Fixd-Rlaiv-Dadlin Schduling of Hard Ral-Tim Tasks wih Slf-Suspnsions Jian-Jia Chn Dparmn of Informaics TU Dormund Univrsiy, Grmany jia.chn@u-dormund.d Absrac In many ral-im sysms, asks may xprinc slf-suspnsion dlays whn accssing xrnal dvics. Th problm of schduling such slf-suspnding asks o m hard dadlins on a uniprocssor is known o b N P-hard in h srong sns. Currn soluions including h common suspnsionoblivious approach of raing all suspnsions as compuaion can b qui pssimisic. This papr shows ha anohr cagory of schduling algorihms, namly fixd-rlaiv-dadlin (FRD) schduling, may yild br prformanc han classical joblvl fixd-prioriy schdulrs such as EDF and RM, for ralim asks ha may xprinc on slf-suspnsion during h xcuion of a ask insanc. W analyz a simpl FRD algorihm, namly, and driv corrsponding psudo-polynomial-im and linar-im schdulabiliy ss. To analyz h qualiy of and is schdulabiliy ss, w analyz hir rsourc augmnaion facors, wih rspc o h spd-up facor ha is ndd o nsur h schdulabiliy and fasibiliy of h rsuling schdul. Spcifically, h spd-up facor of is and 3, whn rfrring o h opimal FRD schduling and any fasibl arbirary schduling, rspcivly. Morovr, h spdup facor of h proposd linar-im schdulabiliy s is.787 and.875, whn rfrring o h opimal FRD schduling and any fasibl arbirary schduling, rspcivly. Furhrmor, xnsiv xprimns prsnd hrin show ha our proposd linarim schdulabiliy s improvs upon prior approachs by a significan margin. To our bs knowldg, for h schduling of slf-suspnding asks, hs ar h firs rsuls of any sor ha indica i migh b possibl o dsign good approximaion algorihms. Inroducion In many ral-im sysms, asks may slf-suspnd whn accssing xrnal dvics such as disks and GPUs. Th rsuling suspnsion dlays ypically rang from a fw microsconds (.g., a rad opraion on a flash driv [7]) o a fw sconds (.g., accssing a GPU [8], using compuaion offloading for spding-up []). Such suspnsion dlays caus inracabiliis in hard ral-im (HRT) schdulabiliy analysis []. Th unsolvd problm of fficinly supporing slf-suspnsions has impdd rsarch progrss on many rlad rsarch opics such as analyzing and implmning I/O-innsiv applicaions in muliprocssor sysms as wll as compuaion offloading in ral-im sysms. Cong Liu Dparmn of Compur Scinc Th Univrsiy of Txas a Dallas cong@udallas.du Du o h fac ha h problm of schduling HRT slfsuspnding ask sysms on a uniprocssor is N P-hard in h srong sns [], i is unlikly o dsign opimal polynomialim soluions. To rsolv h compuaional complxiy issus in many of hs N P-hard schduling problms in ral-im sysms, approximaion algorihms, and in paricular, approximaions basd on rsourc augmnaion hav aracd much anion (.g., ral-im ask pariioning on muliprocssors [], [5]). If an algorihm A has a spd-up facor ρ, hn i guarans ha h schdul drivd from h algorihm A is always fasibl by running a spd ρ, if h inpu ask s admis a fasibl schdul on a uni-spd procssor. In ohr words, by aking h ngaion of h abov samn, if an algorihm A has a spd-up facor ρ, hn i guarans ha if h schdul drivd from h algorihm A is no fasibl, hn h inpu dos no admi a fasibl schdul by running a spd ρ. Thrfor, dsigning schduling algorihms and schdulabiliy ss wih boundd spd-up facors (rsourc augmnaion facors, quivalnly) also nsurs hir qualiis for such N P-hard problms. Rsourc-augmnaion-basd approximaions assum a crain spdup of h procssor. For ordinary ral-im ask sysms wihou slf-suspnsions, i has bn shown in [8] ha wll-known prioriy-basd ral-im schduling algorihms,.g., arlis-dadlin-firs (EDF) schduling policy, which hav poor prformanc on a muliprocssor from an absolu wors-cas prspciv, ar good whn allowing modraly fasr rsourcs. Unforunaly, for h slf-suspnding ask schduling problm, hr dos no xis any of such approximaion rsuls. Bsids h fac ha currn chniqus [9] [3], [6], [7], [9], [] for daling wih slf-suspnsions can b qui pssimisic, non of h xising work provids us a good undrsanding on how o quanify h qualiy of such schduling algorihms. In his papr, w sudy h fundamnal problm of schduling an HRT slf-suspnding ask sysm on a uniprocssor. Classical job-lvl and ask-lvl fixd-prioriy schduling algorihms such as EDF and Ra-Monoonic (RM) may no b suiabl for schduling slf-suspnding ask sysms. Svral wll-known ngaiv rsuls hav bn shown on supporing h singl-sgmn-suspnding ask modl, whr ach ask conains wo compuaion phass wih on suspnsion phas in bwn, undr job- or ask-lvl fixdprioriy schduling algorihms including EDF and RM []. Our ky obsrvaion hrin is ha whn considring slfsuspnsions, job prioriis should b drmind no solly by radiional paramrs such as priods or dadlins, bu also h suspnsion lngh. Considr, for xampl, h uniprocssor ask sysm, schduld by EDF or RM, shown in Fig.. This sysm consiss of wo asks: ask is an ordinary sporadic ask wih an xcuion cos of im uni, a priod and a rlaiv dadlin of 5 im unis; ask is a slf-suspnding ask wih a priod and a rlaiv dadlin of im unis ha
firs xcus for im uni, hn slf-suspnds for 8 im unis, and finally xcus for anohr im uni. As sn from h figur, misss is dadlin undr ihr EDF or RM. Howvr, if w prioriiz h firs compuaion phas of ovr h compuaion phas of, hn boh asks can m hir dadlins. Alhough h job of has a longr rlaiv dadlin han h job of, is firs compuaion sgmn acually mus m an invisibl hard dadlin a im in ordr for h nir job o m is dadlin. Moivad by his, w show in his papr ha for schduling singl-sgmn-suspnding ask sysms, anohr cagory of schduling algorihms, namly fixd-rlaiv-dadlin (FRD) schduling, may yild br prformanc han radiional joblvl or ask-lvl fixd prioriy schdulrs. An FRD schdulr assigns a spara rlaiv dadlin o ach compuaion phas of a ask and prioriizs diffrn compuaion phass by hs rlaiv dadlins. W furhr obsrv and prov ha, surprisingly, a rahr simpl FRD schduling policy, namly qualdadlin assignmn () ha assigns rlaiv dadlins qually o boh compuaion phass of a slf-suspnding ask and uss EDF for schduling h compuaion phass, yilds good prformanc w.r.. spd-up facor. Ovrviw of rlad work. Rcnly h problm of schduling sof ral-im (wih guarand boundd rspons ims) slf-suspnding ask sysms on muliprocssor has rcivd much anion [], []. For h HRT cas, bsids h suspnsion-oblivious approach of raing all suspnsions as compuaion [6], svral schdulabiliy ss hav bn prsnd for analyzing priodic singl-sgmn-suspnding asks on a uniprocssor [9], [], [7], [9], []. Unforunaly, hs ss ar rahr pssimisic as hir chniqus involv sraighforward xcuion conrol mchanisms, which divid a slf-suspnding ask ino wo subasks wih approprialy shornd dadlins and modifid rlas ims (ofn known as h nd-o-nd approach [6]). Such chniqus inviably suffr from significan capaciy loss du o h arificial shorning of dadlins. For h mor gnral slf-suspnding ask modl whr a ask is allowd o suspnd mulipl ims, [5] prsns a uniprocssor uilizaion-bas s undr RM and a muliprocssor uilizaion-basd s undr pariiond approach whr RM is applid as h pr-procssor schdulr. Howvr, h analysis chniqus and h ss prsnd in [5] only applis o synchronous priodic ask sysms wih harmonic priods. On muliprocssors, [3] prsns h only xising global suspnsion-awar analysis for priodic slfsuspnding ask sysms schduld undr global EDF and global fixd-prioriy schdulrs. In anohr rcn work ha is undr submission [], w considr h gnral cas of h slf-suspnsion ask modl, whr no rsricion is placd on h numbr of pr-job suspnsion sgmns and h compuaion and suspnsion parn. W dvlop a gnral inrfrnc-basd analysis framwork ha can b applid o driv sufficin uilizaion-basd ss for h gnral slf-suspnding ask modl. This papr is complly diffrn from h abov-mniond work [], w.r.. h argd problm, h proposd analysis chniqu, and h forma of h soluion. Daild diffrncs ar givn in Appndix. Conribuions. W answr an imporan qusion in his rsarch: for a givn sporadic singl-sgmn-suspnding ask compuaion suspnsion Fig. : Moivaing xampl. sysm, if a fasibl schdul xiss upon a uni-spd procssor, can w dsign a schduling algorihm ha will lad o a fasibl schdul whn allowd modraly fasr rsourcs? To answr his qusion, w firs show ha classical uniprocssor schdulrs including EDF and RM yild poor Efficin Dsign, Analysis, and Implmnaion of Complx Muliprocssor Ral-Tim Sysms prformanc for schduling slf-suspnding ask sysms, as hy yild a rsourc augmnaion bound, ha is infini. As an alrnaiv, w obsrv and prov ha a rahr simpl FRD schdulr,, yilds non-rivial rsourc-augmnaion prformanc guarans w.r.. any FRD schdulr and any arbirary schdulr. Spcifically, w driv a psudo-polynomialim schdulabiliy s for ha is xac and has a rsourc-augmnaion bound of and 3 w.r.. any FRD schdulr and any arbirary schdulr, rspcivly. To rduc h im complxiy, w furhr prsn a linar-im schdulabiliy s for, which yilds a rsourc-augmnaion bound of.787 and.875, w.r.. any FRD schdulr and any arbirary schdulr, rspcivly. Furhrmor, xprimns prsnd hrin show ha our proposd schdulabiliy ss improv upon prior ss by a larg margin in all cass. Morovr, our linar-im schdulabiliy s undr achivs lil or vn no uilizaion loss in many cass. Sysm Modl W considr h problm of schduling a s T = {,,..., n } of n indpndn sporadic slf-suspnding (SSS) asks on on procssor. Each ask is rlasd rpadly, wih ach of such invocaions calld a job. Jobs alrna bwn compuaion and suspnsion phass. W assum ha ach job of i conains a mos wo compuaion phass sparad by on suspnsion phas. This suspnding ask modl (as considrd in numrous prior work [9], [6], [9], []) acually covrs a larg s of ral-world applicaions ha involv slf-suspnsion bhaviors. For xampl, in many mulimdia applicaions, a common ask is o iniializ h vido procssing cod, hn fch vido daa from h disk (which can b modld by a slf-suspnsion phas), and finally procss h vido daa. Each ask i is characrizd by h following paramrs. minimum inr-arrival im (also calld priod), wors-cas xcuion im C i, on h firs compuaion phas, wors-cas suspnsion im S i of a ask insanc, and wors-cas xcuion im C i, on h scond compuaion phas, rlaiv dadlin D i. If C i, is, i mans ha hr is only on compuaion phas for ask i. If S i is, w also implicily assum ha C i, is and ask i dos no slf-suspnd.
compuaion C, S C, suspnsion Fig. : An xampl SSS ask wih C i, =, S =, C i, = 3, D i = =, and a uilizaion of.5. An SSS ask s s said o b an implici-dadlin ask s if D i = holds for ach i. Du o spac consrains, w limi our anion o implici-dadlin SSS asks in his papr. Efficin Dsign, Analysis, and Implmnaion of Complx Muliprocssor Ral-Tim Sysms According o our SSS ask modl, ach ask i can b considrd as wo subasks rprsning h wo compuaion phass, dnod i, and i,. An xampl SSS ask is givn in Fig.. Th j h job of i, dnod j i, is rlasd im rj i and has a dadlin a im D j i = rj i + D i. Similarly, ach job j i consiss of wo subjobs (sparad by a suspnsion), dnod j i, and j i,. Succssiv jobs of h sam ask ar rquird o xcu in squnc. No ha, whn a job of a ask misss is dadlin, h rlas im of h nx job of ha ask is no alrd. Th uilizaion of i is dfind as U i = C i /, and h uilizaion of h ask s T as U sum = U i. W rquir C i, + S i + C i, D i, U i, and U sum ; ohrwis, dadlins can b missd. W dno max{c i,, C i, } as C i,max and min{c i,, C i, } as C i,min, rspcivly. Th fixd-rlaiv-dadlin schduling has a fasibl schdul if and only if h wors-cas rspons im of h firs (scond, rspcivly) compuaion phas of ask i is no mor han D i, (D i,, rspcivly). To nsur h fasibiliy of h rsuling schdul, such a schduling policy has o nsur ha D i, + D i, + S i. Thorm. For a givn ask s T, dciding whhr hr xiss a rlaiv-dadlin assignmn for fixd-rlaiv-dadlin schduling is N P-compl in h srong sns. Proof: Th rducion, from h 3-PARTITION problm, is h sam as h proof for h prmpiv cas in Thorm in []. 3. Rsourc Augmnaion Facor Du o Thorm, a common approach for quanifying h qualiy of schduling algorihms and schdulabiliy ss is o quaniaivly bound h dgr o which h algorihm undr considraion may undr-prform a hypohical opimal on. To obain such a bound, w adop h concp of h rsourc augmnaion facor [], [8]. Whn h spd of h sysm is f, h wors-cas xcuion ims C i, and C i, bcom Ci, f and Ci, f, rspcivly. Howvr, S i rmains h sam. Typically, h rsourc augmnaion facor is dfind, by rfrring o any arbirarily fasibl schdul: 3 Our Schduling Policy and Spd-Up Facors 3. Fixd-Rlaiv-Dadlin Schduling As discussd in Sc., i is no wis o schdul SSS asks undr EDF and RM. A major rason is bcaus h pr-job dadlin paramr canno accuraly rprsn h urgncy of a slf-suspnding job. For an ordinary sporadic ask i, h im availabl for is complion is givn by D i. Howvr, for a slf-suspnding ask i, h im availabl for is complion is acually givn by D i S i. For insanc, for h ask sysm shown in Fig., has only wo im unis availabl for is complion of is wo subasks. Subjob, nds o b compld by im in ordr for job, o m h dadlin, which implis ha, has a dadlin a im on. Moivad by h abov obsrvaion, a br alrnaiv is o s fixd rlaiv dadlins for ach subask. An FRD policy is o s rlaiv dadlins D i, and D i, for h xcuions of h firs subask and h scond subask of i, rspcivly. Whn a job of ask i arrivs a im, h rlas im and h absolu dadlin of h firs subjob (i.., h firs compuaion phas) ar and +D i,, rspcivly, h suspnsion has o b finishd bfor + D i, + S i, h rlas im and h absolu dadlin of h scond subjob (i.., h scond compuaion phas) is +D i, +S i and + D i, + S i + D i,, rspcivly. For h rs of his papr, w call such a schduling policy a fixd-rlaiv-dadlin schdulr (FRD schdulr). Afr h rlaiv dadlins ar assignd, w will us EDF o schdul h subjobs by using dynamic-prioriy schduling. Schduling algorihm wih rspc o arbirary schduls: For noaional brviy, w call such a facor h arbirary spd-up facor. Providd ha h ask s T can b fasibly schduld, an algorihm A is said o hav an α arbirary spd-up facor whn algorihm A guarans o driv a fasibl schdul by spding up h sysm wih a facor α. Schdulabiliy s wih rspc o arbirary schduls: A schdulabiliy s is wih an α arbirary spd-up facor for a schduling algorihm A: if h s fails, i.., h s rurns infasibiliy, hn h ask s is also no schdulabl (undr any schduling policy) by slowing o run a spd α. Morovr, h abov dfiniion can also b xndd by rfrring o h opimal fixd-rlaiv dadlin schduls: Schduling algorihm wih rspc o FRD schduls: For noaional brviy, w call such a facor h FRD spd-up facor. Providd ha h ask s T can b fasibly schduld undr a fixd-rlaiv-dadlin schdul, an algorihm A is said o hav an α FRD spd-up facor whn algorihm A guarans o driv a fasibl fixdrlaiv-dadlin schdul by spding up h sysm wih a facor α. Schdulabiliy ss wih rspc o FRD schduls: A schdulabiliy s is wih an α FRD spd-up facor for an FRD schduling algorihm A: if h s fails, i.., h Th proof for h only-if par of h prmpiv cas in Thorm in [] is no prcis Hr, w us h sam rminologis in [] o pach h proof. Th proof dos no considr a possibl cas, in which four subasks sar in on block k bu only 3 subasks ar compld in his block. In such a cas, in h block k +m, h workload ha can b procssd is sricly lss han h block siz B. Th conradicion in h proof for h only-if samn rmains. 3
s rurns infasibiliy, hn h ask s is also no schdulabl undr any FRD schduls by slowing o run a spd α. Ti Ti - Si 3.3 Spd-Up Facors of EDF and RM Classical prioriy-basd dynamic schduling algorihms such as EDF and RM ar abl o dlivr good prformanc on a uniprocssor for ordinary ral-im ask sysms wihou slfsuspnsions (.g., EDF is opimal on a uniprocssor). Unforunaly, h following horm shows ha hs algorihms yild rahr poor prformanc in h prsnc of slf-suspnsions w.r.. spd-up facors if w do no s individual rlaiv dadlins for h compuaion phass. Thorm. Th arbirary and h FRD spd-up facor undr EDF and RM is. Proof: Th proof is similar o h proof of Thorm 6 in [] wih minor changs. Th dails ar in h Appndix. Ncssary Condiions for Schdulabiliy This scion prsns h ncssary condiions for h schdulabiliy of FRD schduls and any arbirary schduls. Th following lmma givs h ncssary condiion of any FRD schdul (by considring h wors-cas job arrivals). Lmma. No mar how D i, and D i, ar assignd undr h condiion D i, + D i, S i, h ncssary condiion for h schdulabiliy of any FRD schdulr is >, dbf i (), () i T whr < S i dbf i() = (C i, + C i,) S i + (Ti S i ) (C i, + C i,) Proof: This lmma is provd in skch by valuaing h compuaion dmand incurrd by h FRD schdulr in any givn im inrval lngh. I is no difficul o prov ha w only hav o considr h cas ha D i, + D i, = S i for valuaing h ncssary condiion for h schdulabiliy of h FRD schduling policy. Now, l us considr a fixd im insan x. Assum ha h firs job of ask i is rlasd a im x (D i, +S i ) and all h corrsponding jobs ar rlasd as soon as h minimum inr-arrival im consrain is m. Thrfor, w know ha ask i has o finish dbf i () in im inrval (x, + x] for any >. As a rsul, h ncssary condiion is provd. Fig. 3 illusras how h dmand bound funcion in () is dfind for any FRD schdulr. Nx w provid a ncssary condiion for schduabiliy undr any schduling policy. Lmma. Th ncssary condiion for h schdulabiliy undr any schduling policy is >, (). (3) i T () i r k i, k i, r k i, +Di, r k i, =r k i, +Di,+Si k i, r k i, +Di, = r k i, +Ti k+ i, r k i, +Ti+Di, Tim Fig. 3: An illusraion of () as a ncssary condiion for h schdulabiliy undr any FRD schduling policy. dbf i () dbf i () ncssary condiion for a fasibl FDR schdul (usd as a lowr bound) ncssary condiion for any fasibl schdul (usd as a lowr bound) Equaion () Equaion () () ncssary and sufficin condiion of Algorihm Equaion (7) (usd as an uppr bound) Efficin Dsign, Analysis, and Implmnaion of () sufficin condiion of Algorihm wih a Equaion () Complx Muliprocssor Ral-Tim Sysms linar approximaion (usd as an uppr bound) () ransformaion of dbf i () for analyzing h Equaion () spd-up facors of h linar approximaion TABLE I: Varian symbols rlad o h dmand bound funcions, whr is an FRD algorihm sudid in Sc. 5. whr < S i () = C i,max+ () (Ti S i ) (C i, + C i,) S i Proof: Th proof is similar o h proof of Lmma, bu has o considr h arbirary schduling policy. L T b h ask of asks in n which C i, > C i, and T b T \ T. Now, l s fixd a im insan x. For ach ask i, l h firs job of ask i b rlasd a im x if i is in T, and a im x S i C i, if i is in T. Morovr, all h corrsponding jobs ar rlasd as soon as h minimum inr-arrival im consrain is m. Thrfor, w know ha ask i has o finish a las () in im inrval (x, + x] for any >. As a rsul, h ncssary condiion is provd. Thrfor, h funcions dbf i () and () provid h lowr bounds for fasibl FRDand arbirary schduls. According o Lmma, if max i T () > > f, h ask s T canno b fasibly schduld by any FRD schdul if h spd is lowr han or qual o f. Similarly, du o Lmma, if max i T dbf i () > > f, T canno b fasibly schduld by any schdul if h spd is lowr han or qual o f. Noaions Rlad o DBF Th rs of his papr will us svral dfiniions rlad o h dmand bound funcions ha rprsn h minimum or h maximum dmand ha has o b finishd wihin a spcifid im inrval lngh. To xplain h diffrnc among hs rms, w provid a abl (in Tab. I) and an illusraiv summary (in Fig. ) hr. 5 An FRD Schduling Algorihm In his scion, w prsn an qual-dadlin assignmn () algorihm ha assigns qual rlaiv dadlins o subasks of ach ask and all subjobs ar schduld by EDF. W hn driv a psudo-polynomial-im schdulabiliy s for his algorihm and prov corrsponding non-rivial rsourcaugmnaion prformanc guarans.
8 7 6 5 3 9 8 7 6 5 3 () dbf i () 8 6 8 3 36 8 5 56 6 (a) 8 7 6 5 3 9 8 7 6 5 3 8 6 8 3 36 8 5 56 6 (b) () () dbf i () Fig. : An xampl of h lisd funcions in Tab. I, whn C i, = 3, C i, =, S i =, and =. For clariy, wo sub-figurs ar prsnd o avoid ovrlapping of h curvs. 5. Algorihm For ach ask i, D i, + D i, S i mus hold. maks h following assignmn D i, = D i, = S i. (5) As D i, = D i,, for noaional brviy, w dno D i, and D i, as i. Th following horm givs a schdulabiliy s for ha can b chckd in psudo-polynomial im. Thorm 3. Algorihm gnras a fasibl fixdrlaiv-dadlin schdul for h inpu ask s if and only if >, (), (6) whr i T () = < S i C i, + C i, ( C i,max (Ti S i ) ( (Ti S i ) )+ S i < T i S i = S i + )(C i, + C i,) > S i. Proof: Du o spac limiaion, w only provid h concps bhind h s. Th funcion dscribd in (7) is basically similar o ha in () xcp h rangs [ Ti Si + v, S i + v ) for any non-ngaiv ingr v,.g., in Fig. a. Such a rang is du o h fac ha assigns hs wo subasks of ask i wih h sam rlaiv dadlin S i, in which on of hm rquirs C i,max amoun of im for xcuion and h ohr rquirs C i,min. For ask i in any inrval lngh in [ Ti Si + v, S i + v ), in h wors cas, hr ar v + subjobs wih xcuion im C i,max and v subjobs wih xcuion im C i,min. Ths subjobs of ask i do no hav any ovrlap in hir rlas ims and absolu dadlins. Thrfor, h if and only-if condiions can b provd by using h sam sragy o prov h xac schdulabiliy s for h ordinary sporadic ral-im asks in [3]. Thorm 3 also implis ha h condiion in (6) is an xac (7) schdulabiliy s, ha rquirs psudo-polynomial im, of Algorihm. 5. Quaniaiv Evaluaion of W now offr a quaniaiv valuaion of h fficacy of Algorihm. Spcifically, w driv propris ha ar usd o provid a quaniaiv masur w.r.. rsourc augmnaion facors of how ffciv Algorihm is compard o an opimal fixd-ralaiv-dadlin schdulr (Thorm ) and an opimal arbirary schdulr (Thorm 5). Lmma 3. For any for a ask i T, w hav () dbf i () C i,max and () () C i,max. Lmma. For any for a ask i T, w hav dbf i () (). Proof: Th proofs of Lmma 3 and Lmma ar by simpl arihmics and h dfiniions of hs hr funcions. Spcifically, h proof of Lmma is in h Appndix. Boh of h abov propris can b also found in Fig.. Thorm. Th FRD spd-up facor of Algorihm is. Proof: Suppos ha a im >, w hav i T ( ) >. L us now classify h asks in no hr ask ss: T : if < Ti Si, ask i is in T. T : if S i > Ti Si, ask i is in T. T 3 : if S i, ask i is in T 3. Clarly, ach ask i in s ihr in T, T, or T 3. Now suppos ha y. Sinc i T i T dbf i ( ) is x and ( ) >, w hav x + y >. i T dbf 3 i ( ) is 5
By Lmma, w hav i ( ) i T T 3 dbf Thrfor, by (8), w hav max > i T T 3 dbf i ( ). (8) dbf i () dbf i ( ) i T T 3 dbf i ( ) i T T 3 dbf = x + y i ( ) >.5. (9) Thrfor, his implis ha h ask s T canno b fasibly schduld by any fixd-rlaiv-dadlin schdul whn h spd of h sysm is.5. Wih a similar proof, w can also show h arbirary spdup facor for. Thorm 5. Th arbirary spd-up facor of Algorihm is 3. Proof: Th proof sragy is similar o h proof of Thorm. Th major diffrnc in h analysis is o furhr considr a subs of asks in which 3 S i > S i and a propry rvisd from Lmma for funcion dbf i (). Th daild proof is in h Appndix. 6 Linar-Tim Schdulabiliy Ts Sc. 5 prsns and schdulabiliy analysis in psudo polynomial-im. This scion furhr prsns a linar-im schdulabiliy s, undr h assumpion ha h soring of h asks according o S i is don in advanc. 6. Dnsiy-Basd Schdulabiliy Analysis By simpl arihmics, i can b asily shown ha i () for any. Thrfor, o achiv a fas dbf Ci,max ( S i)/ C i,max schdulabiliy analysis, if i T S i, provids a fasibl schdul undr EDF. Howvr, such a schdulabiliy s C i,max i T S i can b vry pssimisic. Thorm 6. Th FRD spd-up facor of h schdulabiliy s by vrifying whhr C i,max i T S i holds for n asks is a las H n, whr H n = n i. Proof: To prov such a lowr bound, w jus hav o provid a concr cas. Considr a spcial inpu insanc wih n asks as follows: C i, = C i, = f, S i = n i + and = n +, for ask i, in which f is Hn +ɛ wih ɛ >. For his inpu insanc, w hav i T f i = n f i = + ɛ >. Th schdulabiliy s hr shows ha his inpu insanc is no schdulabl undr. Howvr, i is vry clar ha h rsuling soluion by is fasibl vn if h sysm is slowd down o run a spd f. 8 7 6 5 3 9 8 7 6 5 3 C i = 5 C i = 3 8 6 8 3 36 8 5 56 6 () () () Fig. 5: An xampl of h linar approximaion by using diffrn C valus, whr C i, = 3, C i, =, S i =, and =. Thrfor, C i, + C i, = 5 and max {C i,max, C i, + C i, U i i } is 3. 6. Linar Approximaion W now prsn a linar-im schdulabiliy s for Algorihm and analyz h spd-up facors for h schdulabiliy s. Th ky ida is o us a linar curv o (uppr) bound h funcion () for all, which has bn usd in [], [5]. Whn h linar curv is oo far away from h funcion (), h rror bcoms oo larg. Thrfor, h dsign philosophy is o minimiz h gap bwn h linar curv and h funcion (). Th incras of h funcion () bcoms priodic whn is larg nough, i.., () = ( ) + C i, +C i,, > S i. Thrfor, w should us h uilizaion U i = Ci,+Ci, as h slop of h linar approximaion, so ha h approximaion bcoms mor prcis whn is larg nough. Tha is, his linar approximaion should b abl o bound h funcion (), undr h slop U i. To provid a saf uppr bound of h funcion () a any inrval lngh, w can sar wih a linar sgmn a inrval lngh i, which is dfind as Ti Si, wih a valu C i = max {C i,max, C i, + C i, U i i }. Tha is, his sgmn is wih a slop quals o U i. Th abov sing of C i coms from wo diffrn spcifid valus in h linar sgmn: () C i,max a lngh i or () C i, + C i, a lngh i (which implis ha h sgmn sars wih C i, +C i, U i i a lngh i ). Sinc max {C i,max, C i, + C i, U i i } C i, + C i,, sing C i o C i, + C i, wih a slop U i, saring a lngh i, is also a fasibl uppr bound of dbf (). Fig. 5 prsns h abov wo diffrn linar approximaions. For h simpliciy of prsnaion for h analysis in Sc. 6., w will considr ha C i is qual o C i, + C i, for h rs of his scion, whil h xprimnal rsuls in Sc. 7 will b basd on C i = max {C i,max, C i, + C i, U i i }. Thrfor, for h rs of his scion, suppos ha Morovr, h linar approximaion is dbf i () = C i = df C i, + C i,. () { < i C i + ( i)u i i. W will us h rm C i o mak disinguishs o C i. () 6
Thn, w hav h following lmma. Lmma 5. For any givn > and i T, dbf i () (). Proof: This is basd on simpl arihmics, as also shown in Fig.. Thorm 7. Algorihm gnras a fasibl fixdrlaiv-dadlin schdul for h inpu ask s if >, (). () i T Proof: This coms dircly from Lmma 5 and Thorm 3. For noaional brviy, for h rs of his scion, w ordr h asks such ha T S T S... T n S n. Tha is, i i+ for i =,,..., n. Thorm 8. Suppos ha i i+ for i =,,..., n. Algorihm gnras a fasibl fixd-rlaiv-dadlin schdul for h inpu ask s if U sum = n U i and l ( l ) l, l =,,..., n. (3) Proof: I is clar ha n U i is a ncssary condiion for fasibl schduls. For noaional brviy, l n+ b. Wih n U i and (3), for any l < l+ and l =,,..., n, w know ha n dbf i () = l dbf i ( l ) + ( l ) l + l =. l Thrfor, w know ha h condiion in () in Thorm 7 holds, which concluds h proof. Th following corollary provids h im complxiy analysis for h schdulabiliy s. Corollary. Th schdulabiliy s in Thorm 8 can b don in linar im providd ha h asks ar ordrd in a nondcrasing ordr wih rspc o i. Proof: Th s can b don by incrmnally valuaing l ( l ) for l =,,..., n. Th dails ar providd in h Appndix. No ha h abov linar approximaion, as prsnd in Fig. 5, has on jump a im i and has a consan slop afr im i. Such a yp of linar approximaion has bn usd o pariion sporadic ral-im asks o idnical muliprocssor sysms [], [5], [6]. Th sam ask pariioning sragy in [], [5], [6] can b usd o assign h slf-suspnd asks o muliprocssor sysms by adoping h linar approximaion hr for sing whhr i is fasibl o assign a slf-suspnd ask ono a procssor. Th dails of h analysis ar no prsnd hr du o h spac limiaions. For h rs of his scion, w will prov h arbirary and FRD spd-up facors of h linar approximaion in h U i schdulabiliy s in Thorm 8. W dno h schdulabiliy s in Thorm 8 as schdulabiliy analysis (s) LA. 6.3 Spd-Up Facor Analysis: Basic Analyss Hr, w will firs provid h basic analysis by using a simplr proof sragy. W dfin h following sp funcion () o rprsn h lowr bound of h linar approximaion dbf i (). { < i () = ) C i + ( i (C i, + C i, ) i () By h abov dfiniion, w hav h following lmma: Lmma 6. For any givn > and i T, dbf i () dbf i (). (5) Proof: I clarly holds whn < i, sinc () =. By h dfiniion of C i, w hav C i C i,+c i,. Morovr, wih h dfiniion of dbf i (), w know ha () dbf i ( + i ). (Ths wo inqualiis happn whn C i is s o max {C i,max, C i, + C i, U i i } insad of C i, + C i,.) Thrfor, whn i, w hav () dbf i (). W can now prov h FRD spd-up facor for h schdulabiliy analysis in Thorm 8. Thorm 9. Th FRD spd-up facor for h schdulabiliy analysis LA (in Thorm 8) is 3. Proof: Th schdulabiliy s in Thorm 8 can fail n by wo cass: () U i > or () l such ha l ( l ) > l. For h formr cas, i is clar ha h ask s is no schdulabl for any schduling policy. To prov h FRD spd-up facor, w only hav o focus on h lar cas, in which h rror coms from h linar approximaion as wll as h schduling policy. Suppos ha l is h smalls indx such ha dbf i ( l ) > l. For i =,,..., l, l δ i b + i. Tha is, δ i is s such ha l i is qual o δi i. As a rsul, ( l ) is qual o l l i dbf i l < (δ i ) + U i ( l δ i ). Thrfor, w hav l dbf i ( l ) ( l ) ( l ) = (δ i ) + U i ( l δ i ) ( l dbf i ( l ) ( l dbf i ( l ) ) ) + ( + ( l U i l ) l U i l ), whr coms from h fac ha δ i l and () 7
is a non-dcrasing funcion of for ach i < l, and coms from Lmma 6. By dividing l in boh sids, w hav ( l ) < dbf i( l ) l + ( U i ). l Thrfor, ihr l U i > 3 or l ( l) l > 3. This also implis ha, by slowing down h sysm o spd 3, hr dos no xis any fasibl FRD schdul sinc h ncssary condiion in Lmma canno b saisfid. Thrfor, w rach h conclusion. Similar o h proofs in Lmma 6 and Thorm 9, h following lmma and horm can b achivd by using a vry similar sragy. Lmma 7. For any givn > and i T, dbf i () dbf i (). (6) Proof: I clarly holds whn < i, sinc () =. By h dfiniion of C i in (), w hav C i C i, + C i,. Morovr, wih h dfiniion of (), w know ha () (+ i)+c i,min. Thrfor, whn i, w hav () () + C i,min (). Thorm. Th arbirary spd-up facor for h schdulabiliy analysis LA (in Thorm 8) is 5. Proof: By using h sam sragy in h proof in Thorm 9 by using () insad of dbf i(), w rach ( l ) l l < ( l ) + ( U i l ). By dividing l in boh sids, w hav ( l ) < dbf i ( l) l + ( U i ). l Thrfor, ihr l U i > 5 or l dbf i ( l) l > 5, which concluds h proof. 6. Spd-Up Facor Analysis: Tighr Analysis Following h proofs in Thorms 9 and, w focus ourslvs o provid ighr analysis in his subscion. Th analysis xnds h analysis dvlopd in [6] for h spd up facor of h linar approxima dmand bound funcion for ordinary sporadic ral-im asks (wihou slf-suspnsions). Wih h sing of C i = C i, + C i, in (), h conncion bwn h wo funcions () and () is idnical o h linar approximaion whn considring normal sporadic asks in [6]. Now, l us look a h linar approximaion in () and h ncssary condiion basd on dbf i () in Thorm () mor closly. Basd on Lmma 5, w can obsrv h following horm. Thorm. Th FRD spd-up facor for h schdulabiliy analysis LA (in Thorm 8) is α if hr xiss > wih i T () > α or U i >.5α. Proof: By Lmma 6, h condiion h xisnc of i T () i T () wih > α implis ha > α. Thrfor, if such a condiion holds or U i >.5α, by dfiniion, h FRD spd-up facor for h schdulabiliy analysis LA is α. For h rs of his subscion, l l b h smalls indx such ha l ( l ) > l. Morovr, δ i is dfind as i + i, as in h proof of Thorm 9. Th ky diffrnc bwn h ighr analyss hr and hos in Sc. 6.3 is du o h rang of for valuaing i T (). In h proofs of Thorms 9 and, w only valua on singl valu, in which = l, whras h ighr analyss hr will adop Thorm and valua h maximum valu of i T () for < l. For noaional simpliciy, w dfin β, k, and k as follows: l C l df l = β (δ i ), (7) l df l ( l δ i ) U i = k (δ i ). (8) k (δ i ) df l = l ( + β) dbf i (δ i ). (9) Tha is, by aking l (δ i ) as h basis, β dfins h raio of C l o l (δ i ), k dfins h raio of h parially rlasd workload l ( l δ i ) U i in im inrval lngh l wih rspc o l (δ i ), and k is dfind o s l = ( + k + β) l (δ i ). l Basd on h abov dfiniion and h fac ha ( l ) > l, w know ha k > k and dbf i max n () l dbf i ( l ) l = + β + k + β. () Lmma 8. If k (.5 ) ( + β), hn.5 { } l max max dbf i() l + β, U i > > ( + k + β). Proof: This coms dircly from h abov analysis in (). Thrfor, whn k is small, h bound in Lmma 8 can b usd. Th following analysis movs furhr by considring largr k, in which h corrsponding proofs ar vry similar 8
o h proofs in [6] and h major diffrncs ar providd in h Appndix, for complnss. Lmma 9. For any non-ngaiv paramrs, k, β, x, α, if x α(+k+β)(.5 ), hn x.5 +k+β dx α x α. Lmma. Suppos ha l is h smalls indx such ha ( l ) > l and α is. If, for l dbf i l k+β +k+β () (.5 ) (.5 ) all < l, w hav α, hn, h oal uilizaion l U i for,,..., l is largr han +k+β dx = α x α. k Lmma. If k > (.5 ) ( + β), hn.5 { } l max max dbf l i().5 (, U i >.5 ) k > ( + k + β). () W can now conclud h analysis by providing h corrsponding spd-up facors for h schdulabiliy analysis in Thorm 8. Thorm. Th FRD spd-up facor for h schdulabiliy analysis LA (in Thorm 8) is (.5 ).5 <.787. +β Proof: By Lmma 8, h funcion (+k+β) is a dcrasing funcion wih rspc o k whn k. By Lmma,.5 (.5 ) k h funcion (+k+β) is an incrasing funcion wih rspc o k. Thrfor, h only inrscion whn k is qual o.5 ( + β) dfins which par should b usd for bounding h.5 spd-up facor. Thr ar wo cass: Whn k.5.5 ( + β), by using Lmma 8, w know ha h FRD spd-up facor is a mos (.5 ).5. Whn k >.5.5.5.5 (+β) +β = ( + β), by using Lmma, w know ha h FRD spd-up facor is a mos (.5 ).5..5.5 (+β) +β = Thrfor, h FRD spd-up facor for h schdulabiliy analysis in Thorm 8 is (.5 ).5 <.787. Thorm 3. Th arbirary spd-up facor for h schdulabiliy analysis in Thorm 8 is (.5 ).5 <.875. Proof: Th proof is idnical o h proof of FRD spd-up facor, bu has o considr. Thrfor, h corrsponding rvision of Thorm for h arbirary spd-up facor bcoms α. By following h sam proof procdur, w can rach h conclusion by considring wo cass, i.., whhr k is largr han.5 ( + β). Wih h sam procdur, w can rach h conclusion..5 Du o spac limiaion, h dails ar omid. 7 Exprimn In his scion, w conduc xnsiv xprimns using randomly-gnrad ask ss o valua h applicabiliy of our linar-im schdulabiliy s (Thorm 8), dnod LA. W valuad h ffcivnss of LA by comparing i o h suspnsion-oblivious approach dnod SC. Tha is, afr ransforming all slf-suspnding asks ino ordinary priodic asks (no suspnsions) using SC, h original ask sysm is schdulabl if h oal uilizaion of h ransformd ask sysm is no grar han. In our xprimns, sporadic slf-suspnding ask ss wr gnrad similar o h mhodology usd in [], [3]. Task priods, i.., s, wr disribud uniformly ovr [ms, ms]. Task uilizaions, i.., U i s, wr disribud diffrnly for ach xprimn using four uniform disribuions. Th rangs for h uniform disribuions wr [.5,.] (ligh), [.,.3] (mdium), [.3,.5] (havy), and [.5,.5] (uniform). Task xcuion coss wr calculad from priods and uilizaions. Suspnsions lnghs of asks wr also disribud using four uniform disribuions: [. ( U i ),. ( U i ) ] (shor suspnsion lngh), [. ( U i ),.3 ( U i ) ] (modra suspnsion lngh), [.3 ( U i ),.6 ( U i ) ] (long suspnsion lngh), and [. ( U i ),.6 ( U i ) ] (uniform suspnsion lngh). 3 For ach combinaion of ask uilizaion disribuion, suspnsion lngh disribuion, and U sum,, ask ss wr gnrad. Each such ask s was gnrad by craing asks unil oal uilizaion xcdd h corrsponding uilizaion cap, and by hn rducing h las ask s uilizaion so ha h oal uilizaion quald h uilizaion cap. Th obaind schdulabiliy rsuls ar shown in Fig. 6 (h organizaion of which is xplaind in h figur s capion). Each curv plos h fracion of h gnrad ask ss h corrsponding approach succssfully schduld, as a funcion of oal uilizaion. As sn, in all sd scnarios, LA clarly improvs upon SC by a subsanial margin. For xampl, as sn in Fig 6(a), whn ask uilizaions ar ligh, LA can achiv % schdulabiliy whn U sum.8, U sum.76, U sum.6, and U sum.5 wih shor, modra, uniform, and havy suspnsion lnghs, rspcivly; whil SC fails o do so whn U sum mrly xcds.36,.,.,., rspcivly. Morovr, as sn in all four ins of Fig 6, whn ask suspnsion lnghs ar long or uniform, SC fails o schdul mos of h gnrad ask ss whil LA is abl o dlivr good prformanc. For insanc, as sn in Fig 6(c), whn ask uilizaions ar havy, LA is abl o achiv % schdulabiliy whn U sum.8 wih havy or uniform suspnsion lnghs; whil SC fails o do so whn U sum mrly xcds.3. This is bcaus in hs cass, h uilizaion loss du o h convrsion of long suspnsions ino compuaion undr SC is oo significan. Anohr inrsing obsrvaion is ha boh mhods prform br whn ask uilizaions ar havir. This is du o h fac ha wih havir ask uilizaions, a lss numbr of slf-suspnding asks can b gnrad and h suspnsion lnghs of such asks ar shorr compard o h cas wih lighr ask uilizaions. This clarly allvias h ngaiv impac du o suspnsions in h schdulabiliy. 3 No ha any S i is uppr-boundd by ( U i ) 9
LA-s LA-m LA-l LA-u SC-s SC-m SC-l SC-u LA-s LA-m LA-l LA-u SC-s SC-m SC-l SC-u LA-s LA-m LA-l LA-u SC-s SC-m SC-l SC-u LA-s LA-m LA-l LA-u SC-s SC-m SC-l SC-u % 8 % 6 % % % %...6.8 (a) ligh ask uilizaion % 8 % 6 % % % %...6.8 (b) mdium ask uilizaion % 8 % 6 % % % %...6.8 (c) havy ask uilizaion % 8 % 6 % % % %...6.8 (d) uniform ask uilizaion Fig. 6: Schdulabiliy rsuls. In all four graphs, h x-axis dnos h ask s uilizaion cap and h y-axis dnos h fracion of gnrad ask ss ha wr schdulabl. Each graph givs four curvs pr sd approach for h cass of shor, modra, long, and uniform ask suspnsion lnghs, rspcivly. As sn a h op of h figur, h labl LA-s(m/h/u) indicas h approach of LA assuming shor (modra/long/uniform) ask suspnsion lnghs. Similar SC labls ar usd for SC. 8 Conclusion For a givn sporadic slf-suspnding ask sysm, if a fasibl schdul xiss upon a uni-spd procssor, can w dsign a schduling algorihm ha will lad o a fasibl schdul whn allowd modraly fasr rsourcs? To answr his qusion, w prsn an FRD schduling algorihm, ha assigns rlaiv dadlins qually o compuaion phass of slf-suspnding asks. W prov ha yilds non-rivial rsourc-augmnaion prformanc guarans. Spcifically, w driv a psudo-polynomial-im schdulabiliy s for ha is xac and yilds a rsourc-augmnaion bound of and 3 w.r.. any FRD schdulr and any arbirary schdulr, rspcivly. To rduc h im complxiy, w furhr prsn a linar-im schdulabiliy s for, which yilds a rsourc-augmnaion bound of.787 and.875, w.r.. any FRD schdulr and any arbirary schdulr, rspcivly. Furhrmor, xprimns prsnd hrin show ha our proposd schdulabiliy ss improv upon prior ss by a larg margin. Basd on Thorms and 5, w can also adop h approach proposd in [] o provid approxima schdulabiliy in polynomial-im complxiy. Tha is, w ak a prdfind numbr of (diffrn) discr valus in (7) a bginning and us a linar approximaion afr h las discr valu. Wih such an approach, i is also no difficul o show ha h sudid problm also admis schdulabiliy ss wih FRD and arbirary spd-up facors +ɛ and 3+ɛ wih polynomial im complxiy proporional o O( ɛ ), rspcivly. Whil w hav assumd implici-dadlin slf-suspnding ask sysms, w obsrv ha our rsuls can b dircly applid o h consraind-dadlin cass, and can also b xndd o apply o h arbirary-dadlin cas. Th inuiiv rason is bcaus our analysis chniqus do no rly on h assumpion ha D i = holds for any ask i. For fuur rsarch, w plan o furhr ighn h analysis, and considr mor gnral slf-suspnding ask modls. Rfrncs [] K. Albrs and F. Slomka. An vn sram drivn approximaion for h analysis of ral-im sysms. In ECRTS, pags 87 95,. [] S. Baruah and N. Fishr. Th Pariiond Muliprocssor Schduling of Sporadic Task Sysms. In Proc. of h 6h IEEE Ral-Tim Sysms Symp., pags 3 39, 5. [3] S. K. Baruah, A. K. Mok, and L. E. Rosir. Prmpivly schduling hard-ral-im sporadic asks on on procssor. In IEEE Ral-Tim Sysms Symposium, pags 8 9, 99. [] S. Chakrabory, S. Künzli, and L. Thil. A gnral framwork for analysing sysm propris in plaform-basd mbddd sysm dsigns. In DATE, pags 9 95, 3. [5] J. Chn and S. Chakrabory. Rsourc Augmnaion Bounds for Approxima Dmand Bound Funcions. In Proc. of h 3nd IEEE Ral-Tim Sysms Symp., pags 7 8,. [6] J.-J. Chn and S. Chakrabory. Rsourc augmnaion for uniprocssor and muliprocssor pariiond schduling of sporadic ral-im asks. Ral-Tim Sysms, 9():75 56, 3. [7] W. Kang, S. Son, J. Sankovic, and M. Amirijoo. I/O-Awar Dadlin Miss Raio Managmn in Ral-Tim Embddd Daabass. In Proc. of h 8h IEEE Ral-Tim Sysms Symp., pags 77 87, 7. [8] S. Kao, K. Lakshmanan, A. Kumar, M. Klkar, Y. Ishikawa, and R. Rajkumar. RGEM: A Rsponsiv GPGPU Excuion Modl for Runim Engins. In Proc. of h 3nd IEEE Ral-Tim Sysms Symp., pags 57 66,. [9] I. Kim, K. Choi, S. Park, D. Kim, and M. Hong. Ral-im schduling of asks ha conain h xrnal blocking inrvals. In Procdings of h nd Inrnaional Workshop on Ral-Tim Compuing Sysms and Applicaions, pags 5 59, 995. [] K. Lakshmanan and R. Rajkumar. Schduling slf-suspnding ral-im asks wih ra-monoonic prioriis. In Procdings of h 6h IEEE Ral-Tim and Embddd Tchnology and Applicaions Symposium, pags 3,. [] C. Liu and J. Andrson. Task schduling wih slf-suspnsions in sof ral-im muliprocssor sysms. In Procdings of h 3h Ral-Tim Sysms Symposium, pags 5 36, 9. [] C. Liu and J. Andrson. An O(m) analysis chniqu for supporing ral-im slf-suspnding ask sysms. In Procdings of h 33h IEEE Ral-Tim Sysms Symposium, pags 373 38,. [3] C. Liu and J. Andrson. Suspnsion-awar analysis for hard ralim muliprocssor schduling. In Procdings of h 5h EuroMicro Confrnc on Ral-Tim Sysms, pags 7 8, 3. [] C. Liu and J. Chn. Bursy-inrfrnc analysis chniqus for analyzing complx ral-im ask modls. In undr submission o RTSS. [5] C. Liu, J. Chn, L. H, and Y. Gu. Analysis chniqus for supporing harmonic ral-im asks wih suspnsions. In Procdings of h 6h Euromicro Confrnc on Ral-Tim Sysms. IEEE,. [6] J. Liu. Ral-Tim Sysms. Prnic Hall,. [7] J. C. Palncia and M. G. Harbour. Schdulabiliy analysis for asks wih saic and dynamic offss. In Procdings of h 9h IEEE Ral-Tim Sysms Symposium, pags 6 37, 998. [8] C. Phillips, C. Sin, E. Torng, and J. Win. Opimal im-criical schduling via rsourc augmnaion. In Proc. of h 9h ACM Symposium on Thory of Compuing, pags 9, 997. [9] R. Rajkumar. Daling wih Suspnding Priodic Tasks. Tchnical rpor, IBM T. J. Wason Rsarch Cnr, 99. [] F. Ridouard, P. Richard, and F. Co. Ngaiv rsuls for schduling indpndn hard ral-im asks wih slf-suspnsions. In Procdings of h 5h IEEE Ral-Tim Sysms Symposium, pags 7 56,. [] K. Tindll. Adding im-offss o schdulabiliy analysis. Tchnical Rpor, Univrsiy of York, 99. [] A. Toma and J.-J. Chn. Compuaion offloading for fram-basd ralim asks wih rsourc rsrvaion srvrs. In ECRTS, pags 3, 3.
Appndix Diffrncs of his papr from []. Th diffrncs can b imizd in dail as follows. ) Th analysis chniqus prsnd in [] focus on driving sufficin uilizaion-basd schdulabiliy ss; whil h analysis chniqus prsnd in his papr focus on analyzing h rsourc augmnaion bound. ) [] focuss on analyzing h ask-lvl fixd-prioriy RM schduling algorihm; whil his papr focuss on analyzing an FRD schdulr. 3) [] considrs a mor gnral slf-suspnding ask modl whr a ask is allowd o suspnd for mulipl ims; whil his papr considrs h singl-sgmn-suspnding ask modl. Proof of Thorm. Considr a spcial inpu insanc wih wo asks as follows: C, = T ε, S = C, =, T = T ε, and C, = ε, S = T ε, C, =, T = T, whr ε can b arbirarily small and T > is a consan. I is clar ha his ask s is fasibl if w assign highr prioriy o jobs of ovr. Howvr, undr EDF or RM, s jobs may g highr prioriis and hus caus jobs of o miss dadlins. In ordr for his ask sysm o b schdulabl undr EDF or RM, on a α-spd procssor, h following mus hold: T ε α + ε α ε. By rarrangmns, w hav α T. Thus, α as ε ε. Proof of Lmma. W considr four cass: If < Ti Si, w hav () =, and h inqualiy dbf i () () holds sinc dbf i (). If Ti Si < S i, w hav () = C i,max and dbf i () C i, + C i, C i,max. Thrfor, h inqualiy dbf i () () holds. If S i < 3Ti Si, w hav () = C i, + C i, and dbf i () C i, + C i, (). If 3Ti Si, w hav dbf i() dbf i( + 3Ti Si ) dbf i( + ) = 3 dbf i() + C i, + C i, ( () C i,max) + C i, + C i, 5 (), whr h inqualiy coms from h dfiniion ha 3 S i, h inqualiy coms from h fac ha > S i, h qualiy = coms from h dfiniion of dbf i (), and h inqualiy coms from Lmma 3. Lmma. For any and any ask i T, w hav ( + ) (). Proof: Th proof is similar o h proof of Lmma. By dfiniion, ( + ) () + C i,max () C i,max + C i,max = (). Lmma 3. { { y inf max x+y+z>, y + x + z }} > 3. Proof: Suppos ha x+y +z = +ɛ wih ɛ >. W can hn rwri y + x+z o y + +ɛ y = +ɛ y. I is clar ha y is an incrasing funcion wih rspc o y, whil +ɛ y is a dcrasing funcion wih rspc o y. Thrfor, h infimum happns whn y +ɛ y is qual o, i.., whn y is +ɛ 3. Thrfor, h infimum is +ɛ 3, which provs h lmma. Proof of Thorm 5. Suppos ha a im >, w hav i T ( ) >. L s now classify h asks in T ino four ask ss: T : if < Ti Si, ask i is in T. T : if S i > Ti Si, ask i is in T. T 3 : if 3 S i > S i, ask i is in T 3. T : if 3 S i, ask i is in T. Clarly, ach ask i in s ihr in T, T, T 3, or T. Similarly, w know ha i T dbf ( ) =. Now suppos ha and w hav i T dbf i ( ) i T dbf i ( ) is x, is z. Sinc i T z + y + x >. i i T dbf 3 i ( ) is y dbf i ( ) >, Basd on h dfiniion of () in () and 3 S i > S i for any ask i in T 3, w know ha ( ) = i T 3 i T 3 (C i, + C i, ) i T 3 C i,max = i T 3 dbf i ( ). () Thrfor, by (), w know ha max dbf i () dbf i ( ) > i T 3 ( ) For a ask i in T, w know ha i T 3 dbf i ( ) = y. (3) i T, dbf i ( ) dbf i ( + 3 S i) dbf i ( + ) dbf i ( ), () whr h firs inqualiy coms from h dfiniion ha 3 S i, h scond inqualiy coms from h fac ha
> S i, and h hird inqualiy coms from Lmma. Thrfor, w hav ( ) ( ). (5) i T i T Morovr, sinc S i > Ti Si for ask i in T, w know ha ( ) = C i,max i T i T = ( S i ) i T ( ). (6) i T By (), w also know ha ( ) ( ) i T 3 i T 3 i T 3 dbf i ( ) (7) By combining h abov inqualiis in (5), (6), and (7), w hav max dbf i () > dbf i ( ) i T T 3 T ( ) i T T ( ) + i T 3 ( ) = z + x + y. (8) By (3) and (8), w conclud h proof by showing ha max dbf i () > { y max, z + x + y } { { y inf max y+x+z>, z + x + y }} > 3, (9) Proof of Lmma 9. Sinc +k+β is non-ngaiv for x α h givn non-ngaiv paramrs, k, β, x, α in h ingraion rang, w hav =α x + k + β x dx α α(+k)(.5 ).5 ( log + k + β x dx α + k + β + k + β (+k+β)(.5 ).5 ) =.5α Proof of Lmma. Th proof is basically idnical o h proofs of Lmmas, 5, and 7 in [6]. Th conncion bwn h wo funcions () and () is idnical o h linar approximaion whn considring normal sporadic asks in [6]. L W b h parially rlasd workload, in which W = l ( l δ i )U i = k (δ i ). (3) df l Th proof can b don by rducing h rlaiv dadlins (Lmma in [6]) and incrasing h minimum inr-arrival im proprly (Lmma 5 in [6]) o incras h valu of W. Undr h assumpion of α, w know ha k is qual o α(+k+β)(.5 ). Thrfor, as in h proof of Lmma 9, w.5 know ha k +k+β dx is qual o α x α. Morovr, basd on Lmma 9, i can b provd (as in Lmma 7 in [6]) ha h uilizaion l U i mus b largr han k +k+β x α dx; ohrwis W will no b sufficin o mak h schdulabiliy s in Thorm 8 fail. Proof of Lmma. This{ coms dircly from Lmma and l Thorm in which max max () >, } l U i is largr han α. Morovr, w furhr drop β in h numraor in (), du o h fac ha α is.5 (.5 ) (k+β) (+k+β).5 (.5 ) (k) (+k+β). whr h las inqualiy coms from Lmma 3. Proof of Corollary. Th schdulabiliy s in (3) rquirs only o s n + im poins, i..,,,..., n,. Morovr, sinc l+ ( l+ ) = dbfl+ ( l+ )+ (( l+ l ) l U i) + ( l ( l )), calculaing l ( l ) can b don in O() im complxiy by soring l U i incrmnally and rfrring o l ( l ). Thrfor, h im complxiy is O(n), undr h assumpion ha h asks ar ordrd in a nondcrasing ordr wih rspc o i.