Semi-Fixed-Priority Scheduling: New Priority Assignment Policy for Practical Imprecise Computation

Transcription

1 Semi-Fixed-Prioriy Scheduling: New Prioriy Assignmen Policy for Pracical Imprecise Compuaion Hiroyuki Chishiro, Akira Takeda 2, Kenji Funaoka 2 and Nobuyuki Yamasaki School of Science for Open and Environmen Sysems Keio Universiy, Yokohama, Japan {chishiro,yamasaki}@ny.ics.keio.ac.jp 2 Toshiba Corporaion, Japan akira.akeda@oshiba.co.jp, funaoka@isl.rdc.oshiba.co.jp Absrac This paper proposes semi-fixed-prioriy scheduling o achieve boh low-jier and high schedulabiliy. Semi-fixedprioriy scheduling is for he exended imprecise compuaion model, which has a wind-up par as a second mandaory par and schedules he par of each exended imprecise ask wih fixed-prioriy. This paper also proposes a novel semi-fixed-prioriy scheduling algorihm based on Rae Monoonic (), called Rae Monoonic wih Wind-up Par (). limis execuable ranges of wind-up pars o minimize jier. The schedulabiliy analysis proves ha one ask se is feasible by if he ask se is feasible by. Simulaion resuls show ha has boh lower jier and higher schedulabiliy han.. Inroducion Real-ime scheduling algorihms have used Wors Case Execuion Time (WCET) o schedule real-ime asks. However, he analysis of he WCET is difficul on curren realime sysems due o boh hardware and sofware complexiies. Moreover, Acual Case Execuion Time (ACET) in real-ime sysems such as auonomous mobile robos [6, 8] ends o change from ime o ime, because heir behaviors depend on heir environmens. In order o make use of he remaining ime (WCET - ACET), he imprecise compuaion model [3] was presened. The imprecise compuaion model is one of he echniques used o cope wih such uncerainy. The crucial poin is ha he compuaion is spli ino wo pars: mandaory This research was suppored by CREST, JST and by Gran in Aid for he Global Cener of Excellence Program for Cener for Educaion and Research of Symbioic, Safe and Secure Sysem Design from he Minisry of Educaion, Culure, Spor, and Technology in Japan. par and opional par. A mandaory par affecs he correcness of he resul and an opional par only affecs he qualiy of he resul. By resricing he execuion of he opional par o only afer he compleion of he mandaory par, real-ime applicaions based on he imprecise compuaion model can provide he correc oupu wih lower qualiy, by erminaing he opional par. However, he imprecise asks require he processings o oupu he resuls o heir acuaors. When he imprecise asks erminae or complee heir opional pars, he imprecise compuaion model canno guaranee o complee hem by heir deadlines. In order o overcome he weakness of he imprecise compuaion model, we use he exended imprecise compuaion model [, 2] wih a second mandaory par, called wind-up par. Exended imprecise asks for auonomous mobile robos such as MPEG decoder, objec deecion and pah search can be adaped o he exended imprecise compuaion model because hey mus guaranee o complee heir windup pars by heir deadlines. In real-ime scheduling of exended imprecise asks, Mandaory-Firs wih Wind-up Par () [, 2] has high-jier of he shores period ask due o Earlies Deadline Firs (EDF) [4] based dynamicprioriy scheduling [5]. Hence, canno be adaped o auonomous mobile robos because he jier-sensiive ask such as he moor conrol ask wih he shores period in auonomous mobile robos requires he minimized jier o achieve precise moions. Unforunaely, fixed-prioriy scheduling such as Rae Monoonic () [4] wih lowjier of he shores period ask canno be adaped o he exended imprecise compuaion model because one ask may miss is deadline due o he overrun of he opional par. This paper proposes semi-fixed-prioriy scheduling o achieve boh low-jier and high schedulabiliy. Semi-fixedprioriy scheduling is for he exended imprecise compuaion model and schedules he par of each exended imprecise ask wih fixed-prioriy. This paper also proposes

2 Discarded Mandaory par Opional par Compleed Wind-up par τ Terminaed Figure. Exended imprecise compuaion model r, f, r,2 f,2r,3 f,3 Mandaory par Opional par Wind-up par Figure 2. Relaive finishing jier a novel semi-fixed-prioriy scheduling algorihm based on, called Rae Monoonic wih Wind-up Par (). limis execuable ranges of wind-up pars o minimize jier. The schedulabiliy analysis proves ha one ask se is feasible by if he ask se is feasible by. Simulaion resuls show ha has boh lower jier and higher schedulabiliy han. The conribuion of his paper is o achieve real-ime scheduling wih low-jier for pracical imprecise compuaion, called semi-fixed-prioriy scheduling. We believe ha as well as and EDF is a basic scheduling algorihm and of wide applicaion. The remainder of his paper is organized as follows: Secion 2 describes he sysem model. Secion 3 reviews relaed work. Secion 4 presens semi-fixed-prioriy scheduling and. The effeciveness of is evaluaed in Secion 5. Finally we offer concluding remarks in Secion Sysem Model Figure shows he exended imprecise compuaion model [, 2]. The exended imprecise compuaion model adds he wind-up par o he imprecise compuaion model [3]. The imprecise compuaion model assumes ha he processing o erminae or complee he opional par is no required. However, moor conrol asks in auonomous mobile robos require he processings o oupu he resuls o heir acuaors. They mus guaranee o complee hem by heir deadlines so ha he exended imprecise compuaion model has he wind-up par. We assume ha he sysem has one processor and a ask se Γ consised of n asks. Task τ i is represened as he following uple (T i,d i,od i,m i,o i,w i ): where T i is he period, D i is he deadline, OD i is he opional deadline, m i is he WCET of he mandaory par, o i is he Required Execuion Time (RET) of he opional par and w i is he WCET of he wind-up par. The RET of each opional par ends o be τ τ2 Mandaory par Opional par Wind-up par Figure 3. Opional deadline underesimaed or overesimaed from ime o ime because auonomous mobile robos run in uncerain environmens. The relaive deadlines D i of each ask τ i is equal o is T i. The j h insance of τ i is called job τ i,j. The uilizaion of each periodic ask is defined as U i =(m i + w i )/T i. The reason why U i does no include o i is because he opional par of τ i is a non-real-ime par so ha compleing i is no relevan o scheduling he ask se successfully. Hence, he uilizaion of he sysem wihin n asks can be defined as U = n i= U i. All asks are ordered by prioriy and T T 2... T n are ordered such ha. In addiion, we define he following symbols as follows. Γ s : he group of successfully scheduled asks o i,j : he acual case RET of τ i,j r i,j : he release ime of τ i,j s i,j : he sar ime of τ i,j f i,j : he finishing ime of τ i,j R i (): he remaining execuion ime of τ i a ime We define jier as Relaive Finishing Jier (RFJ) [4]. RFJ is he maximum deviaion of he finishing ime of wo consecuive jobs: RF J i = max (f i,j+ r i,j+ ) (f i,j r i,j ). j We describe RFJ in Figure 2. In his case, he RFJ of τ is he maximum of (f,2 r,2 ) (f, r, ) and (f,3 r,3 ) (f,2 r,2 ). Reducing jier means reducing RFJ. An opional deadline is a ime when an opional par is erminaed and a wind-up par is released. Each windup par is ready o be execued afer each opional deadline and can be compleed if each mandaory par is compleed by each opional deadline. Figure 3 shows he opional deadline of each ask. Solid up arrow, solid down arrow and doed down arrow represen release ime, deadline and opional deadline respecively. Task τ complees is mandaory par by OD and execues is opional par unil OD. Afer OD, hen τ execues is wind-up par. In conras, ask τ 2 does no complee is mandaory par by OD 2. When τ 2 complees is mandaory par, τ 2 sars o execue is wind-up par, no o execue is opional par. 3. Relaed Work Scheduling wih Liu and Layland s model [4], called general scheduling, such as and EDF do no consider

3 Ri() general scheduling mi+wi semi-fixed-prioriy scheduling m Ti Ti Spli mi wi w ODi Ti Mandaory par Wind-up par Figure 4. Spli one exended imprecise ask ino wo general asks ACET, which is usually less han WCET because WCET ends o be overesimaed [7]. These algorihms canno use he remaining ime so ha he imprecise compuaion model is an effecive echnique in uncerain environmens. In he imprecise compuaion model, Mandaory-Firs wih Earlies Deadline (M-FED) [3] is based on EDF and has high-jier of he shores period ask. Opimizaion wih Leas-Uilizaion [2] requiring known he WCET of each opional par canno be adaped o auonomous mobile robos requiring ha he WCET of each opional par is unknown. Mandaory-Firs wih Wind-up Par () [, 2] adaps M-FED o he exended imprecise compuaion model. However, as well as M-FED has highjier of he shores period ask. The jier-sensiive ask such as he moor conrol ask wih he shores period in auonomous mobile robos requires he minimized jier o achieve precise moions. Unforunaely, fixed-prioriy scheduling such as wih low-jier of he shores period ask canno be adaped o he exended imprecise compuaion model because one ask may miss is deadline due o he overrun of he opional par. 4. Semi-Fixed-Prioriy Scheduling Semi-fixed-prioriy scheduling fixes he prioriy of each par in he exended imprecise ask and changes he prioriy of each exended imprecise ask only in he wo cases: when he exended imprecise ask complees is mandaory par and begins is opional par and when he exended imprecise ask erminaes or complees is opional par and begins is wind-up par. Semi-fixed-prioriy scheduling splis one exended imprecise ask ino wo general asks. The wo general asks have same periods and same or differen release imes, canno be execued simulaneously and are scheduled by fixed-prioriy in Figure 4. Task τ i m and w are he mandaory par and he wind-up par of τ i. The release imes of he firs jobs of m and w are and OD i general semi-fixedprioriy mi mi+wi ODi ODi+wi Di Mandaory par Wind-up par Figure 5. General scheduling and semi-fixedprioriy scheduling Prioriy Scheduler Mandaory par Opional par Wind-up par Sleep Empy Figure 6. Task queue RTQ SQ NRTQ respecively. When here is no ask which is ready o execue is mandaory or wind-up par, he opional par of each ask is execued. Figure 5 shows he difference beween general scheduling such as and EDF wih Liu and Layland s model [4] and semi-fixed-prioriy scheduling. In general scheduling, when τ i is released a, R i () is se o m i + w i and monoonically decreasing unil R i () becomes a m i +w i. In semi-fixed-prioriy scheduling, when τ i is released a, R i () is se o m i and monoonically decreasing unil R i () becomes a m i. When R i () is a m i, hen τ i sleeps unil OD i. When τ i is released a OD i, hen R i () is se o w i and monoonically decreasing unil R i () becomes a OD i + w i. If τ i does no complee is mandaory par by OD i, hen R i () is se o w i a he ime when τ i complees is mandaory par. In boh schedulings, τ i complees is wind-up par by D i. 4. Algorihm is one of semi-fixed-prioriy scheduling algorihms wih he exended imprecise compuaion model o achieve boh low-jier and high schedulabiliy. As shown

4 . When τ i becomes ready, se R i() o m i, dequeue τ i from SQ and enqueue τ i o RTQ. If τ i has he highes prioriy in RTQ, preemp he curren ask. 2. When τ i complees is mandaory par: (a) If OD i expired, se R i() o w i. (b) Oherwise se R i() o o i, dequeue τ i from RTQ and enqueue τ i o NRTQ. If here are one or muliple asks in RTQ or NRTQ which have higher prioriy han τ i, preemp τ i. 3. When τ i complees is opional par, dequeue τ i from NRTQ and enqueue τ i o SQ. 4. When OD i expires: (a) If τ i is in RTQ and does no complee is mandaory par, do nohing. (b) If τ i is in NRTQ, erminae and dequeue τ i from NRTQ, se R i() o w i and enqueue τ i o RTQ. If τ i has he highes prioriy in RTQ, preemp he curren ask. (c) If τ i is in SQ, dequeue τ i from SQ, se R i() o w i and enqueue τ i o RTQ. 5. When τ i complees is wind-up par, dequeue τ i from RTQ and enqueue τ i o SQ. 6. When here are one or muliple asks in RTQ, perform in RTQ. 7. When here is no ask in RTQ and here are one or muliples asks in NRTQ, perform in NRTQ. Figure 7. algorihm in Figure 6, manages hree ask queues: Real-Time Queue (RTQ), Non-Real-Time Queue (NRTQ) and Sleep Queue (SQ). RTQ holds asks which are ready o execue heir mandaory or wind-up pars in order. One ask is no ready o execue is mandaory and wind-up pars simulaneously. NRTQ holds asks which are ready o execue heir opional pars in order. Every ask in RTQ has higher prioriy han ha in NRTQ. SQ holds asks which complee heir opional pars by heir opional deadlines or wind-up pars by heir deadlines. Figure 7 shows algorihm. execues seven scheduling evens when heir condiions are me. In order o execue hese evens, each ask has he opional deadline. We nex describe how o calculae he opional deadline and analyze he schedulabiliy of wih general and harmonic ask ses. 4.2 wih General Task Ses An opional deadline is a ime when an opional par is erminaed and a wind-up par is released. Each wind-up par is ready o be execued afer each opional deadline and τk Ti Mandaory par Wind-up par Figure 8. Case of wors case inerference ime can be compleed if each mandaory par is compleed by he opional deadline. Each opional deadline is se o he ime as lae as possible o expand he execuable range of each opional par. The wind-up par of each ask mus no miss he deadline if he sysem is idle or execues lower prioriy asks beween he ime when he mandaory par is compleed and he wind-up par is released. In order o calculae he opional deadline, we firs esimae he wors case inerference ime Ik i (i<k) which is he upper bound ime when τ k is inerfered by τ i. Theorem (Wors Case Inerference Time by Higher Prioriy Tasks). The wors case inerference ime Ik i (i<k) which is he upper bound ime when τ k is inerfered by τ i is Ik i Tk = (m i + w i ). () T i Proof. If he opional deadline of each ask is equal o in Figure 8, he inerference ime of ask τ k inerfered by τ i is equal o equaion. Moreover, here is no case ha he inerference ime of ask τ k inerfered by τ i is more han equaion. By heorem, we nex calculae he opional deadline wih general ask ses. Theorem 2 (Opional Deadline wih General Task Ses). The opional deadline OD k of ask τ k is k OD k = D k w k Ik i. (2) i= Proof. I is clear ha ask τ k complees is wind-up par by is deadline if τ k complees is mandaory par by is opional deadline by heorem. can calculae opional deadlines by heorem 2. In conras, in, analyzing ha wha job maximizes he wors case inerference ime Ik i is oo complex due o dynamic-prioriy scheduling. Therefore, in order o calculae OD k easily, is a semi-fixed-prioriy scheduling algorihm and only considers asks, prioriies of which are higher han he prioriy of τ k. can delay he release ime of he wind-up par unil he ime when he windup par does no miss he deadline if he mandaory par is Tk

5 τ τ2 τ τ (a) Schedule successfully by deadline miss! (b) Schedule unsuccessfully by Mandaory par Opional par Wind-up par Figure 9. Example of schedule generaed by and compleed by he opional deadline. We nex analyze ha is a leas as effecive as. Theorem 3 ( is a leas as effecive as ). One ask se is feasible by if he ask se is feasible by. Proof. This proof is shown by conraposiion. We show ha if one ask se is no feasible by, he ask se is no feasible by. By heorem 2, i is clear ha τ i complees is wind-up par by is deadline if τ i complees is mandaory par by is opional deadline. Task τ i misses is deadline only if τ i execues is mandaory par afer is opional deadline OD i. In his case, τ i execues is mandaory and windup pars coninuously wihou execuing is opional par. In, ask τ i also misses is deadline because of execuing is mandaory and wind-up pars coninuously. Hence, his heorem holds. Theorem 4 (Leas Upper Bound of wih General Task Ses). For a se of n asks wih semi-fixed-prioriy assignmen, he leas upper bound of wih general ask ses is U lub = n(2 /n ). Proof. is a leas as effecive as by heorem 3 and generaes he same schedule as in he case of wors case inerference ime by heorem. Therefore, he leas upper bound of is he same as ha of [4]. Figure 9 shows an example of schedule generaed by and. The following ask se Γ = {τ = (,, 7, 3,, 3),τ 2 =(5, 5,, 3,, 2)} is scheduled by and in Figure 9(a) and 9(b) respecively. Each opional deadline is calculaed by heorem 2. This example shows ha here is a leas one ask se which is feasible by RT A OODH(Γ) { while (τ k Γ) { A k = D k w k k i= Ii k ; I =; do { } } OD = I + A k ; I = k i= ( OD T i m i + } while (I + A k >OD); OD k = OD; OD ODi T i w i ); Figure. Pseudo code of RTA-OODH and is no feasible by. Moreover, in, job τ, and τ,2 execues is opional par in [4, 5) and [26, 27) respecively. 4.3 wih Harmonic Task Ses The opional deadline by heorem 2 is pessimisic wih general ask ses. In order o improve he execuable range of each opional par, we exend Response Time Analysis (RTA) [] for Opimal Opional Deadline wih Harmonic ask ses (RTA-OODH). The opimal opional deadline OD k of ask τ k is defined as he ime when he assignable ime of τ k in [OD k,d k ) is equal o w k. Tha is o say, he opimal opional deadline is se o he ime when he opional par of τ k is no erminaed or discarded, hough here is ime o execue is opional par, if he ACET of τ k is always equal o is WCET. Also, we show ha he opimal opional deadline by RTA-OODH is more han or equal o ha by heorem 2. We firs calculae he assignable ime A k of ask τ k excep w k by heorem. Theorem 5 (Assignable Time wih Harmonic Task Ses). The assignable ime A k of ask τ k excep w k is k A k = D k w k Ik. i (3) i= Proof. The hyperperiod in τ k and higher prioriy asks han τ k is equal o T k (D k ) wih harmonic ask ses. Moreover, he wors case inerference ime of each job is consan. Hence, he assignable ime A k of ask τ k excep w k is equal o equaion 3. We nex calculae he wors case inerference ime I k of ask τ k wih harmonic ask ses in [,OD k ).

6 τ τ2 τ3 5 I3 A Mandaory par Opional par Wind-up par OD3 Figure. Example of schedule generaed by wih RTA-OODH Theorem 6 (Wors Case Inerference Time in [,OD k )). The wors case inerference ime I k of τ k in [,OD k ) is k ( ODk I k = i= T i m i + ODk OD i T i w i ). Proof. We consider ha one exended imprecise ask τ i is spli ino wo general asks m and w in Figure 4. The release ime of each firs job in m and w is equal o and OD i respecively and he period of eack ask is equal o T i. Hence, in [,OD k ), τ k is inerfered by m in OD k /T i imes and by w in (OD k OD i )/T i imes. By heorem 5 and 6, we presen RTA-OODH. Theorem 7 (RTA-OODH). The opimal opional deadline OD k of ask τ k wih harmonic ask ses is OD k = A k + I k. (4) Proof. In equaion 4, he opional deadline OD k and he assignable ime A k are he response ime and he wors case execuion ime in RTA [] respecively. The opional deadline by equaion 4 is a ime when he opional par of ask τ k is no erminaed or discarded, hough here is ime o execue is opional par, if he ACET of τ k is always is WCET. Moreover, he assignable ime of each job is consan wih harmonic ask ses. Hence, he assignable ime A k of τ k excep w k in [OD k,d k ) is equal o w k so ha he opional deadline by equaion 4 is opimal. The assignable ime A k of ask τ k excep w k by equaion 3 is equal o he opional deadline by equaion 2 wih harmonic ask ses. I is clear ha he opimal opional deadline by heorem 7 is more han or equal o he opional deadline by heorem 2. Figure shows he pseudo code of RTA-OODH. RTA-OODH calculaes he opimal opional deadline by he ieraion, which is similar wih RTA []. We nex analyze he leas upper bound U lub of wih harmonic ask ses by heorem 7. Theorem 8 (Leas Upper Bound of wih Harmonic Task Ses). The leas upper bound of wih harmonic ask ses is U lub =. Proof. By heorem 7, i is clear ha ask τ k does no miss is deadline if τ k complees is mandaory par by is opional deadline. Moreover, is a leas as effecive as by heorem 3 and generaes he same schedule as in he case of wors case inerference ime by heorem. Therefore, he leas upper bound of is equal o ha of. Tha is o say, he leas upper bound of wih harmonic ask ses is U lub =. Figure shows an example of schedule by using RTA-OODH wih he following ask se Γ = {τ = (5, 5, 4,,, ),τ 2 = (,, 8, 2,, ),τ 3 = (2, 2, 4, 2, 2, 2)}. Every OD i is calculaed by heorem 7. For example, we calculae OD 3 of τ 3 and he process of OD 3 hrough ime is shown in he boom par of Figure. The opional deadline OD 3 by heorem 2 is 4. In conras, he opimal opional deadline OD 3 by heorem 7 is 4. Therefore, by he opimal opional deadline, job τ 3, can execue is opional par in [7, 8) and [3, 4). 5. Simulaion Sudies This secion sudies he effeciveness of using five performance merics. These merics describe deails in Secion 5.. The simulaion uses, ask ses and compares wih boh and. In auonomous mobile robos, here are asks which have various periods. Therefore, he period T i of each ask τ i is seleced wihin [, 2, 3,..., 3] wih general ask ses and [, 2, 4, 8, 6, 32] wih harmonic ask ses. Each U i is seleced from [.2,.3,.4,...,.25] and splis U i ino wo uilizaions which are assigned o m i and w i respecively. The CPU uilizaion U is seleced from [, 5,,...,.]. The simulaion lengh of he k h ask se is H k which is he hyperperiod of he k h ask se. The CPU uilizaion of o i,j is wihin he range of [o i.5,o i +.5], where o i is seleced wihin [.,.2, ],

7 represened such as -, -2 and -3, compued a every ask release, because auonomous mobile robos run in uncerain environmens so ha each o i,j is flucuaed. If he CPU uilizaion of o i,j is always equal o, he resul is represened as. Also, we consider he asks, he ACETs of which end o flucuae from ime o ime in auonomous mobile robos so ha we evaluae hree cases where ACET/WCET is uniformly varies in he range of [,.], [.75,.] and.. 5. Performance Merics We use five performance merics o evaluae he effeciveness of from various perspecives. The performance merics are defined as he following equaions. Success Raio = Reward Raio = Swich Raio = RFJ Raio = SPJ Raio = # of successfully scheduled ask ses # of scheduled ask ses T i o i,j k i H k j o i # of asks in successfully scheduled ask ses k # of conex swiches H k # of successfully scheduled ask ses k RF J i i T i # of asks in successfully scheduled ask ses RF J k T # of successfully scheduled ask ses In he above, ask τ i is in he group of Γ s. If he success raio of he CPU uilizaion is, he resuls of he CPU uilizaion evaluaed by oher performance merics are omied. Moreover, he resuls of -, -2 and -3 in success raio and RFJ raio are he same as and hose of -, -2 and - 3 in success raio are he same as so ha hey are omied. The reason why we evaluae no only RFJ raio bu also SPJ raio is because he jier-sensiive ask such as he moor conrol ask wih he shores period in auonomous mobile robos is imporan o achieve precise moions. In he evaluaion wih harmonic ask ses, we do no show he resul of he success raio because i is clear ha he leas upper bounds of, and are equal o by heorem 8 and in [9] and [4] respecively. 5.2 Simulaion Resuls wih General Task Ses Figure 2, 3, 4, 5 and 6 show simulaion resuls wih general ask ses. In Figure 2(a), 2(b) and 2(c), he success raio of M- FWP is always. In conras, boh success raios of and drop when he CPU uilizaion is higher han. The success raio of is always higher han or equal o ha of by heorem 3. The range of ACET/WCET is wider and wider, he success raios of and are improved. In Figure 3(a), 3(b) and 3(c), -, - 2 and -3 ouperform -, -2 and -3 respecively because ses each opional deadline saically and calculaes he assignable ime of each opional par dynamically. In Figure 4(a), 4(b) and 4(c), he swich raio of is higher han ha of because splis one exended imprecise ask ino wo general asks. In -, -2 and -3, he RET of each opional par is more and more, he swich raio drops because of execuing each par coninuously more frequenly. The swich raio of is higher han ha of because when he opional par of each ask is compleed by each opional deadline, he ask sleeps unil he opional deadline in. In conras, in, when he opional par of each ask is compleed, he wind-up par of each ask is execued immediaely. In Figure 5(a), 5(b) and 5(c), ouperforms and by limiing execuable ranges of wind-up pars. The RFJ raio of -, -2 and M- FWP-3 are dramaically he differen resuls o he RFJ raio of because hey calculae he assignable ime of each opional par when each mandaory par is compleed. If he RET of each opional par is more han, he opional par is ready o be execued. Oherwise he wind-up par is ready o be execued immediaely. Therefore, he RFJ raios of -, -2 and -3 are dramaically higher han ha of. On he oher hand, can minimize jier regardless of he assignable ime of each opional par by each opional deadline and is lower jier han boh and. In Figure 6(a), 6(b) and 6(c), he SPJ raios of M- FWP-, -2 and -3 are also dramaically higher han as he resuls in Figure 5. Ineresingly, he SPJ raio of is lower han ha of in Figure 6(b) and 6(c). In, if he RET of each opional par is always, generaes he same schedule as EDF. dominaes EDF in he jier of he highes prioriy ask [5], which is rue only if ACET of each ask is always equal o is WCET in Figure 6(a). If ACET of each ask is less han or equal o is WCET in Figure 6(b) and 6(c), EDF dominaes. 5.3 Simulaion Resuls wih Harmonic Task Ses Figure 7, 8, 9 and 2 show simulaion resuls wih harmonic ask ses. In Figure 7(a), 7(b) and 7(c), - slighly ouperforms - by heorem 7. Moreover, he reward raios of -2 and -3 are he same as

8 (a) ACET/WCET =. Success Raio.2 Success Raio Success Raio (b) ACET/WCET = [.75,.] (c) ACET/WCET = [,.] Figure 2. Success raio wih general ask ses Reward Raio Reward Raio Reward Raio (a) ACET/WCET = (b) ACET/WCET = [.75,.] (c) ACET/WCET = [,.] Figure 3. Reward raio wih general ask ses (a) ACET/WCET = Swich Raio Swich Raio Swich Raio (b) ACET/WCET = [.75,.] (c) ACET/WCET = [,.] Figure 4. Swich raio wih general ask ses hose of -2 and -3 respecively. Therefore, ouperforms wih harmonic asks ses hough ouperforms wih general ask ses. In Figure 8(a), 8(b) and 8(c),, and are he ascending order of he swich raio as well as he resuls in Figure 4. In Figure 9(a), he RFJ raios of and and are he lowes. In Figure 9(b) and 9(c), he RFJ raio of is he lowes and boh he RFJ raios of MFWP and are higher han. Unforunaely, he RFJ raios of -, -2 and -3 are dramaically higher han oher algorihms. In Figure 2(a), 2(b) and 2(c), i is ineresing ha he SPJ raios of and are he lowes in evaluaed algorihms., which generaes he same schedule as EDF only if he RET of each opional par is always, also dominaes. 5.4 Discussion Considering he simulaion resuls, we discuss which algorihm is well suied o auonomous mobile robos. does no consider he case ha he ACET of each ask is less han is WCET. However, in auonomous mobile robos, he ACET of each ask is usually less han is WCET so ha is no well suied o auonomous mobile robos. In and, we discuss he rade-offs beween reward raios and jiers o realize auonomous mobile robos. MFWP is suied o applicaions requiring high schedulabiliy wih non-jier sensiive asks because has higher success raio han. In conras, is suied o auonomous mobile robos having he jier-sensiive ask such as he moor conrol ask wih he shores period in uncerain environmens because no only is he jier of no affeced by he execuion ime of each opional par, bu also has lower jier han. More-

9 RFJ Raio -2.7 RFJ Raio RFJ Raio (a) ACET/WCET =..7.9 (b) ACET/WCET = [.75,.] (c) ACET/WCET = [,.] Figure 5. RFJ raio wih general ask ses (a) ACET/WCET = SPJ Raio.2-2 SPJ Raio SPJ Raio (b) ACET/WCET = [.75,.] (c) ACET/WCET = [,.] Figure 6. SPJ raio wih general ask ses (a) ACET/WCET = Reward Raio Reward Raio Reward Raio (b) ACET/WCET = [.75,.].7.9 (c) ACET/WCET = [,.] Figure 7. Reward raio wih harmonic ask ses over, he reward raio of ouperforms ha of MFWP by heorem 7 wih harmonic ask ses and he reward raio of ouperforms ha of wih general ask ses. Our goal is o achieve real-ime scheduling wih low-jier and high schedulabiliy for pracical imprecise compuaion so ha is well suied o auonomous mobile robos. 6. Concluding Remarks This paper proposed semi-fixed-prioriy scheduling o achieve boh low-jier and high schedulabiliy. Semi-fixedprioriy scheduling is for he exended imprecise compuaion model and schedules he par of each exended imprecise ask by fixed-prioriy. This paper also proposed a novel semi-fixed-prioriy scheduling algorihm, called. The advanage of is ha is a leas as effecive as and has higher success raio han and lower jier han boh and. The disadvanage of is ha has lower success raio han and higher swich raio han and. However, he jier of is no affeced by he execuion ime of each opional par. Therefore, is well suied o auonomous mobile robos. In fuure work, we will implemen o RT-Fronier [], a real-ime operaing sysem supporing he exended imprecise compuaion model. References [] N. C. Audsley, A. Burns, M. F. Richardson, K. Tindell, and A. Wellings. Applying New Scheduling Theory o Saic Prioriy Pre-empive Scheduling. Sofware Engineering Journal, 8(5): , Sep [2] H. Aydin, R. Melhem, D. Mosse, and P. Mejfa-Alvarez. Opimal Reward-Based Scheduling of Periodic Real-Time Tasks. In Proceedings of he 2h IEEE Real-Time Sysems Symposium, pages 79 89, Dec. 999.

10 Swich Raio Swich Raio.5 Swich Raio (a) ACET/WCET =..7.9 (b) ACET/WCET = [.75,.] (c) ACET/WCET = [,.] Figure 8. Swich raio wih harmonic ask ses RFJ Raio RFJ Raio RFJ Raio (a) ACET/WCET =..7.9 (b) ACET/WCET = [.75,.] (c) ACET/WCET = [,.] Figure 9. RFJ raio wih harmonic ask ses SPJ Raio SPJ Raio (a) ACET/WCET =..9-2 SPJ Raio (b) ACET/WCET = [.75,.].7.9 (c) ACET/WCET = [,.] Figure 2. SPJ raio wih harmonic ask ses [3] S. K. Baruah and M. E. Hickey. Compeiive On-line Scheduling of Imprecise Compuaions. IEEE Transacions on Compuers, 47:27 33, 996. [4] G. C. Buazzo. HARD REAL-TIME COMPUTING SYSTEMS: Predicable Scheduling Algorihms and Applicaions. Springer, 2nd ediion, 24. [5] G. C. Buazzo. Rae Monoonic vs. EDF: Judgmen Day. Real-Time Sysems, 29():5 26, 25. [6] F. R. T. Cener. hp:// [7] A. Ermedahl. A Modular Tool Archiecure for Wors-Case Execuion Time Analysis. PhD hesis, Uppsala Universiy, June 23. [8] F. Kanehiro, H. Hirukawa, and S. Kajia. OpenHRP: Open Archiecure Humanoid Roboics Plaform. The Inernaional Journal of Roboics Research, 23(2):55 65, 24. [9] H. Kobayashi. REAL-TIME SCHEDULING OF PRACTICAL IMPRECISE TASKS UNDER TRANSIENT AND PERSISTENT OVERLOAD. PhD hesis, Keio Universiy, Mar. 26. [] H. Kobayashi and N. Yamasaki. An Inegraed Approach for Implemening Imprecise Compuaions. IEICE ransacions on informaion and sysems, 86():24 248, 23. [] H. Kobayashi and N. Yamasaki. RT-Fronier: A Real-Time Operaing Sysem for Pracical Imprecise Compuaion. In Proceedings of he h IEEE Real-Time and Embedded Technology and Applicaions Symposium, pages , May 24. [2] H. Kobayashi, N. Yamasaki, and Y. Anzai. Scheduling Imprecise Compuaions wih Wind-up Pars. In Proceedings of he 8h Inernaional Conference on Compuers and Their Applicaions, pages , Mar. 23. [3] K. Lin, S. Naarajan, and J. Liu. Imprecise Resuls: Uilizing Parial Compuaions in Real-Time Sysems. In Proceedings of he 8h IEEE Real-Time Sysems Symposium, pages 2 27, Dec [4] C. Liu and J. Layland. Scheduling Algorihms for Muliprogramming in a Hard Real-Time Environmen. Journal of he ACM, 2:46 6, 973.