group Release of the hgh-crtcalty Absolute deadlne Fg. : A schedulng round of a task group. t + T Phase I (low-crtcalty mode): In ths phase, the current ob of τ has not yet reached ts -WCET and s thus n lowcrtcalty mode. All obs n the task group are scheduled to ensure the low-crtcalty guarantee. Phase II (swtchng mode): In ths phase, the current ob of τ can potentally exceed ts -WCET and thus swtch to hgh-crtcalty mode durng the current perod of the task group. All low-crtcalty tasks n the task group are suspended to ensure the hgh-crtcalty guarantee. Phase III (low-crtcalty or hgh-crtcalty mode): In ths phase, the current ob of τ has ether fnshed or exceeded ts -WCET, so t s known whether the task group s n lowor hgh-crtcalty mode. If the ob has ndeed fnshed, all the low-crtcalty tasks are scheduled; otherwse the group s n hgh-crtcalty mode, and the low-crtcalty tasks contnue to be suspended. When the hgh-crtcalty ob completes (n phase II or III) and the task group has not yet used up ts budget, any suspended low-crtcalty obs wll contnue to be scheduled; however, f they do not complete by the end of the round (.e., when a new hgh-crtcalty ob s released) or by ther deadlnes, whchever s sooner, they wll be dropped. Schedulng mechansm for each round. Recall that the budget of the task group T G n each TG-perod s B and the number of task group budget replenshments n any perod of τ (.e., n each schedulng round of T G ) s h. Wthout loss of generalty, we suppose that Phase I lasts for k TG-perods,.e., the current ob of τ ether swtches to the hgh-crtcalty mode or has completed ts executon n the (k + )th TGperod, where k h. In each schedulng round, tasks are scheduled based on ther pre-computed budget values, x,

4 b and b ( n) as follows. (The method for computng these parameters s presented n Secton IV-D.) Phase I (the frst k TG-perods): To acheve the lowcrtcalty guarantee, n each of the frst k TG-perods, tasks are scheduled as follows: f there are some pendng lowcrtcalty obs, the current ob of τ s executed frst for a specfc perod of tme x (at most) and then the lowcrtcalty obs are executed n ascendng order of the task ndex, where each low-crtcalty task τ s allocated a budget of b. If there are no pendng low-crtcalty obs, the current ob of τ s executed untl t completes or the budget of the task group s used up. The budget parameters should satsfy two condtons: Frst, the total budget allocated to all tasks n the task group should not exceed the budget B of the task group,.e., x + n = b B () Second, snce the current ob of the hgh-crtcalty task s always n the low-crtcalty mode n ths phase, the total budget receved by τ n ths phase should not exceed ts -WCET,.e., k x C τ () () Phase II (the (k + )th TG-perod): To acheve the hghcrtcalty guarantee, the current ob of τ s executed frst wth an allocated budget of x. If ths ob completes ts executon, the server wll schedule the low-crtcalty tasks n the same manner as n the prevous phase. Otherwse, the ob wll swtch from the low-crtcalty mode to the hgh-crtcalty mode and wll contnue to be executed untl t completes or the budget of the task group s used up; n the former case, the server wll schedule the low-crtcalty obs n the same manner as before untl the budget of the task group s used up (some low-crtcalty obs may be dropped). As above, nequalty () should hold to ensure the budget x for τ and the budget b for each low-crtcalty task τ. Moreover, snce the current ob of τ wll swtch to the hghcrtcalty mode f t cannot complete wth the budget x, the budget receved by τ before the mode swtch should be at least equal to ts -WCET. Therefore, (k + ) x C τ () (3) Phase III (the last h k TG-perods): To acheve the hgh-crtcalty guarantee, the schedulng n each of the last h k perods of the task group s as follows: f the current ob of τ has not completed n the consderng TGperod, t wll be executed untl t completes or the budget of the task group s used up. Otherwse, the low-crtcalty tasks are scheduled n ascendng order of the task ndex, where each task τ s allocated a budget of b. Because all budget of the task group wll be allocated to the low-crtcalty tasks n the task group f the task group s n the low-crtcalty ob mode n ths last phase, we can allocate more budget to each low-crtcalty task. In other words, b b (4) Further, the total budget allocated to all the low-crtcalty tasks should not exceed the budget of the task group,.e., n b B () = TABLE I: An example task group: task parameters Task ζ C () C () T U () U () τ τ.8.8 τ Fg. : A schedule of the example n the low-crtcalty mode Fg. 3: A schedule of the example n the hgh-crtcalty mode. We llustrate the schedulng mechansm usng an example. Example. Consder the task group T G depcted n Table I, wth perod T T G = GCD(,3,3) =. Suppose the budget gven to T G s B =.8 and the schedulng parameters are k =, x =, b =, b =., b = and b =. Fgures and 3 each show two schedulng rounds for T G : one n whch the hgh-crtcalty task always exhbts low-crtcalty behavor, and one n whch t exhbts hgh-crtcalty behavor. Note that each round contans h = T τ /T T G = 3 TG-perods and conssts of only Phase II and Phase III (snce k = ). Frst, consder Fgure. In the frst TG-perod (Phase II), the hgh-crtcalty task s frst executed for x = tme unts and completes; hence, the low-crtcalty task τ s then executed for b = tme unts (τ s not executed snce b = ). Snce the task group s n low-crtcalty mode, the low-crtcalty tasks τ and τ are executed n the second and thrd TG-perod (Phase III) for b =. and b = tme unts, respectvely. The second schedulng round works smlarly, except that n the last TG-perod, τ fnshes ts executon before usng all of ts allocated budget. Now consder Fgure 3. Here, τ exceeds ts -WCET n the frst round, so the task group s swtched to hgh-crtcalty mode, and both low-crtcalty tasks are suspended (untl τ completes). In the second round, however, τ remans n lowcrtcalty mode, so τ and τ are executed as usual, based on ther allocated budgets. (Note that n the frst phase of ths round,.e., tme nterval [3,4], nether τ nor τ s executed. Ths s because the current ob of τ has been dropped at tme 3 when τ s released, whereas the budget allocated to τ s b =.) We observe that only some obs of the low-crtcalty tasks mss ther deadlnes: the low-crtcalty obs released n the frst round are suspended when τ s n the hgh-crtcalty mode, but the low-crtcalty obs released n the second round are stll schedulable. We can also valdate that the task group s mxed-crtcalty schedulable n the example schedules.

5 C. Schedulablty Analyss Consder any task group T G = { τ, τ,..., τ n, τ }. We frst establsh the mxed-crtcalty schedulablty condtons for T G under the three-phase schedulng strategy wth a gven budget B and gven schedulng parameters k,x,b,b ( n), where x,b and b are non-negatve real numbers and k s a non-negatve nteger. We then derve the schedulablty condtons for a set of task groups on a processor. Recall that l = T τ /T T G s the number of task group budget replenshments durng each perod of τ, and h = /T T G s the number of task group budget replenshments T τ n each perod of the hgh-crtcalty task τ (.e., n a schedulng round). Lemma IV. gves the schedulng condtons for each low-crtcalty task τ : Lemma IV.. Each low-crtcalty task τ ( n) can be successfully scheduled f all of the nequaltes () () hold, where the nequalty () s gven by N maxol b + (l NmaxOL ) b C τ (), () where N maxol = l h (k + ) + mn{l mod h,k + } s the maxmum number of TG-perods n a perod of τ that overlap wth phases I and II of a schedulng round. Example. Consder the task group depcted n Table II, wth budget B =, k =, b = and b =. The maxmum number of TG-perods that overlap wth the frst k + = TG-perods durng a perod of the hgh-crtcalty task s mn{3 mod, + } =. As demonstrated n Fgure 4, there are two overlappng TG-perods for the st and 4th releases of τ, one for the nd and 3rd releases, and zero for the th release. We can valdate that all obs of τ can fnsh before ther deadlnes. TABLE II: An example task group: task parameters Task ζ C () C () T U () U () τ. τ 3.. N overlap = N overlap N overlap = = N overlap = N overlap = Fg. 4: Number of overlappng TG-perods n Example. Lemma IV.. Every ob of the hgh-crtcalty task τ receves a budget of at least C τ () when t exhbts lowcrtcalty behavor and at least C τ () when t exhbts hgh- crtcalty behavor by ts deadlne f the nequaltes (), (3) and the followng nequalty hold: k x + (h k ) B C τ () (7) The next theorem s derved drectly from Lemmas IV. and IV.. Theorem IV.. Suppose each task group T G = { τ, τ,..., τn, τ } s guaranteed to receve a budget of B n each of ts perod. Then, T G s mxed-crtcalty schedulable under the three-phase schedulng strategy wth schedulng parameters k,x,b and b ( n) f all of the nequaltes () (7) are satsfed. Remarks. We note that Theorem IV. holds regardless of when exactly the task group s scheduled, because all tasks are guaranteed to receve ther allocated budgets as long as the task group as a whole receves a budget of B n each perod. In addton, the hgh-crtcalty behavor of a ob of τ only affects some low-crtcalty obs that need to be executed durng the perod of ths τ ob but not the ones that are released after ts deadlne. Moreover, snce each round of a task group corresponds to a ob of ts hgh-crtcalty task, each task group can swtch from the hgh-crtcalty mode to the low-crtcalty mode ndependently of each other, and the system can swtch to the low-crtcalty mode once all hghcrtcalty obs n the system exhbt low-crtcalty behavor. Let S k be the set of task groups scheduled on a processor k. Recall that each task group T G S k s scheduled wth other task groups n S k as a conventonal mplct-deadlne perodc task wth perod T T G and executon tme B under EDF. Based on the schedulable utlzaton bound of EDF [9], all task groups n S k are schedulable under EDF f ther total utlzaton s less than or equal to,.e., U k = T G S k B /T T G. In other words, every task group T G s guaranteed a budget of B n each of ts perod f U k. The next theorem follows drectly from ths observaton and Theorem IV.. Theorem IV.. Let S k be the set of task groups scheduled on processor π k (accordng to some task groupng algorthm). If for all T G = { τ, τ,..., τ n, τ } n S k, there exst a nonnegatve budget value B R + and non-negatve schedulng parameters k N (k < h ) and x,b,b R + for all n such that U k = T G S k B /T T G and the nequaltes () (7) are satsfed, then all the tasks assgned to processor k are mxed-crtcalty schedulable. We note that, although Theorem IV. only gves a suffcent analyss, our smulaton shows that our parttoned schedulng based on task groupng outperforms an exstng parttoned algorthm based on EDF-VD, and EDF-VD has been shown to be optmal (f all deadlnes are reduced by the same rato) on sngle processors for two-level MC systems from the perspectve of speedup bound []. Next, we present an algorthm based on Mxed Integer Nonlnear Programmng (MINLP) for groupng tasks nto task groups, as well as for computng the budget and schedulng parameters of each task group. D. Mxed Integer Nonlnear Programmng for Task Groupng Let τ(π k ) = { τ,...,τ n,τ },...,τ m be the set of tasks that are assgned to the processor π k based on some tasks-toprocessors packng algorthm. The goal of the task groupng s to determne (a) for each hgh-crtcalty task τ ( m) a subset of the low-crtcalty tasks n τ(π k ) that wll be grouped wth τ nto the task group T G, and (b) for each obtaned task group T G, the values of the task group s budget B and the followng schedulng parameters: k, the number of TGperods n schedulng Phase I; x, the schedulng budget of the hgh-crtcalty task τ ; and b and b, the schedulng budgets for each low-crtcalty task τ n T G. Recall that the perod T T G of T G s a common dvsor of the perods of all tasks

6 n T G ; to reduce the computaton complexty, we fx T T G to be the greatest common dvsor of all tasks n τ(π k ). Thus, l = T τ /T T G and h = T τ /T T G also become constants. To ths end, we defne a MINLP for task groupng based on the schedulablty condtons of the task groups. Snce the task groups T G ( m) are scheduled under EDF as conventonal mplct-deadlne perodc tasks wth perods T T G and budgets B, the smaller ther total utlzaton m = B /T T G s, the better ther schedulablty wll be. Therefore, the obectve of the proposed MINLP s to mnmze the total utlzaton, m = B /T T G, of the task groups so as to mprove ther schedulablty. The constrants of the MINLP are derved drectly from the mxed-crtcalty schedulablty condtons of the task groups (c.f. Theorem IV.). To enable a low-crtcalty task to use the overbookng of multple hgh-crtcalty tasks, we allow each low-crtcalty task τ to appear n multple task groups,.e., each ob of τ can receve budget from more than one task group and the ob s executed whenever t s scheduled n a task group that t belongs. Snce all task groups are executed sequentally on a sngle processor, τ s never executed at the same tme by more than one task group. Hence, f the total budget that τ s guaranteed to receve n each of ts perods from all of ts task groups s greater than or equal to ts -WCET (= -WCET), then τ s schedulable. Based on the above dea, we use m pars of non-negatve real varables (b,,b, ) for each low-crtcalty task τ ( n, m) to represent the budgets that τ receves from task group T G n the three schedulng phases of T G (nstead of usng a sngle par of varables (b,b )). These varables are also used to ndcate whether τ belongs to the task group T G : f b, = b, =, then τ does not belong to T G, otherwse t does. Based on Theorem IV., the task set τ(π k ) s mxedcrtcalty schedulable f all of the nequaltes () (7) are satsfed for every task group T G. Due to the above encodng of low-crtcalty tasks, the constrants for each task group T G can be obtaned by replacng b wth b,, b wth b,, N maxol wth N, maxol n the nequaltes and rewrttng the nequalty (7) as m ( = b, N, maxol + b, (l N, maxol ) ) C τ (). To reduce the computaton complexty, nstead of usng the nequalty () (.e., x + n = b, ), we set B = x + n = b,. By mergng all constrants, we obtan the MINLP formulaton shown n Fgure. Constrants () (7) correspond to nequaltes () (7), and the remanng constrants are condtons on the varables, as was dscussed earler. Gven an MC task set, f the total task group utlzaton obtaned from soluton of the MINLP s more than one, then ths task set s unschedulable by the task group based schedulng and wll be reected by the scheduler. Complexty. If n and m are the number of low- and hghcrtcalty tasks n the task set, respectvely, then the MINLP has mn + m constrants, m nteger varables and m + mn real varables. Hence, the number of varables and constrants s polynomally bounded n the sze of the nput problem (.e., the task set sze), and t can be solved by a fully polynomaltme approxmaton scheme. The latter s therefore suffcent to obtan all the schedulng parameters. Mnmze m = (( n = b, + x )/T T G ) Subected to () B = x + n = b, () k x C τ () (3) (k + ) x C τ () ( =,...,m) ( =,...,m) ( =,...,m) (4) b, b, ( =,...,n; =,...,m) () n = b, B ( =,...,m) () m (, = b N, maxol + b, (l NmaxOL, ) ) C τ () ( =,...,n; =,...,m) (7) k x + (h k ) B C τ () ( =,...,m) (8) k h ( =,...,m) (9) x R + ( =,...,m) () b, R + ( =,...,n; =,...,m) () b, R + ( =,...,n; =,...,m) () k N ( =,...,m) Fg. : MINLP formulaton of the task groupng Dscussons. The MINLP formulaton gven n Fgure can easly be modfed to consder other types of constrants as well e.g., constrants on the number of task groups that each low-crtcalty task can belong, constrants on the number of low-crtcalty tasks n each task group, etc. Due to space constrants, we omt them here. E. EDF-based Two-phased Schedulng Strategy The proposed budget-drven three-phased schedulng algorthm wthn a task group (c.f. Secton IV-B) may lead to hgh context swtch overhead when there s a large number of low-crtcalty tasks n a task group. To avod ths drawback, we propose a two-phased varant of the algorthm that schedules lowcrtcalty tasks based on EDF. (Ths verson of the algorthm s used n our evaluaton.) Gven a task group T G wth budget B and schedulng parameters k and x, the schedulng durng each round (.e., perod of the hgh-crtcalty task τ n the task group) works as follows: Phase I (the frst k ( k h ) TG-perods): In each TG-perod, f there are some pendng low-crtcalty obs n the task group, the current ob of τ s executed frst for a specfc perod of up to x, and then the low-crtcalty obs n the task group are executed under the EDF polcy. Otherwse, the current ob of τ s executed untl t completes or the budget of the task group expres. Phase II (the last h k perod(s) of task groups): In each TG-perod, the ob of τ s executed frst untl t fnshes and then the pendng low-crtcalty obs n the task group are executed under the EDF polcy. If the current ob of τ exhbts hgh-crtcalty behavor, the low-crtcalty obs that cannot fnsh when ther deadlnes are reached or when a new ob of τ s released are dropped. Note that, unlke n the budget-drven three-phased schedulng strategy, the (k + )th TG-perod and the last h k TGperods are consdered as one phase n the EDF-based twophased schedulng strategy. The reason for ths s that under the three-phased strategy, dfferent budgets (b and b ) may be used for each low-crtcalty task τ n the (k + )th TGperod and n the last h k TG-perods, whereas under the EDF-based schedulng strategy, low-crtcalty tasks are always scheduled usng the EDF polcy. The schedulablty under the EDF-based two-phased schedulng strategy s gven by the next theorem.

7 Theorem IV.3. If a task group T G = { τ, τ,..., τ n, τ } s mxed-crtcalty schedulable under the three-phased schedulng strategy wth budget B and schedulng parameters k,x,b,b, then T G s also mxed-crtcalty schedulable under the EDF-based two-phased schedulng strategy wth budget B and schedulng parameters k and x. Example 3. Consder the task group shown n Table III, wth perod, budget.9, k = and x =. As llustrated n Fgure, all low-crtcalty tasks can be successfully scheduled by the three-phased server wth budgets b =, b =., b =. and b =. From Fgure 7, we can see that an EDF-based schedule for the low-crtcalty tasks can be obtaned by nterchangng some executons of τ and τ, resultng n a smaller number of context swtches. TABLE III: An example task group: task parameters Task ζ C () C () T U () U () τ τ.8.8 τ Fg. : A budget-drven schedule Fg. 7: An EDF-based schedule obtaned by transposton. Due to Theorem IV.3, we can determne the task groups and ther schedulng parameters for the EDF-based two-phased schedulng algorthm by solvng the MINLP formulated n the prevous secton. A smple case study that llustrates the endto-end behavor and the benefts of our task-groupng-based schedulng algorthm compared to an exstng algorthm based on vrtual deadlne [7] s provded n [8]. Remarks. The context swtch overhead of EDF-based twophased schedulng can further be mtgated by consderng the slcng number of low-crtcalty tasks as a constrant n the MINLP formulaton and by extendng analyss to allow other values for the task group perod and/or to determne the tme of the task group budget replenshment dynamcally. In addton, we can also enable servce guarantee for low-crtcalty tasks usng a constrant on the number of task groups that each lowcrtcalty task can belong n the MINLP formulaton. V. TASK-GROUP-BASED PARTITIONED MC SCHEDULING In ths secton, we present a parttonng algorthm for assgnng a set of mxed-crtcalty tasks τ to processors. We consder a platform wth p dentcal, unt-capacty processors, denoted as π = {π,..., π p }. Our algorthm ams to balance the loads across all processors, whle selectng hgh-crtcalty (lowcrtcalty) tasks to assgn based on decreasng hgh-crtcalty (low-crtcalty) utlzaton. Let τ = {τ,...,τ n } be the set of hgh-crtcalty tasks and τ = {τ n +,...,τ n } be the set of low-crtcalty tasks of the task set τ. Algorthm s our parttonng algorthm, wrtten n pseudocode. In ths algorthm, the set of tasks allocated to each processor π s denoted by τ(π ). The low-crtcalty utlzaton U (τ ) and the hgh-crtcalty utlzaton U (τ ) of a task τ are defned as the task s -WCET and -WCET dvded by ts perod, respectvely. For brevty, we smply refer to the hgh-crtcalty utlzaton of a hgh-crtcalty task (and, smlarly, the low-crtcalty utlzaton of a low-crtcalty task) as that task s utlzaton. Further, the cumulatve task group utlzaton for a set of tasks s the total utlzaton (.e., budget dvded by perod) of all task groups of the task set (determned usng the MINLP method n Secton IV-D). Algorthm Task group based MC parttoned algorthm. : τ(π ) /, for all =,..., p. : Sort hgh-crtcalty tasks τ n decreasng order of hgh-crtcalty utlzaton and then n decreasng order of low-crtcalty utlzaton. 3: Sort low-crtcalty tasks τ n decreasng order of low-crtcalty utlzaton and then n ncreasng order of perod. 4: whle τ / and τ / do : τ selected NIL : f τ / then 7: τ selected The frst task n τ 8: f τ / then 9: τ selected The frst task n τ : f τ selected ( = NIL or τselected NIL and U (τ selected ) < U (τ selected )) then : τ selected τ selected : Remove τ selected from ts set 3: Fnd a processor π π that has a mnmal cumulatve task group utlzaton for the tasks τ(π ) {τ selected } 4: f the cumulatve task group utlzaton for tasks τ(π ) {τ selected } does not exceed then : τ(π ) τ(π ) {τ selected } : else 7: return FAILURE 8: return SUCCESS The algorthm starts by ntalzng τ(π ) to null (Lne ) and by sortng hgh-crtcalty tasks and low-crtcalty tasks (Lnes 3). Hgh-crtcalty tasks are sorted n decreasng order of hgh-crtcalty utlzaton, and tasks wth the same hghcrtcalty utlzaton are further ordered by decreasng lowcrtcalty utlzaton (Lne ). Low-crtcalty tasks are sorted n decreasng order of low-crtcalty utlzaton, and tasks wth the same low-crtcalty utlzaton are further ordered by ncreasng perod (Lne 3). Our reason for ths sortng s as follows: for hgh-crtcalty tasks wth the same hgh-crtcalty utlzaton, a task wth a hgher low-crtcalty utlzaton typcally has less overbookng; and for low-crtcalty tasks wth the same low-crtcalty utlzaton, the task wth a smaller perod tends to have a lower utlzaton rate of overbookng, snce the task group perod s chosen as the common dvsor of the perods of the tasks wthn the task group. However, the algorthm can easly be extended to other sortng methods (whch we plan to nvestgate n future work). The algorthm then selects the task τ selected that has the largest utlzaton (Lnes ), and removes t from ts task set (Lne ). Next, the algorthm selects a processor π for ths task, such that the cumulatve task group utlzaton for ths task along wth the exstng tasks on ths processor s mnmal (Lne 3); ths s done to balance the load across

8 dfferent processors. If the resultng mnmum cumulatve task group utlzaton on the selected processor π does not exceed one, τ selected s assgned to π (Lnes 4 ). Otherwse, the algorthm aborts wth a falure (Lne 7). If every task s successfully allocated to a processor, the algorthm reports success (Lne 8). Based on Theorem IV., the resultng task allocaton obtaned by Algorthm always ensures that tasks on each processor can be successfully scheduled by the task group based schedulng strategy. It should be noted that the crtcalty levels of the tasks on a processor may be the same; n ths case, all tasks on the processor are drectly scheduled under EDF. VI. EVALUATION In ths secton, we expermentally compare the performance of our proposed scheme, TG-PEDF, wth the followng parttoned mxed-crtcalty schedulng algorthms: DC-RM: a parttoned schedulng algorthm from [9] that s based on RM prorty assgnment; DC-Audsley: a parttoned schedulng algorthm from [9] that s based on Audsley s optmal prorty assgnment; MC-Partton: a parttoned schedulng algorthm based on EDF-VD (from []); EY-FF: a straghtforward extenson of EY [] for parttoned schedulng wth the FF packng strategy from [4]; MPVD: an extenson of EY for parttoned schedulng wth the hybrd packng strategy from [4]; MPVD-HA: MPVD wth the heavy low-crtcalty task aware allocaton polcy from [4]; and MPVD-HA-BF: MPVD-HA wth the optmzed vrtual deadlne tunng from [4]. Obectves. We have three man goals for our evaluaton: () to compare the performance of TG-PEDF n terms of mxedcrtcalty schedulablty to that of the above algorthms; () to evaluate how well TG-PEDF can protect low-crtcalty obs when a hgh-crtcalty task s n hgh-crtcalty mode; (3) to evaluate the deadlne mss ratos of low-crtcalty tasks under dfferent frequences of hgh-crtcalty tasks beng n hghcrtcalty mode; (4) to evaluate the computatonal complexty of TG-PEDF; and () to evaluate the effect of overbookng by allowng a low-crtcalty task to appear n multple task groups n terms of schedulablty on a unprocessor. For Obectve (), our metrc s the mpact rato of lowcrtcalty tasks wth respect to dfferent percentages of hghcrtcalty tasks beng n the hgh-crtcalty mode. We defne the mpact rato of low-crtcalty tasks as the fracton of such tasks for whch at least one ob s dropped or msses ts deadlne because a hgh-crtcalty task s n hgh-crtcalty mode. The deadlne mss rato of low-crtcalty obs s defned as the fracton of low-crtcalty obs that mss ther deadlnes. Due to space constrants, some addtonal results are presented n our techncal report [8]. A. Workload For our experments, we used randomly generated task sets, whch we created usng the algorthm from [7]. We set the probablty pcrtcalty that a task s a hgh-crtcalty task to.. The perod (relatve deadlne) T of each task τ was an nteger that was drawn unformly at random from the nterval [, ]. The low-crtcalty WCET C () of task τ was drawn from [. T, T ] and, f τ was a hghcrtcalty task, ts hgh-crtcalty WCET C () was drawn from [ C (), 4 C ()]. We defne the normalzed average utlzaton U avg (τ) of a task set τ to be: U avg (τ) = U (τ) +U (τ) (8) p where U (τ) and U (τ) are the cumulatve low-crtcalty utlzaton and the cumulatve hgh-crtcalty utlzaton of a task set τ, respectvely. p s the number of processors. Usng the above parameters, we generated the tasks one at a tme untl the followng condtons on system utlzaton were satsfed: () Uavg. U avg (τ) Uavg +.; () U (τ) p; and () U (τ) p, where Uavg {,,.7,.7,.8,.8,.9,.9,.97} and p {4,8,}. For each Uavg and each p value, we generated, task sets for each of the above eght algorthms. Snce the evaluaton of the mpact rato of low-crtcalty tasks and the deadlne mss rato of low-crtcalty obs nvolves expensve smulatons, we used for ths purpose task sets randomly chosen from the generated task sets wth Uavg {,.7,.8,.9} that can be scheduled by TG-PEDF. We ran each smulaton for,, tme unts. To evaluate the effect of resource overbookng on the schedulablty for dfferent maxmum numbers of hgh-crtcalty task groups that a low-crtcalty task can belong to, we set pcrtcalty to.8, and we randomly drew the low-crtcalty task utlzaton from [., ] to generate more hgh-crtcalty tasks. B. Results Mxed-crtcalty schedulablty. Fgures 8a 8c show the fracton of schedulable task sets versus the normalzed average utlzaton of the dfferent algorthms n 4-processor, 8-processor, and -processor systems, respectvely. We begn wth a few observatons about the exstng algorthms. Frst, the results for 8- and -processor systems show that the performance of EY-FF decreased as the number of processors ncreased. Ths s because the space for EY to tune the vrtual deadlnes s reduced due to the unbalanced allocaton of hgh-crtcalty tasks. Second, the results n Fgure 8c show that, n -processor systems, MPVD, MPVD-HA and MPVD-HA-BF outperformed EY-FF by balancng the demand of hgh-crtcalty tasks among dfferent processors. MPVD- HA enhances MPVD by assgnng the heavy low-crtcalty tasks to processors before the hgh-crtcalty ones, but the performance of the two algorthms s smlar when the task sets contan few low-crtcalty tasks. MPVD-HA-BF ams to mprove schedulablty usng a new vrtual deadlne tunng algorthm, and ts schedulablty s ndeed better than MPVD and MPVD-HA n most cases; however, the vrtual deadlne tunng algorthm does not take the low-crtcalty tasks nto account, so t can affect schedulablty even f the demand n low-crtcalty mode s low. For ths reason, MPVD-HA-BF dd not outperform EY-FF on the systems wth 4 and 8 processors, and t performed worse than MPVD and MPVD-HA on - processor systems wth hgher average utlzaton. Our man result s that TG-PEDF consstently outperformed all exstng parttoned MC schedulng algorthms. The performance gap tends to wden as the average utlzaton of the

9 Fracton of schedulable task sets DC RM DC AUDSLEY MC PARTITION EY FF MPVD MPVD HA MPVD HA BF TG PEDF Fracton of schedulable task sets DC RM DC AUDSLEY MC PARTITION EY FF MPVD MPVD HA MPVD HA BF TG PEDF Fracton of schedulable task sets DC RM DC AUDSLEY MC PARTITION EY FF MPVD MPVD HA MPVD HA BF TG PEDF Normalzed average utlzaton (a) 4-processor systems Normalzed average utlzaton (b) 8-processor systems Fg. 8: Fracton of mxed-crtcalty schedulable task sets for dfferent algorthms Normalzed average utlzaton (c) -processor systems Impact rato of low crtcalty tasks U avg = U avg =.7 U avg =.8. U avg = Rato of hgh crtcalty tasks wth hgh crtcalty behavor Fg. : Impact rato of low-crtcalty tasks. Deadlne mss rato of low crtcalty obs U = avg U =.7 avg U =.8 avg U avg = Hgh crtcalty ob behavor probablty Fg. : Deadlne mss rato of low-crtcalty obs. Fracton of schedulable task sets N max sub = N max sub = N sub = 3. N sub = Normalzed average utlzaton Fg. 3: Effect of overbookng on schedulablty. task sets ncreases; the reason s that, as the number of tasks ncreases, there are more and more opportuntes for TG-PEDF to fnd an approprate task group for each low-crtcalty task. Impact of hgh-crtcalty behavor on low-crtcalty tasks. Fgure shows the mpact rato of the low-crtcalty tasks for dfferent percentages of hgh-crtcalty tasks wth hghcrtcalty behavor on 4-processor systems. The results show that only a certan fracton of low-crtcalty tasks are affected, and that ths fracton grows wth the fracton of hghcrtcalty tasks that exhbt hgh-crtcalty behavor. Ths s because, wth TG-PEDF, the hgh-crtcalty behavor of a hgh-crtcalty task can only affect low-crtcalty tasks that are n the same task group wth ths hgh-crtcalty task, but not low-crtcalty tasks n other task groups. We make three addtonal observatons. Frst, the relatonshp s not exactly lnear because some low-crtcalty tasks are grouped together wth more than one hgh-crtcalty task, and can thus be affected by any one of these tasks. Second, when all hgh-crtcalty tasks exhbt hgh-crtcalty behavor (.e., the rato of hgh-crtcalty tasks wth hghcrtcalty behavor s one), there are stll some low-crtcalty tasks that are not affected. The reason for ths s that there can be processors wth only low-crtcalty tasks and no hgh-crtcalty tasks, and these cannot be nfluenced by the hgh-crtcalty behavor on other processors. Fnally, the results are not senstve to dfferent U avg values (the curves correspondng to dfferent system utlzaton values cross). Ths s expected because the fracton of low-crtcalty tasks that are affected depends prmarly on the task groupng and the fracton of hgh-crtcalty tasks that exhbt hgh-crtcalty behavor nstead of the system utlzaton. Real-tme performance of low-crtcalty tasks. Fgure shows the effect of hgh-crtcalty ob behavor on the deadlne mss rato of low-crtcalty obs, usng the 4-processor system as an example. Not surprsngly, there are no deadlne msses when the probablty of hgh-crtcalty ob behavor s zero: when the system exhbts low-crtcalty behavor, the taskgroup-based schedulng can ensure that the system s schedulable. As the probablty of hgh-crtcalty ob behavor rses, the deadlne mss rato rses as well, but only gradually; the reason s, agan, that the hgh-crtcalty behavor of a ob can only affect low-crtcalty obs wth whch t shares a task group, but not low-crtcalty obs n other task groups. Addtonally, snce the solaton between task groups enables the system to swtch back to low-crtcalty mode once all hgh-crtcalty tasks exhbt low-crtcalty ob behavor (or, at a task group level, even when only the local hgh-crtcalty task exhbts low-crtcalty behavor), the deadlne mss rato of the lowcrtcalty obs can be kept low even for long-runnng systems. We also observe that the deadlne mss rato curves correspondng to dfferent averge system utlzaton values cross each other,.e., a task set wth a hgher average utlzaton may have a smaller deadlne mss rato of low-crtcalty obs than that of a task set wth a lower average utlzaton. Ths s not surprsng, because the deadlne mss rato of lowcrtcalty obs can become smaller as the total number of low-crtcalty obs ncreases, whch can be the case when the average utlzaton ncreases. It s worth notng that the exstng mxed-crtcalty algorthms drop all of the low-crtcalty tasks n the system as soon as a hgh-crtcalty ob exhbts hgh-crtcalty behavor, so they provde no real-tme performance guarantee for low-crtcalty tasks n the hgh-crtcalty mode. Effect of resource overbookng on schedulablty. Fgure 3 llustrates the effect of enablng low-crtcalty tasks to use the overbookng of multple hgh-crtcalty tasks on the schedu-

10 lablty of the task set on a unprocessor. Here, Nsub max s the maxmum number of task groups for each low-crtcalty task. The results show that the more hgh-crtcalty tasks that a lowcrtcalty task s executed wth, the better the schedulablty. In addton, the schedulablty gap tends to wden as the average utlzaton of the task set ncreases, whch further ndcates the effectveness of usng resource overbookng of hgh-crtcalty tasks. We note that ths can also lead to more hgh-crtcalty tasks nterferng wth a low-crtcalty task; however, snce the low-crtcalty schedulablty guarantee also reflects the schedulablty of low-crtcalty tasks, the resultng benefts outwegh the addtonal nterference. Computaton complexty. We also evaluated the tme needed per task set to group the tasks and to compute the task groups schedulng parameters n our experments, when usng the APMontor optmzaton sute [3] for solvng the MINLP problem. The results show that the formulated MINLP problem can be solved effcently (by a fully polynomal-tme approxmaton scheme): t took less than ms across all task sets n our experments. VII. CONCLUSION We have proposed TG-PEDF, a parttoned schedulng technque for mxed-crtcalty multprocessor real-tme systems that s based on task groupng. TG-PEDF works by assocatng each hgh-crtcalty task wth a subset of the low-crtcalty tasks and by encapsulatng them n separate task groups. Task groups are scheduled wth other task groups usng the EDF polcy, whle the tasks wthn each group are scheduled usng a server-based strategy that can ensure the mxed-crtcalty guarantees. Ths strategy not only guarantees solaton among hgh-crtcalty tasks, t also enables more low-crtcalty tasks to meet ther deadlnes. TG-PEDF s also compostonal: new tasks can easly be composed wth exstng ones va task groups. We have presented a schedulablty analyss for the system under the proposed schedulng strategy and an MINLP formulaton for the task groupng, as well as a packng algorthm that consders crtcalty n task deployment to optmze resource use. Our evaluaton shows that TG-PEDF consstently outperforms exstng multprocessor MC schedulng algorthms n terms of system schedulablty, whle enablng better tmng performance for low-crtcalty tasks. As a future drecton, we plan to mplement the proposed technque on a real platform and study ts real-tme performance and schedulng overhead, as well as to extend t to multple crtcalty levels. ACKNOWLEDGMENTS The authors would lke to thank Arvnd Easwaran, Nan Guan and Chuanca Gu for ther nputs on the mplementaton of exstng algorthms, Andreas Haeberlen for hs useful feedbacks on the paper, and Guowe Wu for hs support. Ths work was supported n part by the NSF grants CNS 78, ECCS 33 and CNS 39984, the ONR grant N , the Natonal Natural Scence Foundaton of Chna under Grant No and 44, and the Fundamental Research Funds for the Central Unverstes (No. DUT3JS). REFERENCES [] S. Vestal, Preemptve schedulng of mult-crtcalty systems wth varyng degrees of executon tme assurance, n RTSS, 7. [] D. de Nz, K. Lakshmanan, and R. Rakumar, On the schedulng of mxed-crtcalty real-tme task sets, n RTSS, 9. [3] H. L and S. Baruah, Load-based schedulablty analyss of certfable mxed-crtcalty systems, n Proceedngs of the tenth ACM nternatonal conference on Embedded software,. [4] S. K. Baruah, V. Bonfac, G. DAngelo, A. Marchett-Spaccamela, S. Van Der Ster, and L. Stouge, Mxed-crtcalty schedulng of sporadc task systems, n Algorthms ESA,. [] N. Guan, P. Ekberg, M. Stgge, and W. Y, Effectve and effcent schedulng of certfable mxed-crtcalty sporadc task systems, n RTSS,. [] S. Baruah, V. Bonfac, G. D Angelo, H. L, A. 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