Partitioned Mixed-Criticality Scheduling on Multiprocessor Platforms

Transcription

1 Parttoned Mxed-Crtcalty Schedulng on Multprocessor Platforms Chuanca Gu 1, Nan Guan 1,2, Qngxu Deng 1 and Wang Y 1,2 1 Northeastern Unversty, Chna 2 Uppsala Unversty, Sweden Abstract Schedulng mxed-crtcalty systems that ntegrate multple functonaltes wth dfferent crtcalty levels nto a shared platform appears to be a challengng problem, even on sngle-processor platforms. Mult-core processors are more and more wdely used n embedded systems, whch provde great computng capactes for such mxed-crtcalty systems. In ths paper, we propose a parttoned schedulng algorthm to extend the state-of-the-art sngle-processor mxedcrtcalty schedulng algorthm EY to multprocessor platforms. The key dea of s to evenly allocate tasks wth dfferent crtcalty levels to dfferent processors, n order to better explore the asymmetry between dfferent crtcalty levels and mprove the system schedulablty. Then we propose two enhancements to further mprove the schedulablty of. Experments wth randomly generated task sets show sgnfcant performance mprovement of our proposed approach over exstng algorthms. I. INTRODUCTION Mult-core processors are more and more wdely used n modern real-tme embedded systems, whch provde great computng capactes to ntegrate multple functonaltes wth dfferent crtcalty levels nto a shared hardware platform. Such mxed-crtcalty systems brng sgnfcant challenges to the desgn of real-tme systems. Vestal [14] formalzed the sngle-processor mxed-crtcalty schedulng problem. Tradtonal real-tme schedulng technques, such as EDF, may lead to poor schedulablty when appled to mxed-crtcalty systems. Researchers proposed dfferent technques to better explore the asymmetry between dfferent crtcalty levels and mprove the schedulablty [1], [4], [7], [9], [11]. Recently, Ekberg and Y [8] ntroduced a schedulng algorthm, called EY n ths paper, whch exhbts very good schedulablty. EY s based on the dea of EDF- VD (EDF wth vrtual deadlnes) proposed by Baruah et al. [3], whch balances the schedulablty on dfferent crtcalty levels by tunng the vrtual deadlnes and has been extended to multprocessors [2]. In Secton II-B we wll gve a bref ntroducton to EY. The multprocessor schedulng algorthm proposed n ths paper s based on a varant of EY. Tradtonal (sngle-crtcalty) multprocessor schedulng algorthms are usually categorzed nto two paradgms [6]: global schedulng, n whch each task can execute on any avalable processor at run-tme, and parttoned schedulng, n whch each tasks s assgned to a processor beforehand, and at run-tme each task only executes on ths partcular processor /DATE14/ c 14 EDAA Recent work n multprocessor schedulng has shown that parttoned schedulng typcally has better schedulablty than global schedulng for hard real-tme systems [5]. Therefore, n ths paper we use the parttoned approach to schedule mxedcrtcalty systems. The choce of the packng (task allocaton) strategy greatly affects the performance of parttoned schedulng. In parttoned schedulng of tradtonal real-tme systems, the frstft (FF) packng strategy typcally performs better than others. Therefore, t sounds a natural extenson to apply FF to parttoned schedulng of mxed-crtcalty systems. For example, [1] evaluates several combnatons of dfferent packng strateges and task sortng orders, and conclude that the combnaton of FF and crtcalty-decreasng task orderng performs the best among all the evaluated solutons. However, as shown n ths paper, FF actually cannot very well explore the asymmetry between dfferent crtcalty levels, and thus leads to unsatsfactory schedulablty performance. In ths paper, we propose a parttoned schedulng algorthm, whch treats hgh- and low-crtcalty tasks dfferently. frst uses worst-ft (WF) packng strategy to allocate hgh-crtcalty tasks, then uses FF to allocate low-crtcalty tasks. By such a hybrd packng strategy, hgh-crtcalty tasks are evenly allocated to dfferent processors, whch gves us a better chance to use the EY algorthm to balance the workload on dfferent crtcalty levels and mprove system schedulablty. The performance of may degrade when the number of processors s large. To address ths problem, we propose two enhancements to to further mprove the schedulablty. Frstly, we consder the stuaton that heavy low-crtcalty tasks may fal to be allocated because the free capacty of each ndvdual processor s not enough, although there stll remans much total avalable capacty on all processors. To solve ths problem, we preserve resource for heavy low-crtcalty tasks before allocatng hgh-crtcalty tasks. Secondly, we propose an optmzed vrtual deadlnes tunng algorthm n EY to mprove the performance of our parttoned algorthm. Experments wth randomly generated task systems are conducted to evaluate the performance of our proposed algorthms. Experment results show that our new algorthm, especally wth the two enhancements, can sgnfcantly mprove the system schedulablty comparng wth exstng multprocessor mxed-crtcalty schedulng algorthms.

2 II. PRELIMINARIES A. MC System Model and Notatons We adopt the same mxed-crtcalty (MC) task model as n [8]. Smlar wth the tradtonal sporadc task model, we defne an MC sporadc system π as a fnte set of ndependent MC sporadc tasks, each of whch may generate a potentally nfnte sequence of MC jobs J 1, J 2,. Each MC task s characterzed by a 4-tuple τ = (T, D, ζ, C ), where T R + s the mnmal nter-arrval separaton. D R + s the relatve deadlne. ζ {, HI} s the crtcalty level of task τ, where ndcates the low crtcalty and HI ndcates the hgh crtcalty. C : {, HI} R + s the worst-case executon tme (WCET) functon, whch specfes the WCET of the task at each crtcalty level l: C () denotes the estmated WCET of task τ under the low-crtcalty level, and C (HI) denotes the more rgdly estmated WCET under hgh-crtcalty level,.e., C () C (HI). Note that the relatve deadlnes can be arbtrary postve real numbers wthout any restrctons regardng the task perods,.e., D can be larger than, smaller than or equal to T. To descrbe the workload on dfferent crtcalty levels, we use U and U HI to denote the low- and hgh-crtcalty utlzaton of task τ respectvely: U C ()/T and U HI C (HI)/T. We use (π) {τ π ζ = } to denote the lowcrtcalty task subset of π, and use HI(π) {τ π ζ = HI} to denote the hgh-crtcalty subset. We defne the total utlzaton U (π) and U HI (π) as follows: U (π) τ π U and U HI(π) τ HI(π) U HI. The semantc of the MC system s as follows. When the system starts, t runs n the low-crtcalty mode and each task τ π could release jobs n sequence as long as no released jobs execute beyond ther respectve run-tme deadlnes. Otherwse, once any job has executed for ts WCET under low-crtcalty level wthout sgnalng ts completon of executon, the system wll swtch to hgh-crtcalty mode mmedately. In order to guarantee the hgh-crtcalty jobs stll meet ther deadlnes even though they execute for up to the more strct WCET estmated under such hgh-crtcalty mode, all of the low-crtcalty jobs need not meet deadlnes any more, and wll be dscarded totally after the mode swtchng. Therefore, the low-crtcalty jobs wll make no nterference wth the schedulng of hgh-crtcalty jobs. To schedule such an MC system successfully, all of the jobs released from hgh-crtcalty tasks must always meet ther deadlnes both n low-crtcalty and hgh-crtcalty modes, but the jobs released from low-crtcalty tasks only need to meet deadlnes n low-crtcalty mode. B. The EY Approach The parttoned multprocessor schedulng algorthm proposed n ths paper uses EY [8] to test and tune the schedulablty on each ndvdual processor. The man dea of EY can be summarzed as follows. When the system runs n the low crtcalty mode, each hgh-crtcalty task τ uses a vrtual deadlne D () (shorter than ts orgnal deadlne D ) n the EDF schedulng decson. Ths helps the hgh-crtcalty task to fnsh ts low-crtcalty workload earler, and thus have a better chance to fnsh the remanng hgh-crtcalty workload before ts real deadlne f the crtcalty mode-swtch occurs. Consder the example task set llustrated as below: Task T D ζ C () C (HI) U U HI τ HI τ Suppose the hgh-crtcalty task τ 1 releases a job J at t n the low-crtcalty mode. Under EDF schedulng, n the worst-case τ 1 fnshes t low-crtcalty workload one tme unt before ts absolute deadlne t + D 1, as shown n Fgure 1a. If the executon of τ 1 overflows and the system swtches to the hgh-crtcalty mode, t s mpossble to fnsh the remanng workload C 1 (HI) C 1 () before ts absolute deadlne t + D 1, as shown n Fgure 1a. If we set τ 1 s vrtual deadlne D 1 () = 6, we can guarantee that τ 1 fnshes ts low-crtcalty workload no later than t + D 1 (). In ths case, there s enough tme for τ 1 to fnsh ts hgh-crtcalty workload before ts real deadlne t + D 1 after the system runs nto the hgh-crtcalty mode, as shown n Fgure 1b. Resource demand J : C 1() C 1(HI) C 1() t t + D 1 T (a) D 1 () = 1 J : C 1() C 1(HI) C 1() T t t + D 1() t + D 1 dbf HI dbf Tme nterval length (t) (a) D 1 ()=1 (b) D 1 () = 6 Fgure 1. Impact of dfferent vrtual deadlnes. Resource demand dbf HI dbf Tme nterval length (t) (b) D 1 ()=6 Resource demand dbf HI dbf Tme nterval length (t) (c) D 1 ()=4 Fgure 2. Impact on demand of dfferent vrtual deadlnes. Usng shorter vrtual deadlnes s benefcal to the schedulablty n the hgh-crtcalty mode, but at the same tme the system s more dffcult to be scheduled n the lowcrtcalty mode (snce each hgh-crtcalty task must meet a shorter vrtual deadlne). In the above example, f we set D 1 () = 4, the system s not schedulable n the lowcrtcalty mode. In EY, ths phenomenon s formally captured by the hgh-crtcalty demand bound functon dbf HI (τ, ) and low-crtcalty demand bound functon dbf (τ, ). The system s guaranteed to be schedulable n the low-crtcalty mode f t holds

3 t : dbf (τ, t) t. (1) τ π The system s guaranteed to be schedulable n the hghcrtcalty mode f t holds t : dbf HI(τ, t) t. (2) τ π ζ =HI Due to space lmt, we do not recte the computaton of dbf HI (τ, ) and dbf (τ, ), but refer nterested readers to [8] for detals. For each hgh-crtcalty task τ, f ts vrtual deadlne D () s decreased, the hgh-crtcalty demand bound functon dbf HI decreases whle the low-crtcalty demand bound functon dbf ncreases. Fgure 2 shows the demand bound functons when τ 1 uses dfferent vrtual deadlnes. If the vrtual deadlne s too short, the system s non-schedulable n the lowcrtcalty mode as shown n Fgure 2a. If the vrtual deadlne s too long, the system s non-schedulable n the hgh-crtcalty mode, as shown n Fgure 2c. The system s schedulable n both modes wth a proper D () as shown n Fgure 2b. The vrtual deadlnes of hgh-crtcalty tasks are used as the nob to tune the schedulablty n dfferent crtcalty modes. However, t leads to a larger search space to fnd the optmal vrtual deadlne confguraton. EY uses an effectve heurstc to tune the vrtual deadlnes. It tunes each D () from D to C () monotoncally to control the overall tme complexty. At each tunng pont, EY decreases certan D () to D () 1, and EY always greedly selects the one whch gets the maxmal value of the expresson: dbf HI(τ k, D k ()) dbf HI(τ k, D k () 1). (3) III. PARTITIONING ALGORITHM Parttoned schedulng for EDF-VD has been studed n [3]. And n ths secton we wll dscuss the part- tonng schemes for the hgher performance algorthm EY. A. Motvatng the Hybrd Packng Strategy Before ntroducng our proposed parttonng algorthm wth a hybrd packng strategy, we frst dscuss the drawback of extendng EY to multprocessor schedulng wth the FF packng strategy and motvate the hybrd packng strategy used n ths paper. [1] presented a parttonng strategy usng FF packng and decreasng-crtcalty task orderng (called FFDC for short). FFDC packs as many hgh-crtcalty tasks as possble to one processor untl t s full, then pcks the next processor to pack the remanng hgh-crtcalty tasks. After all hghcrtcalty tasks are successfully allocated, FFDC contnues to pack low-crtcalty tasks also wth the FF strategy. FFDC has the best performance among all the parttonng algorthms evaluated n [1]. Unfortunately, the combnaton of the FFDC strategy and the EY approach does not yeld satsfactory performance, the reason of whch s as follows. The strength of EY s the capablty of balancng the schedulablty between hgh- and low-crtcalty levels. In FFDC, the allocaton of hgh-crtcalty tasks s typcally unbalanced, so the room for EY to tune the vrtual deadlnes s relatvely smaller. Consder the followng task set to be scheduled on two processors: Task T D ζ C () C (HI) U U HI τ HI τ HI τ τ By FFDC, the two hgh-crtcalty tasks τ 1 and τ 2 wll be allocated to processor P 1, after whch we allocate lowcrtcalty tasks. However, the low-crtcalty utlzaton sum of τ 1 and τ 2 s already.4, so τ 3 or τ 4 cannot be allocated to P 1, no matter how do we tune the vrtual deadlnes of τ 1 and τ 2 (snce.4+.7>1). On the other hand, t s nfeasble to allocate both τ 3 and τ 4 to P 2 snce.7+.7>1. So the parttonng of ths task set s faled wth the FF packng strategy. If we use worst-ft (WF) packng strategy to partton ths task set, we can allocate τ 1 and τ 3 to P 1, and allocate τ 2 and τ 4 to P 2. By applyng the vrtual deadlne tunng of EY, we see that tasks on both processors are schedulable f the vrtual deadlnes of τ 1 and τ 2 are set to be 6. In summary, WF dstrbutes hgh-crtcalty tasks to dfferent processors more evenly than FF, whch gves us more room to utlze the vrtual deadlne tunng n EY to mprove the schedulablty. In the allocaton of hgh-crtcalty tasks, the vrtual deadlnes of hgh-crtcalty tasks are tuned to guarantee that they are schedulable n the hgh-crtcalty mode. As soon as all hgh-crtcalty tasks are allocated, the vrtual deadlnes of all hgh-crtcalty tasks are fxed. So when we start to allocate low-crtcalty tasks, the remanng capacty on each processor s fxed. So the parttonng of low-crtcalty tasks s smlar to the tradtonal workload parttonng problem (wthout multple crtcalty levels). Snce FF has proven to be the best packng strategy for such a problem, we shall use FF as the packng strategy for low-crtcalty tasks. B. Now we ntroduce our new parttonng algorthm (Mxed-crtcalty Parttonng wth Vrtual Deadlnes) n detal. We frst ntroduce the notaton of (hgh- or low-crtcalty) remanng utlzaton of a processor, whch denotes the dfference between 1 and the total (hgh- or low-crtcalty) utlzaton of tasks that have been allocated to ths processor. For example, f only one task wth utlzaton.3 s allocated to processor P 1, then the remanng utlzaton of P 1, s 1-.3=.7. We use U HI (P 1 ) and U (P 1 ) to denote the hgh- and low-crtcalty remanng utlzaton of processor P 1. works n the followng three steps: 1) Allocate hgh-crtcalty tasks to processors by the WF packng strategy,.e., always select the processor P x wth the maxmal hgh-crtcalty remanng utlzaton U HI (P x ) to allocate tasks. The tasks are sorted n the decreasng order of ther hgh-crtcalty utlzaton U HI. 2) Tune the vrtual deadlnes of hgh-crtcalty tasks allocated to each processor by the tunng algorthm n EY

4 [8] to meet the dbf HI constrant (2),.e., to guarantee that the hgh-crtcalty task subset on each processor s schedulable n the hgh-crtcalty mode. If the tunng algorthm n EY fals on any processor, the parttonng algorthm fals. 3) Allocate low-crtcalty tasks to processors by the FF packng strategy. The tasks are sorted n the decreasng order of ther low-crtcalty utlzaton U. In ths step, we use the dbf constrant (1) to check whether an unallocated low-crtcalty task can be assgned to a canddate processor,.e., whether the tasks already assgned to ths processor and the current task can meet deadlnes n the low-crtcalty mode f they are scheduled together on ths processor. Note that the frst step of only allocates hghcrtcalty tasks to processors, but does not guarantee anythng about the schedulablty. After all hgh-crtcalty tasks are allocated, we tune ther vrtual deadlnes and decde whether they are schedulable n the hgh-crtcalty mode. The vrtual deadlne tunng algorthm n EY s of pseudo-polynomal tme complexty, and n practce rather tme consumng. It wll be extremely neffcent f the vrtual deadlne tunng algorthm has to be nvoked teratvely. only needs to nvoke the vrtual deadlne tunng algorthm once on each processor. At run-tme, the tasks allocated to each processor are scheduled n the same way as EY. When the system s n the low-crtcalty mode, tasks are scheduled by EDF and each hgh-crtcalty task uses ts vrtual deadlne n the EDF schedulng decson. When the system runs nto the hghcrtcalty mode, all the low-crtcalty tasks are abandoned mmedately, and all hgh-crtcalty tasks use ther orgnal deadlnes n the EDF schedulng decson. IV. ENHANCEMENTS TO The parttonng algorthm ntroduced n last secton may stll lead to unsatsfactory performance under certan crcumstances. In ths secton, we enhance by optmzed task allocaton strateges and vrtual deadlne tunng algorthms to further mprove the schedulablty. A. Heavy Low-Crtcalty Task Aware Parttonng One of the most mportant ssues n parttoned schedulng s to avod the stuaton that a heavy (hgh-utlzaton) task cannot be allocated whle the processors stll have relatvely large avalable capactes. The decreasng utlzaton task orderng s an effectve way to avod the above stuaton. Therefore, we choose to use decreasng utlzaton to order tasks n the allocaton of hgh- and low-crtcalty tasks (step 1 and 3) respectvely. However, snce frst allocates all hghcrtcalty tasks before allocatng any low-crtcalty task, t may happen the stuaton that some heavy low-crtcalty tasks cannot be allocated to any ndvdual processor although there remans much total remanng capacty on all processors. For example, consder the task set n Table I to be parttoned on two processors. By, each processor wll be assgned 2 hgh-crtcalty tasks as llustrated n Fgure 3a. Therefore, the Table I AN EXAMPLE TASK SET Task T D ζ C () C (HI) U U HI τ HI τ HI τ HI τ HI τ low-crtcalty task τ 5 cannot be allocated to any of the two processors no matter how do we tune the vrtual deadlnes of the hgh-crtcalty tasks (snce >1). τ 3 τ 1 P 1 (a) τ 4 τ 2 P 2 τ 4 τ 5 P 1 (b) -HA Fgure 3. Task allocaton results n dfferent approaches To address ths problem, we enhance by a heavy low-crtcalty task aware polcy. The man dea s to make a tradeoff between hgh-crtcalty workload balancng and the schedulablty of low-crtcalty tasks by preservng moderate free resources for heavy low-crtcalty tasks, whch are defned as follows: Defnton IV.1 (Heavy Low-Crtcalty Task). Gven a mxedcrtcalty system π, a low crtcalty task τ k s heavy f U k τ 3 τ 2 τ 1 P 2 > 1 U(πHI) m. (4) The enhanced heavy low-crtcalty task aware parttonng algorthm, called -HA for short, dffers from by addng one step before startng : Select all of the heavy low-crtcalty tasks satsfyng (4) and relate each heavy low-crtcalty task to one processor. If a heavy low-crtcalty task τ b s related to processor P x, then we set the ntal hgh-crtcalty remanng utlzaton of P x as U HI(P x) 1 U b. (5) After that, the parttonng algorthm follows the same procedure as. It s easy to see that f the number of heavy low-crtcalty tasks s greater than the processor number m, then the task set cannot be successfully scheduled by any algorthm (snce n that case the total low-crtcalty utlzaton of the task set s larger than m).

5 In the example of Table I, snce 1 U (π HI )/2 =.6 < U5 =.7, τ 5 s a heavy low-crtcalty task and s related to P 1. So U HI (P 1 ) s set to 1.7 =.3. After that, followng the parttonng algorthm, τ 1, τ 2, τ 3 wll be allocated to P 2 and τ 4 wll be allocated to P 1 as llustrated n Fgure 3b. Consequently, P 1 reserves enough free capacty to adopt the heavy low-crtcalty task τ 5 n step 3 of, and the task set s successfully parttoned. B. Improvng the Vrtual Deadlne Tunng Algorthm A major reason for EY to acheve extremely good schedulablty s that t employs a very effectve heurstc algorthm to tune the vrtual deadlnes of hgh-crtcalty tasks. However, the tunng algorthm n EY s not completely sutable to. Recall that when the tunng algorthm s nvoked n step 2 of, only hgh-crtcalty tasks have been allocated and no nformaton of low-crtcalty s known at that moment. If one uses (3) to choose the task to shrnk ts vrtual deadlne, t may lead to very fast ncrease of dbf, whch s harmful to the schedulablty n the low-crtcalty mode. To address ths problem, we use the balance factor as the metrc to choose tasks for vrtual deadlne tunng. Defnton IV.2 (Balance Factor). For a hgh crtcalty task τ k and tme nterval sze t, the balance factor s defned as φ(τ k, t) = dbfhi(τ k, D k ()) dbf HI(τ k, D k () 1) C k ()/(D k () 1) C k ()/D k (). (6) The denomnator C k ()/(D k () 1) C k ()/D k (), ntutvely, represents the cost of the schedulablty n the lowcrtcalty mode when the vrtual deadlne of a hgh-crtcalty task s decreased by 1. Our new tunng algorthm always chooses the task wth the maxmal φ(τ k, t) value, whch mproves the schedulablty n the hgh-crtcalty mode at the mnmal cost of the schedulablty n the low-crtcalty mode. V. EXPERIMENTAL EVALUATION We conduct experments wth randomly generated task systems to compare the performance, n terms of acceptance rato, of the algorthms proposed n ths paper and prevous multprocessor schedulng algorthms for mxed-crtcalty systems. The prevous algorthms evaluated n our experments nclude both global and parttoned schedulng algorthms. Our experments show that the performance of parttoned schedulng s sgnfcantly better than global schedulng (the algorthms n [12], [13]), whch s smlar to the multprocessor schedulng of tradtonal real-tme task systems. Therefore, n ths secton we only report the comparson between the algorthms proposed n ths paper and other parttoned schedulng algorthms. The evaluated algorthms nclude: : the parttonng algorthm n Secton III. -HA: enhanced by the heavy lowcrtcalty task aware allocaton polcy n Secton IV-A. -HA-BF: -HA further enhanced by the optmzed vrtual deadlne tunng n Secton IV-B. : the parttoned schedulng algorthm based on the Audsley approach n [1]. : the straghtforward extenson of EY to parttoned schedulng wth the FF packng strategy as dscussed n the begnnng of Secton III. MC-Partton: the parttoned schedulng algorthm based on EDF-VD approach n [2]. A. Random Task Set Generaton Our experments use dual-crtcalty mplct deadlnes sporadc task model on an m dentcal unt speed multprocessor platform. We use a smlar approach as n [8] to generate random mxed-crtcalty task sets. The random task s generated by four tunable parameters: the probablty P HI of beng of hgh-crtcalty, the maxmal rato R HI between hgh- and low-crtcalty executon tme of each hgh-crtcalty task, the maxmal low-crtcalty executon tme C max and the maxmal perod T max. Each new task τ s generated as follows: ζ = HI wth probablty P HI, otherwse ζ =. C () s a randomly generated nteger unformly drawn from [1, C max ]. C (HI) s a randomly generated nteger unformly drawn from [C(), R HI C ()] f ζ = HI. Otherwse, C (HI) = C (). T s a randomly generated nteger unformly drawn from [C (HI), T max ]. D = T because of the mplct deadlne constrant. Each random task set s generated wth a target normalzed average utlzaton U wth a acceptable range of errors: Umn = U.5 m and Umax = U +.5 m. A random task set s generated by startng wth an empty task set π =, to whch random tasks are successvely added. We generate a new random task and append t to π contnually as long as mn(u (π), U HI (π)) < Umn. If a task s added such that max(u (π), U HI (π)) > Umn, we dscard the whole task set and start wth a new empty task set. If a task s added such that Umn mn(u (π), U HI (π)) and max(u (π), U HI (π)) Umax, the task set s fnshed, unless all tasks n π have the same crtcalty level or U (π), U HI (π) >.99, n whch case the task set s nstead dscarded. B. Results The parameter confguraton of the experments n Fgure 4 s as follows: P HI =.5, R HI = 4, C max = 1 and T max =. Fgure 4a to Fgure 4d show the acceptance rato,.e., the porton of schedulable task sets out of all the random task sets generated at ths utlzaton range, as a functon of normalzed average utlzaton wth dfferent processor numbers. Each pont n any fgure ncludes at least randomly generated task sets. Fgure 4e and Fgure 4f llustrate the effect of varyng random parameters R HI and P HI through the weghted acceptance rato functon of the vared parameters. For each certan vared parameter, we compute acceptance ratos A(U ) wth dfferent U, and let W (U ) > be the weghted factor for target utlzaton U. The weghted acceptance rato s denoted as U W (U ) A(U )/ U W (U ). Each pont n the two fgures ncludes at least randomly generated task sets.

6 1 1 1 Acceptance rato (%) HA-BF -HA MC-Partton Normalzed average utlzaton (a) Result on 2-processor system Acceptance rato (%) HA-BF -HA MC-Partton Normalzed average utlzaton (b) Result on 4-processor system Acceptance rato (%) HA-BF -HA MC-Partton Normalzed average utlzaton (c) Result on 8-processor system Acceptance rato (%) HA-BF -HA MC-Partton Normalzed average utlzaton (d) Result on 16-processor system Weghted acceptance rato (%) HA-BF -HA MC-Partton Varyng R HI (e) Varyng R HI on 4-processor system Fgure 4. Experment results (P HI =.5, R HI = 4, C max = 1andT max = ) Weghted acceptance rato (%) HA-BF -HA MC-Partton Varyng P HI (f) Varyng P HI on 4-processor system Fgure 4 shows that performs better than DC- Partton, and wth fewer number of processors. However, the performance of degrades as the number of processors becomes greater. Wth 16 processors, may fal to partton task sets wth rather low total normalzed average utlzaton. Ths s manly because of the problem we ponted out at the begnnng of Secton III. The enhanced algorthm -HA can solve ths problem and thus steadly exhbts better performance than and. Moreover, wth the optmzed vrtual deadlne tunng algorthm based on the balance factor, the acceptance rato of -HA-BF s further mproved. VI. CONCLUSIONS In ths paper we studed the parttoned schedulng algorthm for mxed-crtcalty systems on multprocessors. We proposed new parttoned schedulng algorthms based on a hybrd task packng strategy and the state-of-the-art sngle-processor mxed-crtcalty schedulng algorthm EY [8]. Experments wth randomly generated task sets showed sgnfcant mprovement of our proposed approach over exstng algorthms. ACKNOWLEDGEMENTS Supported n part by Chna Fundamental Research Funds for the Central Unverstes under grant N141 and N11843; and Chna Research Fund for the Doctoral Program of Hgher Educaton under grant ; and Chna Scence Fund for Youths under grant REFERENCES [1] S. Baruah, H. L, and L. Stouge, Towards the desgn of certfable mxed-crtcalty systems, n RTAS, 1. [2] S. Baruah, B. Chattopadhyay, H. L, and I. Shn, Mxed-crtcalty schedulng on multprocessors, Real-Tme Systems, 13. [3] S. K. Baruah, V. Bonfac, G. D`Angelo, A. Marchett-Spaccamela, S. Van Der Ster, and L. Stouge, Mxed-crtcalty schedulng of sporadc task systems, n Algorthms-ESA, 11. [4] S. Baruah, A. Burns, and R. Davs, Response-tme analyss for mxed crtcalty systems, n RTSS, 11. [5] A. Baston, B. B. Brandenburg, and J. H. Anderson, An emprcal comparson of global, parttoned, and clustered multprocessor edf schedulers, n RTSS, 1. [6] J. Carpenter, S. Funk, P. Holman, A. Srnvasan, J. Anderson, and S. Baruah, A categorzaton of real-tme multprocessor schedulng problems and algorthms, Proceedngs of the IEEE, 4. [7] D. de Nz, K. Lakshmanan, and R. Rajkumar, On the schedulng of mxed-crtcalty real-tme task sets, n RTSS, 9. [8] P. Ekberg and W. Y, Outstandng paper award: Boundng and shapng the demand of mxed-crtcalty sporadc tasks, n ECRTS, 12. [9] N. Guan, P. Ekberg, M. Stgge, and W. Y, Effectve and effcent schedulng of certfable mxed-crtcalty sporadc task systems, n RTSS, 11. [1] O. R. Kelly, H. Aydn, and B. Zhao, On parttoned schedulng of fxedprorty mxed-crtcalty task sets, n TrustCom, 11. [11] H. L and S. Baruah, An algorthm for schedulng certfable mxedcrtcalty sporadc task systems, n RTSS, 1. [12], Outstandng paper award: Global mxed-crtcalty schedulng on multprocessors, n ECRTS, 12. [13] R. Pathan, Schedulablty analyss of mxed-crtcalty systems on multprocessors, n ECRTS, 12. [14] S. Vestal, Preemptve schedulng of mult-crtcalty systems wth varyng degrees of executon tme assurance, n RTSS, 7.