How To Write A Flud Model Based Multprocessor Schedulng Algorthm

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1 Unversty of Pennsylvana ScholarlyCommons Departmental Papers (CIS) Department of Computer & Informaton Scence MC-Flud: Flud Model-Based Mxed-Crtcalty Schedulng on Multprocessors Jaewoo ee Unversty of Pennsylvana, Keu-My Phan Xaozhe Gu Jyeon ee Arvnd Easwaran See next page for addtonal authors Follow ths and addtonal works at: Part of the Computer Engneerng Commons, and the Computer Scences Commons Recommended Ctaton Jaewoo ee, Keu-My Phan, Xaozhe Gu, Jyeon ee, Arvnd Easwaran, Insk Shn, and Insup ee, "MC-Flud: Flud Model-Based Mxed-Crtcalty Schedulng on Multprocessors", IEEE Real-Tme Systems Symposum (RTSS 2014), December IEEE Real-Tme Systems Symposum (RTSS 2014), Rome, Italy, December 2-4, Ths paper s posted at ScholarlyCommons. For more nformaton, please contact repostory@pobox.upenn.edu.

2 MC-Flud: Flud Model-Based Mxed-Crtcalty Schedulng on Multprocessors Abstract A mxed-crtcalty system conssts of multple components wth dfferent crtcaltes. Whle mxed-crtcalty schedulng has been extensvely studed for the unprocessor case, the problem of effcent schedulng for the multprocessor case has largely remaned open. We desgn a flud model-based multprocessor mxedcrtcalty schedulng algorthm, called MC-Flud n whch each task s executed n proporton to ts crtcaltydependent rate. We propose an exact schedulablty condton for MC-Flud and an optmal assgnment algorthm for crtcalty-dependent executon rates wth polynomal-tme complexty. Snce MC-Flud cannot be mplemented drectly on real hardware platforms, we propose another schedulng algorthm, called MC- DP-Far, whch can be mplemented whle preservng the same schedulablty propertes as MC-Flud. We show that MC-Flud has a speedup factor of (1 + 5) /2 (~ 1.618), whch s best known n multprocessor MC schedulng, and smulaton results show that MC-DP-Far outperforms all exstng algorthms. Dscplnes Computer Engneerng Computer Scences Comments IEEE Real-Tme Systems Symposum (RTSS 2014), Rome, Italy, December 2-4, Author(s) Jaewoo ee, Keu-My Phan, Xaozhe Gu, Jyeon ee, Arvnd Easwaran, Insk Shn, and Insup ee Ths conference paper s avalable at ScholarlyCommons:

3 MC-Flud: Flud Model-based Mxed-Crtcalty Schedulng on Multprocessors Jaewoo ee Keu-My Phan Xaozhe Gu Jyeon ee Arvnd Easwaran Insk Shn Insup ee Dept. of Computer and Informaton Scence, Unversty of Pennsylvana, USA Dept. of Computer Scence, KAIST, South Korea School of Computer Engneerng, Nanyang Technologcal Unversty, Sngapore E-mal: Abstract A mxed-crtcalty system conssts of multple components wth dfferent crtcaltes. Whle mxed-crtcalty schedulng has been extensvely studed for the unprocessor case, the problem of effcent schedulng for the multprocessor case has largely remaned open. We desgn a flud model-based multprocessor mxed-crtcalty schedulng algorthm, called MC- Flud, n whch each task s executed n proporton to ts crtcalty-dependent rate. We propose an exact schedulablty condton for MC-Flud and an optmal assgnment algorthm for crtcalty-dependent executon rates wth polynomal complexty. Snce MC-Flud cannot construct a schedule on real hardware platforms due to the flud assumpton, we propose MC-DP-Far algorthm, whch can generate a non-flud schedule whle preservng the same schedulablty propertes as MC-Flud. We show that MC-Flud has a speedup factor of (1 + 5)/2 ( 1.618), whch s best known n multprocessor MC schedulng, and smulaton results show that MC-DP-Far outperforms all exstng algorthms. I. INTRODUCTION Safety-crtcal real-tme systems such as avoncs and automotve are becomng ncreasngly complex. Recently, there has been a growng attenton towards Mxed-Crtcalty (MC) systems n the real-tme communty. These systems ntegrate multple components wth dfferent crtcaltes onto a sngle shared platform. Integrated Modular Avoncs (IMA) [21] and AUTOSAR [2] are good examples of such systems n ndustry. These systems consst of low-crtcal and hgh-crtcal components. A dfferent degree of crtcalty requres a dfferent level of assurance. The correctness of the hgh-crtcal components should be demonstrated under extremely rgorous and pessmstc assumptons. Ths generally causes large worst-case executon tme (WCET) estmates for hgh-crtcal components, and such large WCETs could lead to neffcent system desgns. Whle certfcaton authortes (CAs) are concerned wth the temporal correctness of only hgh-crtcal components, the system desgner needs to consder the tmng requrement of the entre system under less conservatve assumptons. A challenge n MC schedulng s then to smultaneously (1) guarantee the temporal correctness of hgh-crtcal components under very pessmstc assumptons, and (2) support the tmng requrements of all components, ncludng low-crtcal ones, under less pessmstc assumptons. Whle MC schedulng has been extensvely studed for the unprocessor case, the multprocessor case has receved lttle attenton. In non-mc multprocessor schedulng, many optmal schedulng algorthms [6], [12], [16] are based on the flud schedulng model [17], where each task executes n proporton The author was a vstng student at KAIST durng the course of ths work. to a statc rate (.e., task utlzaton). Whle ts proportonal progress s stll applcable on MC systems, a sngle statc rate s neffcent because characterstcs of MC systems change over tme. If we apply the worst-case reservaton approach, n whch tasks are assgned executon rates based on ther gven crtcalty-levels, a resultng rate assgnment s neffcent because t does not consder dynamcs of the MC systems. Usng crtcalty-dependent executon rates, we can fnd an effcent flud schedulng algorthm for MC systems. In ths paper, we propose a flud schedulng algorthm whch can compute executon rate of each task dependng on a system crtcalty-level. As the system crtcalty-level changes at run tme, each task wll be executed wth ts crtcalty-dependent executon rate. A central challenge that we address n ths paper s how to determne crtcalty-dependent executon rates of all the tasks, gven that the tme nstance when the system crtcalty-level changes s unknown. Even though we have no clarvoyance on the change of the system crtcalty-level, we can optmally allocate crtcalty-dependent executon rates to each task wthn the problem doman. Contrbuton. Our contrbutons are summarzed as follows: We present a flud model-based multprocessor MC schedulng algorthm, called MC-Flud, wth crtcaltydependent executon rates for each task (Sec. III) and analyze ts exact schedulablty (Sec. IV). To our best knowledge, ths s the frst work to apply the flud schedulng model nto MC doman. We present an optmal assgnment algorthm of executon rates wth polynomal complexty (Sec. V). We derve the speedup factor of MC-Flud, (1 + 5)/2, best known n multprocessor MC schedulng (Sec. VI). We propose MC-DP-Far schedulng algorthm, whch can generate a schedule for non-flud platforms wth the same schedulablty propertes as MC-Flud (Sec. VII). Smulaton results show that MC-DP-Far sgnfcantly outperforms the exstng approaches (Sec. VIII). Related Work. Snce Vestal [23] ntroduced a mxedcrtcalty schedulng algorthm for fxed-prorty assgnment, the mxed-crtcalty schedulng problem has receved growng attenton, n partcular, on unprocessor [3], [4], [7], [8], [14], [15], [23]. For fnte jobs, Baruah et al. [7] ntroduced a prorty assgnment algorthm, called OCBP (Own Crtcalty-Based Prorty), on dual-crtcalty systems (low- and hgh-crtcalty). It s further extended wth sporadc task systems by Baruah et al. [4]. For dynamc-prorty assgnment, EDF-VD [8] s ntroduced as an EDF-based schedulng algorthm by whch hghcrtcalty tasks are assgned Vrtual Deadlnes (VDs) wth a

4 sngle system-wde parameter. Baruah et al. [3] mproved the VD assgnment scheme and derved ts speedup factor, 4/3, whch s optmal n unprocessor MC schedulng. Ekberg and Y [15] presented a VD assgnment scheme wth task-level parameters and Easwaran [14] mproved schedulablty wth another VD assgnment scheme wth task-level parameters. Wth the use of VD, the schedulablty result of low-crtcalty mode can be ncorporated nto the schedulablty analyss of hgh-crtcalty mode through the noton of carry-over jobs (a job of a task that s executed across dfferent crtcalty mode) [3], [14], [15]. Unlke the unprocessor case, the multprocessor case has not been much studed [1], [5], [19], [20]. Anderson et al. [1] frst consdered multprocessor MC schedulng wth a twolevel herarchcal scheduler. Pathan [20] proposed a global fxed prorty multprocessor schedulng algorthm for MC task systems. et al. [19] ntroduced a global schedulng algorthm wth a speedup factor of Baruah et al. [5] presented a parttoned schedulng algorthm wth a speedup factor of 8/3. II. PREIMINARIES We study the Mxed-Crtcalty (MC) schedulng problem on a hard real-tme system wth m dentcal processors. In ths paper, we consder dual-crtcalty systems wth two dstnct crtcalty levels: HI (hgh) and O (low). Tasks. Each MC task s ether a O-crtcalty task (Otask) or a HI-crtcalty task (HI-task). Each MC task τ s characterzed by (T, C, CH, χ ), where T R + s mnmum nter-job separaton tme, C R + s O-crtcalty WCET (O-WCET), C H R + s HI-crtcalty WCET (HI- WCET), and χ {HI, O} s task crtcalty level. Snce HI-WCETs are based on conservatve assumptons, we assume that 0 < C C H T. A task τ has a relatve deadlne equal to T. Any task can be executed on at most one processor at any tme nstant. Task Sets. We consder a MC sporadc task set τ = {τ }, where a task τ represents a potentally nfnte job release sequence. O-task set (τ ) and HI-task set (τ H ) are defned def def as τ = {τ τ χ = O} and τ H = {τ τ χ = HI}. Utlzaton. O- and HI-task utlzaton of a task τ are def defned as u = C /T and u H = C H/T, respectvely. System-level utlzatons of a task set τ are defned as U def = τ τ u, U H def = τ τ H u, and U H H def = τ τ H u H. System Modes. The system mode s a system-wde varable representng the system crtcalty level (O or HI). In Omode (the system mode s O), we assume that no job executes for more than ts O-WCET. In HI-mode, we assume that no job executes for more than ts HI-WCET. MC-schedulablty. MC-schedulablty ndcates both Oand HI-schedulablty: O-schedulablty mples that jobs of all O- and HI-tasks can complete to execute for ther O-WCETs before ther deadlnes n O-mode; and HIschedulablty mples that jobs of all HI-tasks can complete to execute for ther HI-WCETs before ther deadlnes n HImode. The MC System Scenaro. We assume the followng scenaro: The system starts n O-mode. In O-mode, jobs of all O- and HI-tasks are released. If a job of any HI-task τ τ H executes for more than ts O-WCET (C ), the system swtches the system mode def from O to HI (called mode-swtch). At mode-swtch, the system mmedately dscards all the jobs of O-tasks. After mode-swtch, only the jobs of HI-tasks are released. If a job of any O-task τ τ (lkewse HI-task τ τ H ) executes for more than C n O-mode (lkewse C H n HImode), we regard that the system has a fault and do not consder the case. Therefore, we assume that C = C H for each O-task τ τ wthout loss of generalty. The problem to determne the tme nstant of swtch-back from HI-mode to O-mode s beyond the scope of ths paper because t s rrelevant to schedulablty of MC systems. We can apply the swtch-back procedure n Baruah et al. [3]. III. MC-FUID SCHEDUING FRAMEWORK A. The Flud Schedulng Platform Consder a platform where each processng core can be allocated to one or more jobs smultaneously. Each job can be regraded as executng on a dedcated fractonal processor wth a speed smaller than or equal to one. Ths schedulng platform s referred to flud schedulng platform [17]. Defnton 1 (Flud schedulng platform [17]). The flud schedulng platform s a schedulng platform where a job of a task s executed on a fractonal processor at all tme nstants. Defnton 2 (Executon rate). A task τ s sad to be executed wth executon rate θ (t 1, t 2 ) R +, s.t. 0 < θ (t 1, t 2 ) 1, f every job of the task s executed on a fractonal processor wth a speed of θ (t 1, t 2 ) over a tme nterval [t 1, t 2 ], where t 1 and t 2 are tme nstants s.t. t 1 t 2 1. Schedulablty of a flud platform requres two condtons; () task-schedulablty (each task has an executon rate that ensures to meet ts deadlne) and () rate-feasblty (actve jobs 2 of all tasks can be executed wth ther executon rates). In non-mc multprocessor systems, many optmal schedulng algorthms [6], [12], [16] have been proposed based on the flud platform. These algorthms employ a sngle statc rate for each job of a task τ τ from ts release to ts deadlne: k, θ (r k, dk ) = θ where r k and dk are the release tme and deadlne of a job J k (the k-th job of task τ ), respectvely. They satsfy task-schedulablty by assgnng C /T to θ, whch s the task utlzaton of a non-mc task τ = (T, C ) where C s ts WCET. They satsfy rate-feasblty f τ θ τ m. emma 1 presents rate-feasblty condton for flud model, whch can be reused for MC systems. Task-schedulablty for MC systems s dscussed n Sec. IV. emma 1 (Rate-feasblty, from [6]). Gven a task set τ, all tasks can be executed wth executon rates ff τ τ θ m. B. MC-Flud Schedulng Algorthm The flud schedulng algorthm wth a sngle statc executon rate per task s neffcent n MC systems 3. If we assgn θ := u H for each HI task τ τ H and θ := u for each O task τ τ by the worst-case reservaton approach, the result of rate assgnment can become overly pessmstc because 1 Snce the task cannot be executed on more than one processor, θ 1. 2 An actve job of a task s a job of the task that s released but not fnshed. 3 In non-mc multprocessor systems wth an adaptve task model, Block et al. [10] dentfed neffcency of a sngle statc rate. To mprove soft realtme performance, they adjust the executon rate of a task to explot spare processng cycles.

5 task characterstc of MC system can change substantally at mode-swtch. Accordng to typcal dual-crtcalty system behavors, the system changes task characterstc at modeswtch, from executng all O- and HI-tasks to executng only HI-tasks. Thus, f a schedulng algorthm s allowed to adjust the executon rate of tasks at mode-swtch, t can reduce the pessmsm of the sngle rate assgnment, consderng the dynamcs of MC systems. We propose a flud schedulng algorthm, called MC-Flud, that executes a task wth two statc executon rates, one for O-mode and the other for HI-mode. Defnton 3 (MC-Flud schedulng algorthm). MC-Flud s defned wth O- and HI-executon rate (θ and θ H ) for each task τ τ. For a job J k of a task τ, MC-Flud assgns θ to θ (r k, mn(t M, d k )) and θh to θ (max(t M, r k), dk ) where r k s ts release tme, d k s ts deadlne, and t M s the tme nstant of mode-swtch. Snce all O-tasks are dropped at mode-swtch, θ H s not specfed for all O-tasks τ τ. Informally, MC-Flud executes each task τ τ wth θ n O-mode and wth θ H n HI-mode. Based on Def. 3, processor resources consumng by a job wthn a tme nterval s defned as executon amount. Defnton 4. Executon amounts of a job of a task τ τ n a tme nterval of length t n O- and HI-mode, denoted by E (t) and EH (t), are the total amount of processor resources that the job has consumed durng ths tme nterval n O- and HI-mode, respectvely: E def (t) = θ t and E H def (t) = θ H t. IV. SCHEDUABIITY ANAYSIS In ths secton, we analyze schedulablty of MC-Flud. MC-schedulablty (Theorem 1) conssts of O- and HIschedulablty, whch are task-schedulablty n O- and HImode (Eqs. (1) and (2)) and rate-feasblty n O- and HImode (Eqs. (3) and (4)). Theorem 1 (MC-schedulablty). A task set τ, where each task τ τ has O- and HI-executon rates (θ and θ H ), s MC-schedulable under MC-Flud ff τ τ, θ u, (1) τ τ H, u θ + uh u θ H 1, (2) θ m, (3) τ τ τ τ H θ H m. (4) To prove Theorem 1, we need to derve task-schedulablty condton n O- and HI-mode because we already know ratefeasblty condton (emma 1). Task-schedulablty n O-mode. Task-schedulablty n Omode depends only on O-task utlzaton. emma 2 (Task-schedulablty n O-mode). A task τ τ can meet ts deadlne n O-mode ff θ u. Proof: ( ) Consder a job of the task whch s fnshed n O-mode. We need to show that the executon amount of the job from ts release tme (tme 0) to ts deadlne (tme T ) s greater than or equal to O-WCET (C ). From θ u, θ T u T E (T ) C. (by Def. 4) 0 ww Mode-swtch Tme TT C HH C The executon amount Θ ww Fg. 1. The model of a carry-over job of a task τ τ H where mode-swtch happens at w and the job s executed wth θ and θ H n O- and HI-mode, respectvely (the executon amount of the job untl ts deadlne (T ) should be C H to meet ts deadlne). ( ) We wll prove the contrapostve: f θ < u, then the task cannot meet ts deadlne n O-mode. It s true because E (T ) = θ T < u T = C. Task-schedulablty n HI-mode. In HI-mode, we do not need to consder task-schedulablty of a O-task because t s dropped at mode-swtch. In addton, although a job of a HItask can be fnshed n ether O- or HI-mode, we do not need to consder the job fnshed n O-mode for task-schedulablty n HI-mode. Consder a job of a HI task τ that s fnshed n HI-mode. The job s released n ether O- or HI-mode. The job n the frst case s called carry-over job, where the job s released n O-mode and fnshed n HI-mode as shown n Fg. 1. et w R + be the length of a tme nterval from the release tme of the job to mode-swtch (or an executed tme of the job n O-mode). We do not need to consder the second case because t s a specal stuaton of the frst case when w = 0, whch means that the job s released at mode-swtch. To derve task-schedulablty for a carry-over job of τ, we need to know the relatve tme to mode-swtch (w ). We frst derve task-schedulablty condton wth a gven w. Snce mode-swtch trggers n the mddle of ts executon, the job s executed wth θ before mode-swtch and wth θ H after mode-swtch. A cumulatve executon amount of the job from ts release tme to ts deadlne (T ) conssts of ts executon amount from ts release tme to mode swtch wth θ and ts executon amount from mode swtch to ts deadlne wth θ H: E (w ) + E H (T w ) = θ w + θ H (T w ) by Def. 4. Task-schedulablty condton wth w s that the cumulatve executon amount of the job s greater than or equal to ts HI-WCET (C H ): θ w + θ H (T w ) C H. (5) Although the value of w s requred to derve taskschedulablty, MC system model assumes that tme nstant of mode-swtch s unknown untl runtme schedulng. Thus, we should consder all possble mode-swtch scenaros (any vald w ). Note that 0 w T because mode-swtch can happen any tme nstant between release tme and deadlne of the job. Then, task-schedulablty s Eq. (5) for w n [0, T ]: w, θ w + θ H (T w ) C H. (6) To sum up, task-schedulablty n HI-mode s Eq. (6).

6 For concse presentaton, we want to elmnate the term of w n task-schedulablty condton. Its dervaton s dfferent dependng on whether θ > θ H or θ θ H. emma 3 consders the frst case (θ > θ H ). emma 3. A HI-task τ wth θ > θ H can meet ts deadlne n HI-mode ff t meets ts deadlne n HI-mode when θ = θ H. Proof: ( ) et θ be the value of the orgnal θ where θ > θ H. Snce the task can meet ts deadlne n HI-mode when θ = θ H, from Eq. (6), we have w, θ H w + θ H (T w ) = θ H T C H. (7) To show that the task can meet ts deadlne n HI-mode when θ = θ, we need to show that Eq. (6) holds: w, θ w + θ H (T w ) > θ H w + θ H (T w ) = θ H T, whch s greater than or equal to C H by Eq. (7). ( ) Suppose that the task can meet ts deadlne n HI-mode. Then, we clam that θ H u H. We prove t by contradcton: suppose that θ H < u H. Then, Eq. (6) does not hold when w = 0: θ 0 + θ H (T 0) = θ H T, whch s smaller than C H because θ H T < u H T = C H. However, snce we assume that the task meets ts deadlne n HI-mode, Eq. (6) holds, whch s a contradcton. Thus, we proved the clam (θ H u H ). Next, we need to show that the task can meet ts deadlne n HI-mode when θ = θ H. Then, t s requred to show that Eq. (6) holds: w, θ H w + θ H (T w ) = θ H T, whch s greater than or equal to C H because θ H u H. Usng below corollary, we assume that θ θ H for any task τ τ H n the rest of the paper. Corollary 4. For task-schedulablty of a HI-task τ n HImode, we only need to consder the case where θ θ H. Proof: Task-schedulablty of the task n HI-mode when θ > θ H s equvalent to the one when θ := θ H by emma 3. Thus, ts task-schedulablty n HI-mode s equvalent to the one when θ θ H. emma 5 derves task-schedulablty n HI-mode by usng the assumpton that task-schedulablty n O-mode holds. It s a vald assumpton because we eventually consder task-schedulablty n both O- and HI-mode for MCschedulablty. emma 5 (Task-schedulablty n HI-mode). Gven a HI-task τ satsfyng task-schedulablty n O-mode, the task can meet ts deadlne n HI-mode ff u /θ + (u H u )/θ H 1. Proof: Consder a carry-over job of the task. We frst derve the range of a vald w and derve task-schedulablty n HI-mode by usng the range. Consder the range of w, whch s [0, T ]. We can further reduce the range by usng task-schedulablty n O-mode. The executon amount of the job n O-mode from ts release tme T C C H χ u u H θ θ H τ HI /10 1 τ HI /10 9/10 τ HI /10 1/10 τ O /10 TABE I EXAMPE TASK SET AND ITS EXECUTION RATE ASSIGNMENT. to any tme nstant cannot exceed ts O-WCET (C )4. Thus, ts executon amount from ts release tme to mode-swtch also cannot exceed C : E (w ) C θ w C. (8) Combnng 0 w T and Eq. (8), we have 0 w mn(c /θ, T ). Snce task-schedulablty n O-mode holds, we have θ u by emma 2. Then, θ C /T T C /θ. (multplyng by T /θ ) Thus, the range of vald w s 0 w C /θ. We know that task-schedulablty n HI-mode s Eq. (6), whch s rewrtten to: w, θ w + θ H (T w ) C H w, θ H T (θ H θ ) w + C H θ H T (θ H θ ) C /θ + C H ( θ H θ 0) 5 θ H T C θ H /θ + C H C 1 u θ + uh u θ H. (dvdng by θ H T ) Thus, the task can meet ts deadlne n HI-mode ff Eq. (6) holds ff 1 u /θ + (u H u )/θh. MC-schedulablty. Now, we can prove Theorem 1 by usng emmas 2 and 5 (task-schedulablty n O- and HI-mode). Proof of Theorem 1: ( ) We need to show that the task set s both O- and HI-schedulable. From Eq. (1), task-schedulablty holds n O-mode by emma 2. From Eq. (3), rate-feasblty holds n O-mode by emma 1. Then, the task set s O-schedulable. Snce Eq. (2) and task-schedulablty n O-mode hold, task-schedulablty n HI-mode holds by emma 5. From Eq. (4), rate-feasblty holds n HI-mode by emma 1. Then, the task set s HI-schedulable. ( ) We wll prove the contrapostve: f any of the condtons does not hold, then the task set s not MC-schedulable. If Eq. (1) does not hold, task-schedulablty n O-mode does not hold by emma 2. If Eq. (3) does not hold, ratefeasblty n O-mode does not hold by emma 1. If Eq. (2) does not hold, task-schedulablty n HI-mode does not hold by emma 5. If Eq. (4) does not hold, rate-feasblty n HI-mode does not hold by emma 1. Accordng to Theorem 1, the worst-case stuaton s the one where all jobs of HI-tasks are executed for ther C at modeswtch. On flud platforms, ths stuaton happens because all tasks are always executed wth ther executon rates whenever they are ready. Example 1. Consder a two-processor system where ts task set τ and ts executon rate assgnment s gven as shown n Table IV. To show that t s schedulable, we need to show that Eqs. (1), (2), (3) and (4) hold by Theorem 1. We can 4 The job trggers mode-swtch f the job execute for more than C. 5 Note that we already assumed that θ θ H by Corollary 4.

7 easly check that Eq. (1) holds. We show that Eq. (2) holds: 0.3 6/ for τ 1, 6/ /10 1 for τ 2, and 0.1 1/ for τ 3. We can check that τ τ θ = for Eq. (3) and τ τ H θ H = for Eq. (4). We conclude that the task set s MC-schedulable wth the executon rate assgnment. V. THE EXECUTION RATE ASSIGNMENT We frst defne the optmalty of an executon rate assgnment algorthm, n a smlar way to Davs et al. [13]. Defnton 5. A task set τ s called MC-Flud-feasble f there exsts an executon rate assgnment that the task set s schedulable under MC-Flud wth. An executon rate assgnment algorthm A s called optmal f A can fnd a schedulable assgnment for all MC-Flud-feasble task sets. For brevty, we refer to an executon rate assgnment as an assgnment and say that a task s feasble when the task s MC-Flud-feasble. In ths secton, we construct an effcent and optmal assgnment algorthm. A nave optmal algorthm checkng all combnatons of executon rates s ntractable because possble real-number executon rates are nfnte. Sec. V-A analyzes condtons of an optmal assgnment algorthm and formulates t as an optmzaton problem. Sec. V-B presents a polynomalcomplexty algorthm to solve the optmzaton problem. A. Condtons for an Optmal Assgnment Algorthm emma 6 and Theorem 2 present condtons for an optmal assgnment algorthm of θ and θ H, respectvely. emma 6 (An optmal assgnment of θ ). If a task set τ s feasble, there s a schedulable assgnment where () θ := u for a task τ τ and () θ := u θh for τ θ H τ H. uh +u Proof: Snce the task set s feasble, there exsts a schedulable assgnment (denoted by A) satsfyng Eqs. (1), (2), (3), and (4) by Theorem 1. () Consder a task τ τ and ts O-executon rate (θ ) n A. We wll show that τ s stll schedulable after reassgnment of θ. If we reassgn θ, t only affects Eqs. (1) and (3). et θ be the value of θ n A. Snce A s schedulable, Eq. (1) holds wth θ, whch s θ u. Suppose that we reassgn θ := u. Then, Eq. (1) stll holds because θ u. Eq. (3) stll holds because the decreased θ (from θ to u ) does not ncrease the sum of executon rates. () Consder a task τ τ H and ts O-executon rate (θ ) n A. We wll show that τ s stll schedulable after reassgnment of θ. If we reassgn θ, t only affects Eqs. (1), (2), and (3). et θ and θ H be the value of θ and θ H n A, respectvely. Snce Eq. (2) holds for θ and θ H, we have u H u θ H 1 u θ uh u θ H < 1, (9) snce u /θ u u θ θh θ H u > 0. Eq. (2) wth θ u H + u θ θh θ H θ H u H + u (θ H and θ H s rewrtten to: u H + u ) (multplyng by θ H θ ) θ. ( θ H > u H u by Eq. (9)) Snce A s schedulable, Eqs. (1), (2), and (3) hold wth θ. Suppose that we reassgn θ :=. Then, Eq. (1) u θh θ H u H +u stll holds because θ u. Eq. (2) stll holds because the value of the reassgned θ s the mnmum value satsfyng Eq. (2). Eq. (3) stll holds because the decreased θ does not ncrease the sum of executon rates. Theorem 2 (Condtons for an optmal assgnment of θ H). A task set τ s feasble ff there exsts an assgnment of θ H for τ τ H satsfyng u H θ H, (10) U + UH + u (uh u ) θ H τ τ H u H + u m, (11) θ H m. (12) τ τ H Proof: ( ) To show that the task set s feasble, we need to show that there exsts a schedulable assgnment. Consder an assgnment where θ := u for each task τ τ, θ := u θh for each task τ θ H τ H, and θ uh +u H for each task τ τ H satsfes Eqs. (10), (11), and (12). To show that ths assgnment s schedulable by Theorem 1, we need to show that t satsfes Eqs. (1), (2), (3), and (4). We know that θ for τ τ satsfes Eq. (1) and θ for τ τ H satsfes Eq. (2). We can rewrte Eq. (3) to: θ m θ + θ m τ τ τ τ τ τ H u + u θh θ H τ τ τ τ H u H + u m U + ( u + u (uh u ) ) θ H u H + u m, τ τ H whch s Eq. (11). Then, Eq. (3) holds from Eq. (11). We know that Eq. (12) s Eq. (4). Thus, we showed that the assgnment satsfes Eqs. (1), (2), (3), and (4). ( ) Snce the task set s feasble, there s a schedulable assgnment where θ = u for τ τ and θ = u θh θ H uh +u by emma 6. We need to show that θ H for τ τ H n the assgnment satsfes Eqs. (10), (11), and (12). Consder θ H of a task τ τ H. Snce we already assumed that θ θ H by Corollary 4, we can rewrte θ θ H to: u θh θ H u H + u θ H u θ H u H + u, whch s Eq. (10). Snce the assgnment s schedulable, Eqs. (3) and (4) hold by Theorem 1. We already proved that Eq. (3) s Eq. (11) n Case and knew that Eq. (4) s Eq. (12). Thus, we showed that Eqs. (10), (11), and (12) holds. An optmal assgnment algorthm can determne θ by emma 6. Although we have Theorem 2, an assgnment algorthm cannot easly determne θ H satsfyng the condtons n Theorem 2. So, we present Def. 6, whch formulates an optmzaton problem based on Theorem 2. emma 7 shows that an assgnment algorthm of θ H by Def. 6 s optmal accordng to Def. 5.

8 Defnton 6 (Assgnment Problem). Gven a task set τ, we defne a non-negatve real number X for each task τ τ H such that θ H := u H + X and X s the soluton to the followng optmzaton problem, mnmze subject to τ τ H u (uh u ) X + u X m + UH H 0, τ τ H τ τ H, X 0, (CON1) (CON2) τ τ H, X 1 + u H 0. (CON3) emma 7. An assgnment algorthm based on Def. 6 s optmal accordng to Def. 5. Proof: We clam that f the assgnment by Def. 6 cannot satsfy Eqs. (10), (11), and (12), no assgnment can satsfy those equatons. Then, τ s not feasble by Theorem 2. Suppose that we know a soluton to the optmzaton problem n Def. 6. For each task τ τ H, let X be the value of X n the soluton. et OBJ be the value of the objectve functon wth X for τ τ H. By Def. 6, we have an assgnment where θ H := u H + X to for τ τ H. Then, Eq. (10) holds from CON2 and CON3 and Eq. (12) holds from CON1. Suppose that Eq. (11) does not hold wth the assgnment, meanng OBJ > m U U H. For any set of X satsfyng CON1, CON2, and CON3, we have u (uh u ) τ τ H X +u OBJ by the defnton of the optmzaton problem and thus Eq. (11) does not hold, τ τ H u (uh u ) X + u OBJ > m U UH. Thus, no assgnment can satsfy Eqs. (10), (11), and (12). B. Convex Optmzaton for the Assgnment Problem In Sec. V-A, we formulated the optmzaton problem to construct an optmal assgnment algorthm. In ths secton, we wll solve the optmzaton problem usng convex optmzaton. Any optmzaton problem can be rewrtten n ts dual form usng agrange multpler (Chapter 5, Boyd et al. [11]), a technque to fnd a soluton for a constraned optmzaton problem. Usng agrangan multplers, we can transform any optmzaton problem wth constrants nto ts dual problem wthout constrants. In partcular, when ts objectve functon and constrants are dfferentable and convex, we can apply Karush-Kuhn-Tucker (KKT) condtons (Chapter 5.5.3, [11]), whch s a set of optmalty condtons for an optmzaton problem wth constrants. emma 8 (KKT condtons [11]). et x s a vector of x for = 1,, n where n = x and x be the value of x. Consder an optmzaton problem: mnmze f(x) subject to g j (x) 0 for j = 1,, N where N s the number of g j (x) and all of f(x) and g j (x) for j are dfferentable and convex. Then, x mnmzes f(x) ff there exsts λ s.t. j, g j (x ) 0, (13) j, λ j g j (x ) = 0, (14) f(x, ) + (λ g j (x ) j ) = 0, (15) x x j j, λ j 0, (16) where λ j s a agrange multpler, λ j s the value of λ j, and λ s a vector of λ j. emma 9 apples KKT condtons (emma 8) to the optmzaton problem n Def. 6 because the objectve functon and all the constrants are dfferentable and convex. Then, we only need to fnd the value of agrange multplers satsfyng KKT condton for the optmal soluton. For brevty of ths secton, we abbrevate τ τ H as τ because only HI-tasks are consdered n Def. 6. We use ψ, λ, and ν as agrange multplers for CON1, CON2, and CON3, respectvely. We denote a vector of X, λ, and ν by X, λ, and ν, and denote the value of X, λ, and ν by X, λ, and ν, respectvely. We defne, for each task τ, Cost (x) def = u (uh u ) (x + u )2 where x R +. emma 9. Consder the optmzaton problem n Def. 6. X mnmzes f(x) ff there exst ψ, λ, and ν s.t. τ X m + U H H 0, (17) τ, X 0, X 1 + u H 0, (18) ψ ( τ X m + U H H ) = 0, (19) τ, λ ( X ) = 0, ν (X 1 + u H ) = 0, (20) τ, Cost (X ) + ψ λ + ν = 0, (21) ψ 0 τ (λ 0 ν 0), (22) where f(x) = u (uh u ) τ X +u. Proof: We wll derve KKT condtons for the optmzaton problem n Def. 6: f(x) = u (uh u ) τ X +u and g 1 (X) = τ X m + UH H for CON1; for τ, g 2, (X) = X for CON2 and g 3, (X) = X 1 + u H for CON3. We know that f(x) and all constrants are dfferentable. To show that they are convex, we need to show that ther second dervatves are no smaller than zero: τ, X, 2 f(x) (X ) 2 = 2 u (uh u ) (X + u )3 0 and τ, τ j, X, 2 g 1 (X) (X ) 2 = 2 g 2,j (X) (X ) 2 = 2 g 3,j (X) (X ) 2 = 0. We derve KKT condtons for the problem by emma 8: We denote agrange multplers for g 1 (X), g 2, (X) and g 3, (X) by ψ, λ and ν, respectvely. Eq. (13) for g 1 (X) s Eq. (17). Eq. (13) for g 2, (X) and g 3, (X) s Eq. (18). Eq. (14) for g 1 (X) s Eq. (19). Eq. (14) for g 2, (X) and g 3, (X) s Eq. (20).

9 Eq. (15) s τ, f(x)/ X + ψ λ + ν = 0, whch s Eq. (21). Eq. (16) for g 1 (X), g 2, (X) and g 3, (X) s Eq. (22). By emma 8, X mnmzes f(x) ff there exst ψ, λ, and ν satsfyng Eqs. (17), (18), (19), (20), (21), and (22). Example 2. Consder the system n Example 1. We need to solve the optmzaton problem n Def. 6. Suppose that we have X satsfyng KKT condtons n emma 9 for the system: X1 = 0.2, X2 = 0.2, and X3 = 0. Eq. (17) holds: = 0. We can easly check that Eqs. (18) and (19) hold. To satsfy Eq. (20), we have λ 1 = λ 2 = ν2 = ν3 = 0. By Eq. (21), we have 0.15 ( ) + ψ + ν 2 1 = 0, 0.12 ( ) + ψ = 0, and ψ λ 2 3 = 0. Then, we have ψ = 1/3, ν1 = 4/15, and λ 3 = 1/3. Eq. (22) holds wth ψ, λ, and ν. Thus, X s the soluton to the problem n Def. 6 by emma 9. By Def. 6, we have θ1 H := 1, θ2 H := 0.9, and θ3 H := 0.1. We can assgn θ for τ τ by emma 6. Snce ths calculated assgnment s the same as the assgnment n Example 1, we conclude that the assgnment n Example 1 s optmal. The OERA algorthm. General KKT condtons are not easly solvable because we cannot fnd the feasble values of the agrange multplers for the condtons. We propose the Optmal Executon Rate Assgnment (OERA) algorthm to solve our KKT condtons. In our KKT condtons (emma 9), we note that λ and ν for each task τ depend only on X. Snce only ψ depends on X, f we know ψ satsfyng our KKT condtons, we can ndependently analyze λ, ν, and X for each task τ. Based on ths observaton, we develop a smple greedy algorthm. Before presentng the OERA algorthm, emma 10 shows that ths observaton s correct. emma 10. Gven a task τ and ψ ( 0), f we assgn X := 0 f ψ Cost (0), Cal X (ψ) def = 1 u H f ψ < Cost (1 u H ), u (u H u ) ψ u otherwse, then Eqs. (18), (20), and (21) hold wth some λ ( 0) and ν ( 0). Proof: To satsfy Eq. (18), we assume that 0 X 1 u H. Then, to prove ths lemma, we only need to show that Eqs. (20) and (21) hold n each case of Cal X (ψ). Case (ψ Cost (0)). For some ν 0, Eq. (21) s smplfed to: ψ Cost (X ) + λ, (23) by removng ν. Note that Cost(X ) Cost(0) where 0 X 1 u H. To satsfy Eq. (23), λ should satsfy that λ ψ Cost (X ) ψ Cost (0), whch s greater than or equal to 0 from the assumpton. To satsfy Eq. (20), we assgn ν := 0 and X := 0. Thus, Eqs. (20) and (21) hold. Case (ψ<cost (1 u H )). For some λ 0, Eq. (21) s smplfed to: ψ + ν Cost (X ), (24) by removng λ. Note that Cost(1 u H ) Cost(X ) where 0 X 1 u H. To satsfy Eq. (24), ν should satsfy that ν Cost (X ) ψ Cost (1 u H ) ψ, whch s greater than 0 from the assumpton. To satsfy Eq. (20), we assgn λ := 0 and X := 1 u H. Thus, Eqs. (20) and (21) hold. Case (Cost (1 u H ) ψ < Cost (0)). To satsfy Eq. (20), we assgn λ := 0 and ν := 0. Then, to satsfy Eq. (21), we need to satsfy ψ = Cost (X ), from whch, X can be computed: ψ = u (uh u ) u (X + u X = (uh u ) u )2. ψ Combnng three cases, we showed that Eqs. (18), (20), and (21) hold wth some postve λ and ν. To fnd ψ satsfyng our KKT condton, we dvde the range of ψ. Accordng to emma 10, f ψ s greater than or equal to Cost (0) or smaller than Cost (1 u H ), Cal X (ψ) s a constant; otherwse, Cal X (ψ) s a lnear functon of 1/ ψ. To dvde the range of ψ, we defne an ordered set G def = {x R + x = Cost (0) or x = Cost (1 u H ) for τ }, sorted n ncreasng order. We denote the j-th element of G as G j. In OERA, we utlze the property that Cal X (ψ) for some task τ changes a constant to a lnear functon of 1/ ψ or vce-versa at ψ = G j. We present the OERA algorthm (Algorthm 1). Suppose that ψ satsfyng our KKT condtons s greater than zero. We assgn X := Cal X (ψ ) for τ, whch satsfes Eqs. (18), (20), (21), and (22) by emma 10. To satsfy the remanng condtons, Eqs. (17) and (19), we requre the condton that τ X m+uh H = 0, whch s τ Cal X (ψ ) = m UH H. The HS of the requred condton s a pecewse lnear functon of 1/ ψ. To handle the pecewse functon, we note that t only changes at G j G. By examnng the value of the HS of the condton n each G j, we can fnd the range of ψ where some ψ satsfes the condton (ne 2). If the range s found, the HS of the condton s just a lnear functon wthn the range. Then, we can fnd ψ by solvng the condton (ne 3) and OERA returns X where X = Cal X (ψ ) (ne 4). However, t s possble that there does not exst postve ψ satsfyng our KKT condtons, whch means that ψ satsfyng our KKT condtons s zero because t s nonnegatve. OERA returns X where X = Cal X (0) for τ (ne 7). Theorem 3 proves the correctness of OERA. Algorthm 1 The OERA algorthm Input: τ H, G, and m Output: X satsfyng KKT condtons n emma 9 1: for j := 1 to G 1 do 2: f τ Cal X (G j+1 ) m U H H < τ Cal X (G j ) then 3: fnd ψ by solvng τ Cal X (ψ ) = m U H H 4: return X where X = Cal X (ψ ) 5: end f 6: end for 7: return X where X = Cal X (0) Example 3. Consder KKT condtons n Example 2. We have G = {0.245, 0.6, 0.75, 1.667}. We apply Algorthm 1. In Fg. 2, X-axs represents 1/ ψ. Blue (sold) lne s τ Cal X (ψ), whch s a pecewse lnear functon where ts slope s changed at each 1/ G j where G j G. Red (dashed) lne s a constant functon of m UH H. When

10 Cal_X(ψψ) mm UU HH HH 0 1/ GG / GG 1/ GG / GG / ψψ Fg. 2. The plot of τ Cal X (ψ) n respect to 1/ ψ on Example 3, whch s a pecewse lnear functon (note that there s ψ s.t. τ Cal X (ψ ) = 0.4 where G 1 ψ < G 2 ). G 1 (= 0.245) ψ < G 2 (= 0.6), the condton n ne 2 holds: τ Cal X (G 2 ) m UH H < τ Cal X (G 1 ), whch s < In other words, a crossng pont between blue lne and red lne s located n (1/ G 2, 1/ G 1 ]. Then, we can fnd ψ by solvng τ Cal X (ψ ) = m UH H, where blue lne meets red lne at 1/ ψ. Snce /ψ 0.4 = 0.4, we have 1/ ψ = 3. Fnally, ψ = 1/3, X1 = 0.2, X2 = /3 0.4 = 0.2, and X 3 = 0. We conclude that OERA can fnd X satsfyng the KKT condton n emma 9 for Example 2. Theorem 3. Algorthm 1 (OERA) can fnd X satsfyng KKT condtons n emma 9. Proof: () [ne 1-6] Suppose that there exsts ψ (> 0) satsfyng our KKT condtons. Consder some ψ > 0. We assgn X := Cal X (ψ ) for τ. By emma 10, Eqs. (18), (20), (21), and (22) hold. To satsfy Eqs. (17) and (19), we requre the condton τ X m + UH H = 0, whch s τ Cal X (ψ ) = m UH H. We need to fnd ψ satsfyng the requred condton. Snce the HS of the condton s a pecewse-lnear ncreasng functon of 1/ ψ, we can fnd a unque ψ 0 satsfyng the condton f τ Cal X (ψ ) m UH H where ψ s the maxmum value of ψ and τ Cal X (ψ ) > m UH H where ψ s the mnmum value of ψ. We know that τ Cal X (max j G j ) = 0, whch s smaller than or equal to m UH H. However, we do not know whether τ Cal X (mn j G j ) > m UH H or not. If t holds, the algorthm can fnd a unque soluton of ψ satsfyng the requred condton and return X where X = Cal X (ψ ) satsfyng the KKT condtons. If t does not hold ( τ Cal X (mn j G j ) m UH H ), there s no soluton when ψ > 0. () [ne 7] Suppose that there does not exst ψ (>0) satsfyng the KKT condtons. Then, ne 1-6 cannot fnd a soluton, whch means τ Cal X (mn j G j ) m UH H. Then, we assgn ψ := 0 because ψ s non-negatve from Eq. (22). We assgn X := Cal X (0) for τ. By emma 10, Eqs. (18), (20), (21), and (22) hold. To satsfy Eqs. (17) and (19), we requre the condton τ Cal X (0) m UH H. For a task τ, we clam that Cal X (0) = Cal X (mn j G j ). If mn j G j = 0, we have Cal X (0) = Cal X (mn j G j ). Otherwse, we have mn j G j > 0. When mn j G j Cost (1 u H ), by the second case of emma 10, we know that Cal X (0) = Cal X (mn j G j ). When mn j G j = Cost (1 u H ), we need to apply the thrd case of emma 10. Snce Cal X (ψ) calculates X s.t. ψ = Cost(X ) n the thrd case of emma 10, we have Cal X (Cost (1 u H )) = 1 uh, from whch we have Cal X (mn j G j ) = 1 u H by usng Cost (1 u H ) = mn j G j. Snce Cal X (0) = 1 u H, we have Cal X (0) = Cal X (mn j G j ). Thus, we proved the clam. By the clam above, we have τ Cal X (0) = τ Cal X (mn j G j ), whch s no greater than m UH H from the assumpton. Thus, the requred condton holds. Snce the KKT condtons hold wth ψ = 0, the algorthm returns X where X = Cal X (ψ ) satsfyng the KKT condtons Algorthm Complexty. et n be τ. Algorthm 1 (OERA) can fnd a soluton at most O(n) teraton because G 2 τ H 2n. Snce each teraton takes O(n) tme to solve the equaton n ne 3, OERA has polynomal complexty. VI. THE SPEEDUP FACTOR In ths secton, we quantfy the effectveness of MC-Flud va the metrc of processor speedup factor [18]. In general, the speedup factor of an algorthm A s defned as a real number α ( 1) such that any task set schedulable by an optmal clarvoyant algorthm 6 on m speed-1 processors s also schedulable by A on m speed-α processors. In other words, a task set that s clarvoyantly schedulable on m speed-(1/α) processors s also schedulable by A on m speed-1 processors. We consder ths specal task set (clarvoyantly schedulable on m speed-(1/α)) for speedup factor dervaton. emma 11 shows α 4/3 for non-clarvoyant algorthms. emma 11. No non-clarvoyant algorthm for schedulng dual-crtcalty task set on multprocessors can have a speedup factor better than 4/3. Proof: It follows from Theorem 5 of Baruah et al. [3] snce MC schedulng on unprocessor s a specal case of MC schedulng on m processors. For the specal task set for speedup dervaton, we present a suffcent schedulablty condton wth MC-Flud. emma 12. Gven a task set τ that s clarvoyantly schedulable on m speed-(1/α) processors, the task set s schedulable on m speed-1 processors by MC-Flud f there s x R + s.t. 0 x 1, (25) U + UH + U H H U H m, (α 1)x + 1 (26) UH H + (α 1)UH x m. (27) Proof: To show that τ s feasble by MC-Flud, we need to show that there exsts an assgnment satsfyng Eqs. (10), (11), and (12) by Theorem 2. Snce τ s clarvoyantly schedulable on m speed-(1/α) processors, we assume that U +U H m/α and U H H m/α. Snce the task s assumed to be executable on a speed-(1/α) processor, we assume that u u H 1/α for τ τ. et A be an assgnment where θ H := u H + (α 1)u x for τ τ H satsfyng Eq. (25), (26), and (27). We show that the assgned θ H by x s no greater than 1 from the defnton of the executon rate (Def. 2), for τ τ H, θ H = u H + (α 1)u x u H + (α 1)u (from Eq. (25)) u H + (α 1)u H ( u uh ) 1. (from u H 1/α) 6 In MC systems, a clarvoyant (flud or non-flud) schedulng algorthm s the one that knows the tme nstant of mode-swtch before runtme schedulng.

11 () We show that A satsfes Eq. (10): from Eq. (25), for τ τ H, 0 (α 1)u x u H u H + (α 1)u x u H θ H, whch s Eq. (10). () We show that A satsfes Eq. (11): from Eq. (26), U + U H + U + U H + U H H U H (α 1) x + 1 m τ τ H U + U H + uh u (α 1) x + 1 m u (uh u ) θ H u H + u m, τ τ H ( (α 1)u x + u = θh u H + u ) whch s Eq. (11). () We show that A satsfes Eq. (12): from Eq. (27), U H H + (α 1)U H x m τ τ H ( u H + (α 1)u x ) m τ τ H θ H m, whch s Eq. (12). Now, we consder the range of feasble x satsfyng Eqs. (25), (26) and (27). We wll fnd the lower bound of the feasble x. Eq. (26) can be rewrtten to: U + U H + U H H U H (α 1) x + 1 m U H H U H (α 1) x + 1 m U UH U H H U H m U U H (α 1) x + 1 ( m U U H > m/α U U H 0) U H H + U m m U U H (α 1) x UH H + U m (α 1)(m U U H x. ( α 1 > 0) (28) ) Snce HS of Eq. (28) s non-negatve, the lower bound of x U s H H +U m by Eqs. (25) and (28). ) (α 1)(m U U H We wll fnd the upper bound of the feasble x. Eq. (27) can be rewrtten to x. Snce (α 1)UH + U H H (α 1)U H H + U H H m U H H (α 1)U H m, we have (α 1)UH + UH H m 1 m U H H (α 1)UH. Thus, the upper bound of x s 1 by Eqs. (25) and (27). From the range of the feasble x, we can derve the condton for the exstence of x satsfyng Eqs. (25), (26), and (27): UH H + U m (α 1)(m U U H 1. (29) ) If Eq. (29) holds, MC-Flud can schedule the specal task set for speedup factor dervaton by emma 12. We derve the speedup factor of MC-Flud usng Eq. (29). Theorem 4. The speedup factor of MC-Flud s (1 + 5)/2. Proof: Consder a task set schedulable by any clarvoyant algorthm on m speed-(1/α) processors. To prove ths theorem, we need to show that the task set can be scheduled by MC-Flud on m speed-1 processors and α = (1+ 5)/2. Snce Eq. (29) s a suffcent schedulablty condton of MC-Flud for the task set, we can derve Eq. (30) as another suffcent schedulablty condton of MC-Flud: U H H + U m (α 1)(m U U H) U H H + (α 1)U H + α U α m m/α + (α 1)(m/α U ) + α U α m ( U + U H m/α and U H H m/α) m + U α m m + m/α α m. ( U U + U H m/α) (30) Eq. (30) holds when α (1 + 5)/2 because Eq. (30) s α 2 α 1 0 and α s a postve value. Snce Eq. (30) holds wth α = (1 + 5)/2, MC-Flud can schedule any task set that s clarvoyantly schedulable on m speed-(1/α) processors. Thus, the speedup factor of MC-Flud s (1 + 5)/2. Baruah et al. [3] showed that any non-clarvoyant unprocessor algorthm for a dual-crtcalty task set cannot have a speedup factor better than 4/3. Whle ths result s also applcable for multprocessors, we do not know an exact lower bound on a speedup factor for any non-clarvoyant multprocessor algorthm yet. In ths work, we show that a suffcent lower bound for multprocessors s (1 + 5)/2. VII. MC-DP-FAIR SCHEDUING AGORITHM Many flud-based schedulng algorthms, ncludng MC- Flud, rely on the fractonal (flud) processor assumpton, and ths assumpton makes them nfeasble to construct a schedule on real (non-flud) hardware platforms. Overcomng the lmtaton of flud-based algorthms, several approaches (e.g., [6], [12], [16]) have been ntroduced to construct a nonflud schedule for real hardware platforms, whle holdng an equvalent schedulablty to that of a flud-based schedule. Such approaches dffer n the unt of a tme nterval over whch they enforce the equvalence of flud-based and nonflud schedules. Quantum-based approaches (e.g., [6]) dentfy the mnmal schedulng unt (.e., a tme quantum) n hardware platforms: every tme quantum, they enforce the executon of every task to satsfy that the dfference of the executon amount between the actual schedule and the flud schedule s no greater than 1. Deadlne parttonng approaches (e.g., [12], [16]) enforce every task to meet the flud schedulng requrement only every dstnct deadlne of the system, whch suffces wth respect to schedulablty. DP-Far. We choose DP-Far [16] due to ts smplcty. DP- Far enforces the flud requrement every tme slce, defned as a tme nterval between two consecutve Deadlne Parttons (DPs), where a DP s defned as a dstnct release tme or deadlne from all jobs n the system. For a tme slce, DP-Far ensures that every task gets executed for ts requred executon amount untl the end of the tme slce, whch satsfes the flud requrement wthn the tme slce. The requred executon amount for a job of a non-mc task τ wthn a tme slce of nterval length l s calculated as l δ

12 where δ s densty of the task (δ = C /T where T s ts perod and C s ts WCET). We recaptulate the schedulablty propertes of DP-Far n the followng lemmas. emma 13 (from [16]). Gven a non-mc task set τ and a tme slce, f the task set s scheduled wthn the tme slce under DP-Far and τ δ m, then the requred executon amount of each task wthn the tme slce can be executed untl the end of the tme slce. emma 14 (from [16]). A non-mc task set τ s schedulable under DP-Far ff τ τ δ m. MC-DP-Far. Buldng upon DP-Far, we propose MC-DP- Far schedulng algorthm, whch constructs a non-flud schedule based on MC-Flud. In MC-DP-Far, DP-Far should be extended consderng the characterstcs of MC-Flud: 1) O- and HI-densty should be modfed to satsfy the flud executon requrement of MC-Flud n the end of a tme slce; and 2) the worst-case scenaros of MC-Flud and MC-DP-Far should be the same. Defnton 7 (MC-DP-Far schedulng algorthm). MC-DP- Far s defned wth a per-task vrtual deadlne and a specal DP. For each task τ τ, we defne a vrtual deadlne V R + s.t. 0 < V T. et Γ denote the earlest DP after tme nstant of mode-swtch. Accordng to DP-Far, MC-DP-Far executes each task τ τ wth V before Γ and wth T after Γ as ts deadlne. Accordng to Def. 7, the DP Γ s one of vrtual deadlnes of jobs because deadlne parttonng s performed based on vrtual deadlnes of jobs n the system and Γ s the earlest DP after mode-swtch. Note that schedulng polcy of MC- DP-Far s changed not at mode-swtch but at Γ, whch s the earlest DP after mode-swtch. By ths delayed schedulng polcy swtch, MC-DP-Far has the same worst-case stuaton as MC-Flud does. In O-mode, MC-DP-Far consders, for a task τ τ, δ def = C /V, where V s the vrtual deadlne of τ. Then, the amount of executon requred for a task τ and tme slce length l s l δ and any job of the task can be executed for C untl ts vrtual deadlne n O-mode. In HI-mode, we can derve the densty of task dependng on whether the tme nstant of mode-swtch s a DP or not. We clam that we only need to consder the frst case. Consder that mode-swtch happens n the mddle of a tme slce. Note that Γ ndcates the end of ths tme slce. MC-DP-Far executes HI-tasks for the amount of ther requred remanng executon (calculated based on C ) untl Γ. Then, the second case s equvalent to the case where mode-swtch happens at Γ, whch s a DP. Now, consder the case where mode-swtch happens at a DP. The densty of the job depends on the remanng executon amount to C H and the remanng tme to deadlne (T ) at the DP. We frst calculate the remanng tme to deadlne of the job: T w where w s tme nterval length from ts release tme to the DP. Next, we calculate the amount of remanng executon up to C H by usng emma 13. If we assume that τ τ δ m, we can compute the executon amount of the job from ts release tme to the DP: E (w ) = δ w. We can compute δ H δ H after the DP: def = CH E (w ) = CH δ w. (31) T w T w Then, the amount of executon requred for the job and tme slce length l n HI-mode s l δ H and the job s fnshed ts executon n ts deadlne n HI-mode. Vrtual Deadlne Assgnment. In ths secton, we denote the values of θ and θ H n the optmal assgnment of MC-Flud (by θ and θ H ), respectvely. Intutvely, to utlze MC- Flud analyss, the requred O-densty for a HI-task s no greater than θ and the requred O-densty for a O-task s no greater than ts task utlzaton. Def. 8 proposes a vrtual deadlne assgnment accordng to the optmal assgnment of MC-Flud. emma 15 valdates the correctness of Def. 8. Defnton 8 (Vrtual deadlne assgnment for MC-DP-Far). We assgn V := T for a O-task τ τ and V := C /θ for a HI-task τ τ H where θ s the value of θ n the optmal assgnment for MC-Flud. emma 15. Gven a feasble MC task set τ, f the task set s scheduled by MC-DP-Far wth the vrtual deadlne assgnment by Def. 8, then () δ θ for each task τ τ and () δ H θ H for each task τ τ H. Proof: () We show that δ θ for τ τ: we have δ = C /T = u = θ for τ τ by emma 6 and δ = C /V = θ for τ τ H. () We wll show that δ H θh for τ τ H. Snce δ H vares on w n Eq. (31), we wll derve the maxmum δ H and show that the value s no greater than θ H. Snce we assume τ τ δ m n dervaton of Eq. (31), we need to check that the assumpton holds: t holds because δ θ from Case () and τ τ θ m from the feasble task set. To fnd the maxmum δ H, consder dervatve of Eq. (31): d δ H d w = δ (T w ) + (C H δ w ) (T w ) 2 = CH δ T (T w ) 2, whch s greater than or equal to 0 f u H δ. To show that the dervatve s non-negatve, we show that u H δ. Snce θ δ from Case (), we only need to show u H θ : snce θ = u θh by emma 6, u H u θh θ H u H + u θ H u H +u u H (θ H u H + u ) u θ H θ H (u H u ) u H (u H u ), whch s true because θ H satsfes Eq. (10), whch s θ H u H, by Theorem 2. From the dervatve of δ H, we can fnd the maxmum δh : the maxmum δ H can be found at the maxmum w because d δ H d w 0 and thus δ H s an ncreasng functon of w. Snce the task use vrtual deadlne n O-mode, we have w V. Then, we can calculate δ H when w = V : δ H = CH δ V T V = CH C /V V T C /θ = uh u 1 u, /θ whch s θ H by emma 6. Thus, we conclude δ H θ H.

13 Acceptance Rato Acceptance Rato Acceptance Rato MC-DP-Far PART GO FP Normalzed Utlzaton Bound Normalzed Utlzaton Bound Normalzed Utlzaton Bound (a) m=2 (b) m=4 (c) m=8 Fg. 3. The acceptance rato wth varyng the normalzed utlzaton bound (U b /m) and the number of processors (m). Schedulablty Analyss. Snce MC tasks are subject to dfferent executon tme requrements (and thereby dfferent denstes), we extend emma 14 for MC systems as follows. emma 16. Gven δ and δ H for each task τ τ, a MC task set τ s MC-schedulable ff δ m, (32) τ τ τ τ H δ H m. (33) Proof: Eq. (32) s O-schedulablty by emma 14 wth O-mode. Eq. (33) s HI-schedulablty by emma 14 wth HI-mode. Snce MC-schedulablty mples both O- and HIschedulablty, Eqs. (32) and (33) are MC-schedulablty. Based on emmas 15 and 16, Theorem 5 presents that MC- DP-Far has the same schedulablty as MC-Flud. Theorem 5. A MC task set τ s schedulable by MC-DP-Far wth the vrtual deadlne assgnment by Def. 8 ff τ s MC- Flud-feasble. Proof: ( ) To show that τ s feasble, we need to show that there exsts an assgnment satsfyng Eq. (10), (11), and (12) by Theorem 2. Snce θ H for τ τ H can only volate Eq. (11) accordng to Def. 6, we need to show that Eq. (11) holds. Snce the task set s schedulable by MC-DP-Far wth Def. 8, we know that Eq. (32) holds by emma 16. Eq. (32) s rewrtten to: δ m C /T + C /V m τ τ τ τ τ τ H U + u θh θ H τ τ H u H + u m U + ( u + u (uh u ) ) θ H τ τ H u H + u m U + UH + u (uh u ) θ H u H + u m, τ τ H whch s Eq. (11). ( ) To show that the task set s schedulable, we need to show that Eqs. (32) and (33) hold by emma 16. Snce the task set s feasble, Eq. (11) wth θ H holds by Theorem 2. We already showed that Eq. (11) s Eq. (32) n Case. Snce the feasble task set s scheduled by MC-DP-Far wth Def. 8, we have δ H θ H for each task τ τ H by emma 15. We show that Eq. (33) holds: m, τ τ H δ H τ τ H θ H whch s true because the optmal assgnment satsfes CON1 n Def. 6 wth θ H = X + u H. VIII. SIMUATION In ths secton, we wll evaluate the performance of the MC-Flud framework. We compare the schedulablty of MC- DP-Far (a non-flud algorthm wth the same schedulablty as MC-Flud) wth prevously publshed MC-schedulng approaches on multprocessors: the global fpedf algorthm (GO) [19], the parttoned EDF algorthm (PART) [5], and the global fxed-prorty algorthm (FP) [20]. The speedup factors of MC-DP-Far, GO, and PART are (1 + 5)/2 ( 1.618), ( 3.236), and 8/3 ( 2.667), respectvely. Task Set Generaton. We generate random task sets accordng to the workload-generaton algorthm [19]. et U b be the upper bound of system utlzaton n both O- and HI-mode. Input parameters are U b, m (the number of processors), Z b (the upper bound of task utlzaton), and P c (the probablty of task crtcalty). Intally, m = 2, Z b = 0.7, and P c = 0.5. We wll also evaluate varyng dfferent nput parameters. A random task s generated as follows (all task parameters are randomly drawn n unform dstrbuton): u s a real number drawn from the range [0.02, Z b ]. T s an nteger drawn from the range [20, 300]. R (the rato of u H /u ) s a real number drawn from the range [1, 4]. P (the probablty that the task s a HI-task) s a real number from the range [0,1]. If P < P c, set χ := O and C := u T. Otherwse, set χ := HI, C := u T, and C H := u R T. Repeat to generate a task n the task set untl max(uh + U, U H H) s larger than U b. Then, dscard the task added last. Smulaton Results. Fg. 3 shows the acceptance rato (rato of schedulable task sets) over varyng m {2, 4, 8} and normalzed utlzaton bound U b /m from 0.3 to 1.0 n step of Each data pont s based on 10,000 task sets. The result shows that MC-DP-Far outperforms prevously known approaches. Fg. 4 and 5 show the effect of varyng dfferent parameters (P c or Z b ). We use the weghted acceptance rato [9] to reduce

14 Weghted Acceptance Rato MC-DP-Far PART GO FP Max. Task Utlzaton Fg. 4. The weghted acceptance rato wth varyng the upper bound of task utlzaton (Z b ). Weghted Acceptance Rato MC-DP-Far PART GO FP The Probablty of Task Crtcalty Fg. 5. The weghted acceptance rato wth varyng the probablty of task crtcalty (P c ). the number of dmensons n the plots. et Um b be U b /m and A(Um) b be the acceptance rato for Um. b The weghted acceptance rato W (S) s calculated to: ( U U bm S b m A(Um) ) b W (S) def = U b m S U b m where S s the set of U b /m. The set S s the same as the one for Fg. 3 (S = {0.3, 0.35,, 1.0}). In Fgs. 4 and 5, each data pont s based on 15,000 task sets where 1,000 task sets are expermented for each U b /m n S. Fg. 4 shows the weghted acceptance rato varyng the upper bound of task utlzaton (Z b ). MC-DP-Far and GO are nsenstve to Z b whle the performance of PART decreases as Z b ncreases due to dffculty of schedulng large utlzaton tasks. On the other hand, the performance of FP ncreases as Z b ncreases because nterference-based analyss favors a smaller number of tasks 7. Fg. 5 shows the weghted acceptance rato varyng the probablty of task crtcalty (P c ). MC-DP-Far can schedule all task sets when only O-tasks or only HI-tasks are generated (.e., P c = 0 or P c = 1) snce MC-DP-Far generalzng DP- Far s also optmal for the non-mc task model. IX. CONCUSION We presented a multprocessor mxed-crtcalty schedulng algorthm, called MC-Flud, based on the flud schedulng platform. Gven O- and HI-executon rates per task, we derved an exact schedulablty analyss of MC-Flud on the dual-crtcalty systems. We also presented an optmal rate assgnment algorthm wth polynomal complexty. For standard (non-flud) platforms, we presented MC-DP-Far schedulng 7 Whle FP uses nterference-based analyss, all others use utlzaton-based analyss., algorthm, whch has the same schedulng propertes as MC- Flud. We showed that MC-Flud has a speedup factor of (1+ 5)/2 ( 1.618), whch s best known n multprocessor MC schedulng, and MC-DP-Far outperforms all exstng algorthms n smulaton results. As future work, we plan to derve a tghter speedup factor and apply another schedule generaton algorthm for non-mc platforms (e.g., RUN, whch s based on a weak noton of the flud schedulng model [22]) to reduce preempton overheads, under the MC-Flud framework. We also plan to mprove the MC-Flud framework tself by consderng more than two executon rates for better schedulablty. ACKNOWEDGEMENT Ths work was supported n part by BSRP (NRF , NRF-2012R1A1A ), NCRC ( ), IITP ( , ) and KIAT (M ) funded by the Korea Government (MEST/MSIP/MOTIE). In addton, t was supported n part by ARO W911NF , ONR N , and MoE Ter-2 grant number MOE2013-T REFERENCES [1] J. H. Anderson, S. K. Baruah, and B. B. Brandenburg. Multcore operatng-system support for mxed crtcalty. In Workshop on Mxed Crtcalty, [2] AUTOSAR. AUTomotve Open System ARchtecture. [3] S. Baruah, V. Bonfac, G. D Angelo, H., A. Marchett-Spaccamela, S. Van der Ster, and. Stouge. The preemptve unprocessor schedulng of mxed-crtcalty mplct-deadlne sporadc task systems. In ECRTS, [4] S. Baruah, A. Burns, and R. Davs. Response-tme analyss for mxed crtcalty systems. In RTSS, [5] S. Baruah, B. Chattopadhyay, H., and I. Shn. Mxed-crtcalty schedulng on multprocessors. Real-Tme Systems, [6] S. Baruah, N. K. Cohen, C. G. Plaxton, and D. A. Varvel. Proportonate progress: a noton of farness n resource allocaton. In Symposum on the Theory of Computng (STOC), [7] S. Baruah, H., and. Stouge. Towards the desgn of certfable mxed-crtcalty systems. In RTAS, [8] S. K. Baruah, V. Bonfac, G. D Angelo, A. Marchett-Spaccamela, S. Van Der Ster, and. Stouge. Mxed-crtcalty schedulng of sporadc task systems. In European Symposum on Algorthms (ESA), [9] A. Baston, B. B. Brandenburg, and J. H. Anderson. Cache-related preempton and mgraton delays: Emprcal approxmaton and mpact on schedulablty. In OSPERT workshop, [10] A. Block, J. Anderson, and G. Bshop. Fne-graned task reweghtng on multprocessors. In RTCSA, [11] S. Boyd and. Vandenberghe. Convex Optmzaton. Cambrdge Unversty Press, [12] H. Cho, B. Ravndran, and E. D. Jensen. An optmal real-tme schedulng algorthm for multprocessors. In RTSS, [13] R. Davs and A. Burns. Improved prorty assgnment for global fxed prorty pre-emptve schedulng n multprocessor real-tme systems. Real-Tme Systems, [14] A. Easwaran. Demand-based schedulng of mxed-crtcalty sporadc tasks on one processor. In RTSS, [15] P. Ekberg and W. Y. Boundng and shapng the demand of mxedcrtcalty sporadc tasks. In ECRTS, [16] S. Funk, G. evn, C. Sadowsk, I. Pye, and S. Brandt. DP-Far: a unfyng theory for optmal hard real-tme multprocessor schedulng. Real-Tme Systems, [17] P. Holman and J. H. Anderson. Adaptng pfar schedulng for symmetrc multprocessors. J. Embedded Comput., [18] B. Kalyanasundaram and K. Pruhs. Speed s as powerful as clarvoyance. Journal of the ACM, [19] H. and S. Baruah. Global mxed-crtcalty schedulng on multprocessors. In ECRTS, [20] R. Pathan. Schedulablty analyss of mxed-crtcalty systems on multprocessors. In ECRTS, [21] P. Prsaznuk. Integrated modular avoncs. In Aerospace and Electroncs Conference, [22] P. Regner, G. ma, E. Massa, G. evn, and S. Brandt. RUN: Optmal Multprocessor Real-Tme Schedulng va Reducton to Unprocessor. In RTSS, [23] S. Vestal. Preemptve schedulng of mult-crtcalty systems wth varyng degrees of executon tme assurance. In RTSS, 2007.