Available online at ScienceDirect. Procedia Computer Science 24 (2013 ) 9 14

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1 Avalable onlne at ScenceDrect Proceda Computer Scence 24 (2013 ) th Asa Pacfc Symposum on Intellgent and Evolutonary Systems, IES2013 A Proposal of Real-Tme Schedulng Algorthm based on RMZL and Schedulablty Analyss Ken YANAI a, *, Myungryun YOO a, Taanor YOKOYAMA a a Toyo Cty Unversty, Tamazutsum,Setagaya-u,Toyo ,Japan Abstract Recently, multprocessor platform s generally used n embedded real tme systems. The optmal real tme schedulng algorthms for multprocessor are demanded. Several algorthms based on RM are proposed. In ths study, we propose RMZLPD based on RMZL appled zero-laxty rule to RM. RMZLPD can realze hgh parallelsm. Through smulaton, RMZLPD has shown the hgh schedule success rato. The schedulablty of proposed algorthm also s shown by response tme analyss The Authors. Plshed by Elsever B.V The Authors. Plshed by Elsever B.V. Open access under CC BY-NC-ND lcense. Selecton Selecton and and peer-revew peer-revew under under responsblty responsblty of of the the Program Program Commttee Commttee of IES2013. of IES2013 Keywords: real-tme schedulng algorthm, RMZL, RMZLPD, schedulablty 1. Introducton Real tme systems are characterzed by computatonal actvtes wth tmng constrants. Tmng constrants n real tme applcatons are predomnantly soft n that deadlnes may be mssed as long as the long run fracton of the processng tme allocated to each tas n the applcaton s n accordance wth ts utlzaton. A system desgn that can guarantee that deadlne msses, f any, are bounded by constant amounts s suffcent to provde guarantees on long term processor shares. Hence, schedulng methods that ensure bounded deadlne msses and that can be appled when other methods cannot are of consderable value and nterest 1. Multprocessor schedulng are usually categorzed nto two paradgms: global schedulng, n whch each tas can execute on any avalable processor at runtme, and parttoned schedulng n whch each tass s assgned to a processor beforehand, and at runtme each tas can only execute on ths partcular processor. Parttoned schedulng enjoys relatvely easer desgn and analyss. On the other hand, global schedulng on average utlzes computng * Correspondng author. Tel.: (Ext. 3737). E-mal address: g @tcu.ac.jp The Authors. Plshed by Elsever B.V. Open access under CC BY-NC-ND lcense. Selecton and peer-revew under responsblty of the Program Commttee of IES2013 do: /j.procs

2 10 Ken Yana et al. / Proceda Computer Scence 24 ( 2013 ) 9 14 resource better, and s more robust n the presence of tmng errors 2. Global schedulng algorthms are based on wdely optmal unprocessor schedulng algorthms le RM (Rate Monotonc) and EDF (Earlest Deadlne Frst) by Dhall 3. However, these algorthms aren t optmal on multprocessor systems. Although many schedulng algorthms based on RM or EDF are proposed, optmal algorthms aren t establshed yet. Compared to EDF, RM s easy to mplement and tny jtter. Thus, n ths paper, we focus n global schedulng algorthm based on RM. Taeda et al. proposed RMZL (Rate Monotonc untl zero laxty) based on RM 4 and Nshga et al. proposed LP- RMZL (Lmted Preemptve-RMZL) based RMZL 5. These algorthms show hgher schedule success rato than that of RM. However, there are stll reducble deadlne mss. In ths paper, we propose a new schedulng algorthm, called RMZLPD (RMZL wth Pseudo Deadlne). The proposed algorthm domnates global RM schedulng, RMZL and LP-RMZL. Also, we analyse schedulablty of RMZLPD usng Response Tme Analyss (RTA). The rest of the paper s organzed as follows: In Secton 2, we explan system model. In Secton 3, global RM, RMZL and LP-RMZL are explaned more detal. Secton 4 ntroduces proposed RMZLPD. In secton 5, a schedulablty of RMZLPD s analysed. Then, the expermental results are llustrated and analysed n Secton 6. Fnally, Secton 7 provdes dscusson and suggestons for further wor on ths problem. 2. System Model The notaton descrbed n ths Secton. We consder a set of n perodc tass to be scheduled on m symmetrc processors usng a global algorthm. Each tas = (C, T ) s characterzed by a worst-case computaton tme C, a perod T. The utlzaton of a tas s defned as U = C /T. The system utlzaton s defned as U = (U )/m.a tas s a sequence of jobs J j, where each job s characterzed by an arrval tme r j and a fnsh tme f j. Moreover, each job has an absolute deadlne d j = r j + T. The laxty of a job at tme t s defned as L j = d j t C j (t), where, C j (t) s a remanng executon tme of job J j at tme t. 3. Related Wors 3.1. global RM (Rate Monotonc) Global RM s preemptve fxed prorty schedulng algorthm. Tass wth hgher request rates wll have hgher prortes n global RM. But, owng to low schedulablty, t s nown that global RM whch s appled for multprocessor platforms s not optmal schedulng algorthm RMZL (Rate Monotonc untl zero laxty) global RM RMZL Fg. 1. Example of RM schedule and RMZL schedule

3 Ken Yana et al. / Proceda Computer Scence 24 ( 2013 ) RMZL 4 s based on global RM. Under RMZL, jobs are scheduled accordng to the fxed prorty of ther assocated tas, untl a stuaton s reached where the remanng executon tme of a job s equal to the tme to ts deadlne. Such a job has zero laxty and wll mss ts deadlne unless t executes contnually untl completon. RMZL gves the hghest prorty to such zero-laxty jobs. The schedules produced by RMZL and global RM schedulng are dentcal untl the latter fals to execute a tas wth zero laxty. Such a tas wll ssequently mss ts deadlne. Hence RMZL domnates global RM schedulng, n the sense that all prorty ordered tas sets that are schedulable accordng to global RM schedulng are also schedulable accordng to RMZL. Fgure 1 shows the schedulng example of global RM and RMZL, when three perodc tass, 1 = 2 = 3 = (2, 3) are smtted on two processors. As shown n the fgure 1, 3 msses a deadlne at tme 3 n global RM. On the other hand, all the three tass are successfully scheduled by RMZL, snce the prorty of 3 s promoted to the top at tme 1 due to zero laxty LP-RMZL (Lmted Preemptve-RMZL) LP-RMZL 5 s based on RMZL. Under LP-RMZL, runnng jobs are not preempted by hgher prorty tass except for zero-laxty tass. Compared to RMZL, LP-RMZL reduced preempton and mproved success rato and schedulablty. Fgure 2 shows the schedulng example of RMZL and LP-RMZL, when four perodc tass, 1 = 2 = (1, 2), 3 = (1, 4) and 4 = (6, 8) are smtted on two processors. As shown n the fgure 2, 4 msses a deadlne at tme 8 n RMZL. On the other hand, all the four tass are successfully scheduled by LP-RMZL, snce the 4 s not preempted at tme 2, 4, and 6. RMZL LP-RMZL Fg. 2. Example of RMZL schedule and LP-RMZL schedule 4. Related WorsRMZLPD(RMZL wth Pseudo Deadlne) We propose RMZLPD added pseudo deadlne to RMZL. Under RMZL, jobs are scheduled accordng to RMZL, untl a stuaton s reached where the remanng pseudo executon tme of a job s equal to the tme to ts pseudo deadlne. Such a job has pseudo zero laxty and wll mss ts pseudo deadlne unless t executes contnually untl ts pseudo deadlne. RMZLPD gves the sem hghest prorty to such pseudo zero-laxty jobs untl ts pseudo deadlne. Pseudo Deadlne s set on the half deadlne. Pseudo executon tme also s set on the half executon tme. Fgure 3 shows the schedulng example of LP-RMZL and RMZLPD, when fve perodc tass, 1 = 2 = 3 = (1, 4) and 4 = 5 = (6, 12) are smtted on two processors. As shown n the fgure 3, 3 msses a deadlne at tme 3 n LP-RMZL. On the other hand, all the fve tass are successfully scheduled by RMZLPD, snce 5 has the sem hghest prorty at tme 5 due to pseudo zero laxty.

4 12 Ken Yana et al. / Proceda Computer Scence 24 ( 2013 ) Schedulablty LP-RMZL RMZLPD Fg. 3. Example of LP-RMZL schedule and RMZLPD schedule In ths secton, we derve suffcent schedulablty test for RMZLPD wth Response Tme Analyss (RTA) Interference and Worload To analyze the schedulablty, we defne two parameters: the nterference and the worload. Interference The nterference I (a, b) on over an nterval [a, b] s the cumulatve length of all ntervals n whch s baclogged but cannot be scheduled on any processor due to the contemporary executon of m hgher prorty tass. We also defne the nterference I (a, b) of a tas on a tas over an nterval [a, b] as the cumulatve length of all ntervals n whch s baclogged but cannot be scheduled on any processor, whle s executng. Worload The worload W (a, b) of a tas n an nterval [a, b) represents the amount of computaton that the tas requres n [a, b) on a gven stuaton. As for nterference, the followng lemma 1 s showed. Lemma 1 For any global schedulng algorthm t s: I a, b x mn( a b x I (, ), ) The flow of an algorthm analyss s shown below. Let J * be the job of wth maxmum nterference. The upper bound of I, I, can be calculated over an nterval [r *, r * + R ) from arrval tme to response tme of J *. Also, the upper bound of response tme of, R, s derved from I and the executon tme of. And then, we can calculate the lower bound of laxty of, L lb from the followng equaton (2). The scedulablty of an algorthm s analyzed usng L lb. In here, we have lemma 2 as for I. L lb T Lemma 2 I (a, b) s always smaller than W (a, b). Therefore, I can be calculated by calculatng W. R mx (1) (2)

5 Ken Yana et al. / Proceda Computer Scence 24 ( 2013 ) Schedulablty of RMZLPD Under RMZLPD schedulng, f the laxty or the pseudo laxty of a job reaches zero then t s gven the hghest or the sem hghest prorty. Therefore, s nterfered wth not only hgher statc prorty tass but also lower statc prorty tass. If s hgher statc prorty than, worload of over an nterval [r *, r * + R ) s represented n fgure 4. Thus, W R Fg. 4.The worload on RMZLPD ( > ) C C mn,max 2 0, R R T C f 2 2 (3) If s lower statc prorty than, worload of over an nterval [r *, r * + R ) s represented n fgure 5. Thus, Fg. 5. The worload on RMZLPD ( < ) lb W R n R C mn C, R T C L n R T f (4) T L where n (R ) s the maxmum number of jobs of tas that contrbute all of ther executon tme n the nterval: lb R T L C (5) R T n Theorem 1 (RTA for RMZLPD) An upper bound on the response tme of a tas n a multprocessor system scheduled wth RMZLPD can be derved by the fxed pont teraton on the value R of the followng expresson, startng wth R = C : 1 (6) R C R m I ( ) wth ) mn(, 1). I ( R W R R C Theorem 2 (Schedulablty for RMZLPD) A perodc tas system ={ 1, n }s schedulable by RMZLPD on m symmetrcal processors unless the followng nequalty holds for least m + 1 dfferent tass, and t holds strctly(<)for ) at least one of them:

6 14 Ken Yana et al. / Proceda Computer Scence 24 ( 2013 ) Expermental Evaluaton L l 0 (7) lb In order to see how well the RMZLPD algorthm and the above schedulablty test perform, a seres of experments were conducted wth the same smulaton envronment 4 and we assumed maxmum amount of processor s 16. Fgure 6 shows the result of smulaton about success rato, when 1,000 tas set (system utlzaton 0.3 ~ 1.0) are smtted on four processors. For comparson, three schedulng algorthm mentoned prevously were tested. It ndcates that RMZLPD s superor to other algorthms over an nterval [30%, 100%]. Fg. 6.The result of smulaton Fg. 7. The result of schedulablty. Fgure 7 shows the result of schedulablty, when 1,000 tas set (system utlzaton 0.3 ~ 1.0) are smtted on four processors. RMZLPD has two prorty promoton chances. Therefore, RMZLPD s more nterfered wth statc lower prorty tass than RMZL and LP-RMZL. 7. Conclusons A new tass schedulng algorthm on real tme multprocessor system s proposed n ths paper. The schedulablty of the proposed algorthm, RMZLPD, s analysed usng RTA. RMZLPD has hgh schedule success rato and can realze hgh parallelsm. From the numercal results, the results of the proposed RMZLPD are better than that of other algorthms. However, there are a gap between the schedulablty and the success rato. Ths determnes the next step of our study. We plan to analyss the schedulablty wth another method other than RTA. Acnowledgements Ths wor s supported n part by KAKENHI ( ). References 1. Dev, U.C. and Anderson, J.H. Tardness Bounds under Global EDF Schedulng on a Multprocessor. Proceedngs of the 26th IEEE Internatonal Real-Tme Systems Symposum 2005; Guan, N. and Y, W. Fxed-Prorty Multprocessor Schedulng: Crtcal Instant, Response Tme and Utlzaton Bound. IEEE 26th Internatonal Parallel and Dstrbuted Processng Symposum Worshops & PhD Forum 2012; Dhall, S. K. and Lu, C. L. On a real-tme schedulng problem. Operatons Research 1978; 26: Taeda, A., Kato, S. and Yamasa, N. Real-tme schedulng based on rate monotonc for multprocessors. IPSJ CPSY 2009; 2: Nshga, K., Yoo, M. and Yooyama, T. A Proposal of Real-Tme Schedulng Algorthm based on RM and Schedulablty Analyss. IEICEDE 2012; 95: Bertogna, M. and Crne, M. Response-tme analyss for globally scheduled symmetrc multprocessor platforms. 28th IEEE Internatonal Real-Tme Systems Symposum 2007;