Mixing Real-Time and Non-Real-Time. CSCE 990: Real-Time Systems. Steve Goddard.

Transcription

1 CSCE 990: Ral-Tim Sym Mixing Ral-Tim and Non-Ral-Tim hp:// 1 Ral-Tim Sym Mixd Job - 1 Mixing Ral-Tim and Non-Ral-Tim in Prioriy-Drivn Sym (Chapr 7 of Liu) W dicud mixing ral-im and non-ral-im (apriodic) job in cyclic chdul W now addr h am iu in prioriy-drivn ym. W fir conidr wo raighforward chduling algorihm for priodic and apriodic job. W hn loo a a cla of algorihm calld bandwidh-prrving rvr ha chdul apriodic job in a ral-im ym. Ral-Tim Sym Mixd Job - 2 2

2 Priodic and Apriodic Ta (A rviw of h rminology Liu u ) Priodic a: T i i pcifid by (φ i, p i, i, D i ).»p i i h minimum im bwn job rla. Apriodic a: non-ral-im» Rlad a arbirary im.» Ha no dadlin and i i unpcifid. W aum priodic and apriodic a ar indpndn of ach ohr. Corrc and Opimal Schdul in mixd job ym (mor rm ) A corrc chdul nvr rul in a dadlin bing mid by priodic a. A corrc chduling algorihm only produc corrc chdul. An opimal apriodic job chduling algorihm minimiz ihr» h rpon im of h apriodic job a h had of h quu or» h avrag rpon im of all apriodic job. Ral-Tim Sym Mixd Job - 3 Ral-Tim Sym Mixd Job

3 Schduling Mixd Job W aum hr ar para job quu for ral-im (priodic) and non-ral-im (apriodic) job. How do w minimiz rpon im for apriodic job wihou impacing priodic? Priodic Job Apriodic Job Procor Bacground Schduling Priodic job ar chduld uing any prioriy-drivn chduling algorihm. Apriodic ar chduld and xcud in h bacground:» Apriodic job ar xcud only whn hr i no priodic job rady o xcu.» Simpl o implmn and alway produc corrc chdul. Th low prioriy a xcu job from h apriodic job quu.» W can improv rpon im wihou jopardizing dadlin by uing a lac aling algorihm o dlay h xcuion of priodic job a long a poibl. Thi i h am hing w did wih cyclic xcuiv. Howvr, i i vry xpniv (in rm of ovrhad) o implmn lac-aling in prioriy-drivn ym. Ral-Tim Sym Mixd Job - 5 Ral-Tim Sym Mixd Job

4 Simpl Priodic Srvr (Liu call hi a Polling rvr or h Pollr) Priodic job ar chduld uing any prioriy-drivn chduling algorihm. Apriodic ar xcud by a pcial priodic rvr:» Th priodic rvr i a priod a T p =(p, ). i calld h xcuion budg of h rvr. Th raio u = /p i h iz of h rvr.» Supnd a oon a h apriodic quu i mpy or T p ha xcud for im uni (which vr com fir). Thi i calld xhauing i xcuion budg.» Onc upndd, i canno xcu again unil h ar of h nx priod. Tha i, h xcuion budg i rplnihd (r o im uni) a h ar of ach priod. Thu, h ar of ach priod i calld h rplnihmn im for h impl priodic rvr. Priodic Srvr wih RM Schduling Exampl Schdul: Two a, = (3,1), = (10,4), and a priodic rvr T p = (2.5,0.5). Aum an apriodic job J a arriv a = 0.1 wih and xcuion im of a = 0.8. Th priodic rvr canno xcu h job ha arriv a im 0.1 inc i wa upndd a im 0 bcau h apriodic job quu wa mpy. T p Ral-Tim Sym Mixd Job - 7 Ral-Tim Sym Mixd Job

5 Priodic Srvr wih RM Schduling (xampl coninud) Exampl Schdul: Two a, = (3,1), = (10,4), and a priodic rvr T p = (2.5,0.5). Aum an apriodic job J a arriv a = 0.1 wih and xcuion im of a = 0.8. Th priodic rvr xcu job J a unil i xhau i budg. Priodic Srvr wih RM Schduling (xampl concludd) Exampl Schdul: Two a, = (3,1), = (10,4), and a priodic rvr T p = (2.5,0.5). Aum an apriodic job J a arriv a = 0.1 wih and xcuion im of a = 0.8. Th rpon im of h apriodic job J a i 5.2. T p 0.1 T p Ral-Tim Sym Mixd Job - 9 Ral-Tim Sym Mixd Job

6 Improving h Priodic Srvr Th problm wih h priodic rvr i ha i xhau i xcuion budg whnvr h apriodic job quu i mpy.» If an apriodic job arriv ε im uni afr h ar of h priod, i mu wai unil h ar of h nx priod (p - ε im uni) bfor i can bgin xcuion. W would li o prrv h xcuion budg of h polling rvr and u i lar in h priod o horn h rpon im of apriodic job:» Bandwidh-Prrving Srvr do ju hi! Bandwidh-Prrving Srvr Fir om mor rm:» Th priodic rvr i bacloggd whnvr h apriodic job quu i nonmpy or h rvr i xcuing a job.» Th rvr i idl whnvr i i no bacloggd.» Th rvr i ligibl for xcuion whn i i bacloggd and ha an xcuion budg (grar han zro).» Whn h rvr xcu, i conum i xcuion budg a h ra of on im uni pr uni of xcuion.» Dpnding on h yp of priodic rvr, i may alo conum all or a porion of i xcuion budg whn i i idl: h impl priodic rvr conumd all of i xcuion budg whn h rvr wa idl. Ral-Tim Sym Mixd Job - 11 Ral-Tim Sym Mixd Job

7 Bandwidh-Prrving Srvr Bandwidh-prrving rvr diffr in hir rplnihmn im and how hy prrv hir xcuion budg whn idl. W aum h chdulr rac h conumpion of h rvr xcuion budg and upnd h rvr whn h budg i xhaud or h rvr bcom idl. Th chdulr rplnih h rvr xcuion budg a h appropria rplnihmn im, a pcifid by h yp of bandwidh-prrving priodic rvr. Th rvr i only ligibl for xcuion whn i i bacloggd and i xcuion budg i non-zro. Four Bandwidh-Prrving Srvr Dfrrabl Srvr (1987)» Old and impl of h bandwidh-prrving rvr.» Saic-prioriy algorihm by Lhoczy, Sha, and Sronidr.» Dadlin-drivn algorihm by Ghazali and Bar (1995). Sporadic Srvr (1989)» Saic-prioriy algorihm by Sprun, Sha, and Lhoczy.» Dadlin-drivn algorihm by Ghazali and Bar (1995). Toal Bandwidh Srvr (1994, 1995)» Dadlin-drivn algorihm by Spuri and Buazzo. Conan Uilizaion Srvr (1997)» Dadlin-drivn algorihm by Dng, Liu, and Sun. Ral-Tim Sym Mixd Job - 13 Ral-Tim Sym Mixd Job

8 Dfrrabl Srvr (DS) L h a T DS = (p, ) b a dfrrabl rvr. Conumpion Rul:» Th xcuion budg i conumd a h ra of on im uni pr uni of xcuion. Rplnihmn Rul:» Th xcuion budg i o a im inan p, for 0.» No: Unud xcuion budg canno b carrid ovr o h nx priod. Th chdulr ra h dfrrabl rvr a a priodic a ha may upnd ilf during xcuion (i.., whn h apriodic quu i mpy). Ral-Tim Sym Mixd Job - 15 DS wih RM Schduling Exampl Schdul: Sam wo a, = (3,1), = (10,4), and dfrrabl rvr T DS = (2.5,0.5). Aum an apriodic job J a arriv a im = 0.1 wih and xcuion im of a = 0.8 (again). 0.5 T DS Budg T DS Th DS can xcu h job ha arriv a im 0.1 inc i prrvd i budg whn h apriodic job quu wa mpy Ral-Tim Sym Mixd Job

9 DS wih RM Schduling (xampl concludd) Exampl Schdul: Sam wo a, = (3,1), = (10,4), and dfrrabl rvr T DS = (2.5,0.5). Aum an apriodic job J a arriv a im = 0.1 wih and xcuion im of a = 0.8 (again). Th rpon im of h apriodic job J a i 2.7 I wa 5.2 wih h impl priodic rvr. 0.5 T DS Budg DS wih RM Schduling Anohr Exampl: Two a, = (2,3.5,1.5), = (6.5,0.5), and a dfrrabl rvr T DS = (3,1). Aum an apriodic job J a arriv a im = 2.8 wih and xcuion im of a = 1.7. Th rpon im of h apriodic job J a i 3.7. T DS Budg T DS T DS Ral-Tim Sym Mixd Job - 17 Ral-Tim Sym Mixd Job

10 DS wih EDF Schduling Sam Ta S: Two a, = (2,3.5,1.5), = (6.5,0.5), and a dfrrabl rvr T DS = (3,1). Aum an apriodic job J a arriv a im = 2.8 wih and xcuion im of a = 1.7. Th rpon im of h apriodic job J a i ill 3.7. T DS Budg DS wih EDF and Bacground Schduling Sam Ta S: Two a, = (2,3.5,1.5), = (6.5,0.5), and T DS = (3,1) wih bacground chduling. Aum an apriodic job J a arriv a im = 2.8 wih and xcuion im of a = 1.7. T DS Budg Th DS xhau i budg a im T DS T DS Ral-Tim Sym Mixd Job - 19 Ral-Tim Sym Mixd Job

11 DS wih EDF and Bacground Schduling (xampl concludd) Sam Ta S: Two a, = (2,3.5,1.5), = (6.5,0.5), and T DS = (3,1) wih bacground chduling. Aum an apriodic job J a arriv a im = 2.8 wih and xcuion im of a = 1.7. Howvr, uing bacground chduling, h rpon im of h apriodic job J a i rducd o 2.4. T DS Budg T DS DS wih Bacground Schduling W can alo combin bacground chduling of h dfrrabl rvr wih RM.» For h dfrrabl rvr xampl a, h rpon im don chang. Why? Why complica hing by adding bacground chduling of h dfrrabl rvr? Why no ju giv h dfrrabl chdulr a largr xcuion budg? S h nx lid! Ral-Tim Sym Mixd Job - 21 Ral-Tim Sym Mixd Job

12 DS wih RM Schduling Rviid Modifid Exampl: Sam wo a, = (2,3.5,1.5), = (6.5,0.5), and dfrrabl rvr T DS = (3,1). Aum an apriodic job J a arriv a im 0 = 65 wih and xcuion im of a = 3. A largr xcuion budg for T DS would rul in miing a dadlin. Tim 0 = 65 i a criical inan for hi a. T DS Budg T DS 0 Schdulabiliy and DS Thr ar no nown ncary and ufficin chdulabiliy condiion for a ha conain a DS wih arbirary prioriy. W will why horly. Howvr, w can xnd TDA and Gnralizd TDA o yild ncary and ufficin chdulabiliy whn h DS i h high prioriy a in a priodic (ralworld poradic) a. ju ma i! Ral-Tim Sym Mixd Job - 23 W ar wih a criical inan lmma for ym wih a DS. Ral-Tim Sym Mixd Job

13 Criical Inan in Fixd-Prioriy Sym wih a Dfrrabl Srvr Lmma 7-1: [Lhoczy, Sha, and Sronidr] In a fixd-prioriy ym in which h rlaiv dadlin of vry indpndn, prmpabl priodic a i no grar han i priod and hr i a dfrrabl rvr (p, ) wih h high prioriy among all a, a criical inan of vry priodic a T i occur a im 0 whn all of h following ar ru. 1. On of i job J i,c i rlad a A job in vry highr-prioriy a i rla a h am im. 3. Th budg of h rvr i a 0, on or mor apriodic job ar rlad a 0, and hy p h rvr bacloggd hrafr. 4. Th nx rplnihmn im of h rvr i 0. Dicuion of Lmma 7.1 Th Proof of Lmma 7.1 i a raighforward xnion of h proof w gav for Thorm 6.5. Convinc yourlf of hi! No: W ar no aying ha T DS,,, T i will all ncarily rla job a h am im, bu if hi do happn, w ar claiming ha h im of rla will b a criical inan for T i. W can u h criical inan 0, dfind by Lmma 7.1, o driv ncary and ufficin condiion for h chdulabiliy of a a whn h DS ha high prioriy. Fir, l a a loo a a procor dmand anomaly crad by h bandwidh prrving DS. Ral-Tim Sym Mixd Job - 25 Ral-Tim Sym Mixd Job

14 Dicuion of Lmma 7.1 (con.) All four condiion of Lmma 7.1 hold in h la xampl: T DS Budg T DS 0 ju ma i! 0 = Noic ha h procor dmand crad by h DS in an inrval from [65,68.5] i wic wha i would b if i wr an ordinary priodic a! Thi i bcau w prrv h bandwidh of h DS. TDA wih a DS Obrvaion: Ca (1) and (2) of Lmma 7.1 dfin a criical inan for any fixd-prioriy a. Whn ca (3) and (4) of Lmma 7.1 ar ru, h procor dmand crad by h DS in an inrval of lngh can b a mo ( ) p Thu, TDA and Gnralizd TDA wih blocing rm can b xndd o ym wih a DS ha xcu a h high prioriy. Th TDA funcion bcom: i 1 wi() = i bi p 1 p = for 0< min(d,p ) i i Ral-Tim Sym Mixd Job - 27 Ral-Tim Sym Mixd Job

15 DS wih High Fixd Prioriy Whn h DS i h high prioriy proc in a fixd-prioriy ym:» I may b abl o xcu an xra im uni mor han a normal priodic a in h faibl inrval of a T i, a xprd in Equaion ( ) and h modifid w i (). Thu, h TDA mhod, uing h modifid w i () provid a ncary and ufficin condiion for fixd-prioriy ym wih on DS xcuing a h high prioriy. DS wih Arbirary Fixd Prioriy Whn h DS i no h high prioriy proc:» I may no b abl o xcu h xra im uni xprd in Equaion ( ) and h modifid w i ().» Howvr, h im-dmand funcion of a a T i wih lowr prioriy han an arbirary-prioriy DS i boundd from abov by h modifid w i (). Thu, h TDA mhod provid a ufficin (bu no ncary) condiion for fixd-prioriy ym wih on arbirary-prioriy DS. Ral-Tim Sym Mixd Job - 29 Ral-Tim Sym Mixd Job

16 Mulipl Arbirary Fixd-Prioriy DS W may wan o diffrnia apriodic job by xcuing hm a diffrn priorii. To do hi, w u mulipl DS wih diffrn priorii and a paramr (p,,, ). Th TDA and Gnralizd TDA wih blocing rm can b furhr xndd o h ym. Spcifically, h im dmand funcion w i () of a priodic a T i wih a lowr prioriy han m DS bcom: m i 1 w () b, i = i i 1, 1 p, 1 p = = for 0< min(d,p ) i i Schdulabl Uilizaion wih a Fixd-Prioriy DS W now loo a uilizaion bad chduling for fixd-prioriy ym wih on DS. Thr ar no nown ncary and ufficin chdulabl uilizaion condiion for fixdprioriy ym wih a DS. Howvr, hr do xi a ufficin condiion for RM whn h DS ha h hor priod plu om ohr condiion... Ral-Tim Sym Mixd Job - 31 Ral-Tim Sym Mixd Job

17 RM Schdulabl Uilizaion wih a DS Thorm 7.2 Conidr a ym of n indpndn, prmpabl priodic a who priod aify h inqualii p < p 1 < p 2 < < p n < 2p and p n > p and who rlaiv dadlin ar qual o hir rpciv priod. Thi ym i chdulabl ra monoonically wih a dfrrabl rvr (p, ) if hir oal uilizaion i l han or qual o U RM/DS u 2 (n) = (n1) u 1 1/(n1) whr u i h uilizaion /p of h rvr. 1 Proof: Similar o Thm 6.11 and lf a an xrci! No ha hi i only a ufficin chdulabiliy. Ral-Tim Sym Mixd Job - 33 RM Schdulabl Uilizaion wih a DS and Arbirary Priod Obrv: If p i < p, hn a T i i unaffcd by h DS. If p i > p, hn i may b blocd an xra im uni in a faibl inrval. Thorm: Conidr a ym of n indpndn, prmpabl priodic a who rlaiv dadlin ar qual o hir rpciv priod. Ta T i wih p i > p i chdulabl ra monoonically wih a dfrrabl rvr (p, ) if bi Ui u URM (i 1) p i whr u i h uilizaion /p of h rvr, U i i h oal uilizaion of h a...t i, and b i i h blocing im ncounrd by a T i from lowr prioriy a. Ral-Tim Sym Mixd Job

18 Schdulabiliy of Dadlin-Drivn Sym wih a DS In fixd-prioriy ym, h DS bhav li a priodic a (p, ) xcp ha i could xcu an xra amoun of im (a mo im uni) in h faibl inrval of any lowr prioriy job. In a dadlin-drivn ym, h DS can xcu a mo im uni in h faibl inrval of any job (undr crain condiion). W prn a ufficin (bu no ncary) chdulabiliy condiion for h EDF algorihm. Fir, a bound on h procor dmand crad by a DS. Bounding h Dmand of a DS in an EDF Schduld Sym An inrval (a, b] i po-idl if ihr a = 0, or if no job wih a dadlin in h inrval (a1, b] xcu in h inrval (a 1, a].» Th implicaion of hi dfiniion i ha all job wih dadlin in (a1, b] ar idl during h inrval (a 1, a] in h n ha all job rlad bfor im a wih dadlin in (a1, b] hav compld xcuion bfor im a (i.., ihr h procor i idl in (a 1, a] or a job wih dadlin a im a or afr im b xcu in (a 1, a]). Th following lmma giv u a impl uppr bound for h procor dmand in a po-idl inrval of lngh L. Ral-Tim Sym Mixd Job - 35 Ral-Tim Sym Mixd Job

19 Maximum Dmand of a DS Lmma: Th maximum dmand w DS (L) of a DS= (p, ) during a po-idl inrval of lngh L in an EDF chduld ym of n indpndn, prmpabl priodic a i boundd uch ha w DS (L) u (L p ) whr u i h uilizaion /p of h rvr. Proof: L ( -1, ] b a po-idl inrval. Th maximum dmand for DS occur whn 1. A im -1, i budg i qual o and h rvr dadlin (and budg rplnihd im) i On or mor apriodic job arriv a -1 and h DS i bacloggd unil a la im. 3. Th rvr dadlin -1 i arlir han h dadlin of all h priodic job ha ar rady for xcuion in h inrval ( -1, -1 ]. Proof Coninud Maximum dmand crad by DS in a po-idl inrval undr EDF: -1 p Th DS do no xcu hr -1 inc i dadlin i afr. Obrv ha undr h condiion, h maximum dmand crad by DS in h po-idl inrval ( -1,] i a mo ( 1 ) = p p 1 Ral-Tim Sym Mixd Job - 37 Ral-Tim Sym Mixd Job

20 39 Ral-Tim Sym Mixd Job - 39 Proof Coninud Thu, DS p p ) ( w 1 1 u ) ( u p p p p = = )) ( (p u 1 = Sinc ( -1, ] i a po-idl inrval of lngh L = -1, w DS (L) u (L p ). ) p ( u 1 = 40 Ral-Tim Sym Mixd Job - 40 Schdulabiliy wih a DS Combining hi rul wih Thorm 6.2, w g h following horm from Ghazali and Bar: Thorm 7.3: A priodic a T i in a ym of n indpndn, prmpabl priodic a i chdulabl wih a DS= (p, ) according o h EDF algorihm if whr u i h uilizaion /p of h rvr. 1 D p 1 u ) p, min(d i n 1 = (7.5)

21 41 Ral-Tim Sym Mixd Job - 41 Proof of Thorm 7.3 Suppo Equaion (7.5) hold for a T i bu a dadlin i mid. L d b h arli poin in im a which a dadlin i mid and -1 b h ar of la po-idl inrval ha includ im d. Thu, a dadlin i mid in h po-idl inrval ( -1, d ]. From Thorm 6.2 and h prviou lmma, h dmand in hi inrval i a mo Bcau a dadlin i mid a d, dmand ovr ( -1, d ] xcd d -1. Thu, w hav ( ) i d n 1 i d i d p u ) p, min(d < = ( ) i d n 1 i d p u ),p min(d - = ( ) i d n 1 i d p u ) p, min(d = 42 Ral-Tim Sym Mixd Job - 42 Dividing boh id by ( d -1 ), w g Sinc D i ( d -1 ). Thi conradic our aumpion ha Equaion (7.5) hold. Proof Coninud ( ) i d i d n 1 p u ),p min(d 1 < = = = i d n 1 p 1 u ),p min(d = i n 1 D p 1 u ),p min(d

22 Mulipl DS W may wan o diffrnia apriodic job by xcuing hm a diffrn priorii. To do hi in a dadlin-drivn ym, w (again) u mulipl DS wih diffrn priorii and a paramr (p,,, ). Corollary: A priodic a T i in a ym of n indpndn, prmpabl priodic a i chdulabl wih m a DS= (p,,, ) according o h EDF algorihm if n = 1 min(d, p ) m u, = 1 p 1 whr u, i h uilizaion, /p, of rvr. Th proof i lf a an xrci., D i, 1 DS Punch Lin In boh fixd-prioriy and dadlin-drivn ym, w ha h DS bhav li a priodic a wih paramr (p, ) xcp i may xcu an addiional amoun of im in h faibl inrval of any lowr prioriy job. Thi i bcau, h bandwidh-prrving condiion rul in a chduling algorihm ha i non-wor-conrving wih rpc o a normal priodic a. Ral-Tim Sym Mixd Job - 43 Ral-Tim Sym Mixd Job

23 Sporadic Srvr Sporadic Srvr (SS) wr dignd o ovrcom h addiional blocing im a DS may impo on lowrprioriy job. All poradic rvr ar bandwidh prrving, bu h conumpion and rplnihmn rul nur ha a SS, pcifid a T S = (p, ) nvr cra mor dmand han a priodic ( ral-world poradic) a wih h am a paramr. Thu, chdulabiliy of a ym wih a SS i drmind xacly a a ym wihou a SS. Sporadic Srvr (SS) W will loo a wo SS for fixd-prioriy ym and on for dadlin-drivn ym. Thy diffr in complxiy (and hu ovrhad) du o diffrn conumpion and rplnihmn rul. W aum, a wih a DS, ha h chdulr monior h xcuion budg of h SS. Howvr, in all ca chdulabiliy condiion rmain unchangd from an quivaln ym wihou an SS. Ral-Tim Sym Mixd Job - 45 Ral-Tim Sym Mixd Job

24 Simpl SS in a Fixd-Prioriy Sym Fir om (nw) rm:» L T b a of n indpndn, prmpabl priodic a.» Th (arbirary) prioriy of h rvr T S in T i π.» T H i h ub of a ha hav highr prioriy han T S.» T (T H ) i idl whn no job in T (T H ) i ligibl for xcuion. T (T H ) i buy whn i i no idl.» L BEGIN b h inan in im whn T H raniion from idl o buy, and END b h inan in im whn i bcom idl again (or infiniy if T H i ill buy). Th inrval (BEGIN, END] i a buy inrval.» r i h la rplnihmn im of T S.» r i h nx chduld rplnihmn im of T S.» i h ffciv rplnihmn im of T S.» f i h fir inan afr r a which T S bgin o xcu. Simpl SS in a Fixd-Prioriy Sym Conumpion Rul: a any im afr r, T S conum i budg a h ra of on im uni pr uni of xcuion unil h budg i xhaud whn ihr C1 T S i xcuing, or C2 T S ha xcud inc r and END <. (END < T H i currnly idl.) Rplnihmn Rul: r i o h currn im whnvr h xcuion budg i rplnihd wih im uni by h chdulr. R1 Iniially, r = = 0 and r = p (auming h ym ar a im 0). R2 A im f, if END = f, = max( r, BEGIN). If END < f, = f. Th nx chduld rplnihmn im i r = p. R3 Th nx rplnihmn occur a r xcp (a) If r < f, hn h budg i rplnihd a oon a i i xhaud. (b) If T i idl bfor r and hn bgin a nw buy inrval a b, hn h budg i rplnihd a min( r, b ). Ral-Tim Sym Mixd Job - 47 Ral-Tim Sym Mixd Job

25 Simpl SS wih RM Schduling Exampl Schdul: = (3,0.5), = (4,1), T 3 = (19,4.5), and T S = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. T S Budg Simpl SS wih RM Schduling (xampl coninud) Exampl Schdul: = (3,0.5), = (4,1), T 3 = (19,4.5), and T S = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. T S Budg J a1 J a2 J a1 J a2 T S T S r r = p T 3 r = r = p T 3 Ral-Tim Sym Mixd Job - 49 Ral-Tim Sym Mixd Job

26 Simpl SS wih RM Schduling (xampl coninud) Exampl Schdul: = (3,0.5), = (4,1), T 3 = (19,4.5), and T S = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. T S Budg Simpl SS wih RM Schduling (xampl coninud) Exampl Schdul: = (3,0.5), = (4,1), T 3 = (19,4.5), and T S = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. T S Budg R3b J a1 J a2 J a1 J a2 J a3 T S T S r = r = p T 3 r = r = p T 3 Ral-Tim Sym Mixd Job - 51 Ral-Tim Sym Mixd Job

27 Simpl SS wih RM Schduling (xampl concludd) Exampl Schdul: = (3,0.5), = (4,1), T 3 = (19,4.5), and T S = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. T S Budg T S J a1 J a2 R3b r = T 3 J a3 R3b r = p Corrcn of Simpl SS Th Simpl SS bhav xacly a a ral-world poradic a xcp whn Rul R3b i applid. Rul R3b a advanag of h chdulabiliy for a fixd-prioriy priodic ( ral-world poradic) a T.» W now ha if h ym T raniion from an idl a o a buy inrval, all job will ma hir dadlin--vn if hy ar all rlad a h am inan (a h ar of h nw buy inrval).» Thu Rul R3b rplnih h Simpl SS a hi inan inc i will no affc chdulabiliy! Ral-Tim Sym Mixd Job - 53 Ral-Tim Sym Mixd Job

28 Enhancmn o h Simpl SS W can improv rpon im of apriodic job by combining h Bacground Srvr wih h Simpl SS o cra a Sporadic/Bacground Srvr (SBS). Conumpion Rul ar h am a for h Simpl SS xcp whn h a ym T i idl.» A long a T i idl, h xcuion budg ay a. Rplnihmn Rul ar h am a for h Simpl SS xcp Rul R3b.» Th SBS budg i rplnihd a h bginning of ach idl inrval of T. r i a h nd of h idl inrval. Ohr Enhancmn o h Simpl SS W can alo improv rpon im of apriodic job by rplnihing h rvr xcuion budg in mall chun during i priod rahr han wih a ingl rplnihmn of im uni a h nd.» Thi add o h complxiy of h conumpion and rplnihmn rul, of cour. Sprun, Sha, and Lhoczy propod uch a rvr:» Th SpSL poradic rvr prrv unconumd chun of budg whnvr poibl and rplnih h conumd chun a oon a poibl.» Thu, i mula vral priodic a wih paramr (p,, ) uch ha Σ, =. Ral-Tim Sym Mixd Job - 55 Ral-Tim Sym Mixd Job

29 SpSL in a Fixd-Prioriy Sym Braing of Excuion Budg ino Chun: B1 Iniially, h budg = and r = 0. Thr i only on chun of budg. B2 Whnvr h rvr i upndd, h currn budg, if no xhaud, i bron up ino wo chun. Th fir chun i h porion ha wa conumd during h la rvr buy inrval, 1. I nx rplnihmn im, r1, i h am a h original chun : r. Th rplnihmn amoun will b 1. Th cond chun i h rmaining budg, 2. I la rplnihmn im i naivly o r2 =, which will b r if hi budg i ud bfor r1. Ohrwi, h wo chun will b combind ino on budg a im r1. Conumpion Rul: C1 Th rvr conum budg (whn hr i mor han on budg chun) in h ordr of hir la rplnihmn im. Tha i, h budg wih mall r i conumd fir. C2 Th rvr conum i budg only whn i xcu. Rplnihmn Rul: Th nx rplnihmn im of ach chun of budg i according o rul R2 and R3 of h impl SS. Th budg chun ar combind whnvr hy ar rplnihd a h am im (.g. R3b). Ral-Tim Sym Mixd Job - 57 SpSL Rul R2 and R3 Rplnihmn rul R2 and R3 from h Simpl SS: R2 A im f, if END = f, = max( r, BEGIN). If END < f, = f. Th nx chduld rplnihmn im i r = p. R3 Th nx rplnihmn occur a r xcp (a) If r < f, hn h budg i rplnihd a oon a i i xhaud. (b) If T i idl bfor r and hn bgin a nw buy inrval a b, hn h budg i rplnihd a min( r, b ). Noic ha whn R3b appli, all budg chun will b combind ino a ingl budg again. Ral-Tim Sym Mixd Job

30 SpSL wih RM Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T SpSL = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. Plu job J a4 arriv a 4 = 6.5 wih a4 = 0.5 T SpSL Budg SpSL wih RM Schduling (xampl coninud) Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T SpSL = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. Plu job J a4 arriv a 4 = 6.5 wih a4 = 0.5 T SpSL Budg J a1 J a1 J a4 T SpSL r T SpSL 2 r1 r2 r T 3 r 1 T 3 Ral-Tim Sym Mixd Job - 59 Ral-Tim Sym Mixd Job

31 Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T SpSL = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. Plu job J a4 arriv a 4 = 6.5 wih a4 = 0.5 T SpSL Budg SpSL wih RM Schduling (xampl coninud) Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T SpSL = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. Plu job J a4 arriv a 4 = 6.5 wih a4 = 0.5 T SpSL Budg SpSL wih RM Schduling (xampl coninud) J a1 J a4 J a2 J a1 J a4 J a2 T SpSL 2 r1 = 1 r2 r1 T SpSL r1 = 1 r2 = 2 r1 r2 T 3 T 3 Ral-Tim Sym Mixd Job - 61 Ral-Tim Sym Mixd Job

32 Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T SpSL = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. Plu job J a4 arriv a 4 = 6.5 wih a4 = 0.5 T SpSL Budg SpSL wih RM Schduling (xampl coninud) SpSL wih RM Schduling (xampl coninud) Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T SpSL = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. Plu job J a4 arriv a 4 = 6.5 wih a4 = 0.5 R3b T SpSL Budg J a1 J a4 J a2 J a1 J a4 J a2 T SpSL No: Liu r2 = 2 r1 = 1 r2 r1 Exampl i miing hit 3 xcuion! T SpSL No: Liu r = r Exampl i miing hit 3 xcuion! Ral-Tim Sym Mixd Job - 63 Ral-Tim Sym Mixd Job

33 SpSL wih RM Schduling (xampl coninud) Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T SpSL = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. Plu job J a4 arriv a 4 = 6.5 wih a4 = 0.5 R3b T SpSL Budg SpSL wih RM Schduling (xampl concludd) Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T SpSL = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. Plu job J a4 arriv a 4 = 6.5 wih a4 = 0.5 R3b R3b R3b T SpSL Budg J a1 J a4 J a2 J a3 J a1 J a4 J a2 J a3 T SpSL r = r T SpSL r = r T 3 T 3 Ral-Tim Sym Mixd Job - 65 Ral-Tim Sym Mixd Job

34 Enhancing h SpSL Srvr W can furhr improv rpon im of apriodic job by combining h SpSL wih h Bacground Srvr o cra a SpSL/Bacground Srvr.» Try o wri prcily h conumpion and rplnihmn rul for hi rvr! W can alo nhanc h SpSL by uing a chniqu calld Prioriy Exchang.» Whn h rvr ha no wor, i rad im wih an xcuing lowr prioriy a.» S Liu for dail. Simpl SS in Dadlin-Drivn Sym L T S =(p, ) b a impl poradic rvr (SS) in a a ym T chduld wih EDF. Th rvr i rady for xcuion only whn i i bacloggd and i dadlin d i.» Thu, h rvr i upndd whnvr i dadlin i undfind or whn h rvr i idl. Conumpion Rul: T S conum i budg a h ra of on im uni pr uni of xcuion unil h budg i xhaud whn ihr C1 T S i xcuing, or C2 d i dfind, h rvr i idl, and hr i no job wih a dadlin bfor d rady for xcuion. Ral-Tim Sym Mixd Job - 67 Ral-Tim Sym Mixd Job

35 Simpl SS in Dadlin-Drivn Sym Rplnihmn Rul: r i o h currn im whnvr h xcuion budg i rplnihd wih im uni by h chdulr. R1 Iniially, r = 0, and d ar undfind. R2 Whnvr i dfind, d = p, and r = p. Whn i undfind, i drmind (dfind) a follow: (a) A im whn an apriodic job arriv an h rvr i idl, h valu of i drmind bad on h hiory of h ym bfor a follow: i. If only job wih dadlin arlir han r p hav xcud hroughou h inrval ( r, ), = r. ii. If a job wih a dadlin afr r p ha xcud in h inrval ( r,), =. (b) A rplnihmn im r, i. If h rvr i bacloggd, = r. ii. If h rvr i idl, and d bcom undfind. R3 Th nx rplnihmn occur a r xcp (a) If r < (from R2a), h budg i rplnihd a oon a i i xhaud. (b) Th budg i rplnihd a h nd of ach idl inrval of T. Ral-Tim Sym Mixd Job - 69 Simpl SS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T S = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. T S Budg J a1 arriv a = 3 T S = r r d = p T 3 Ral-Tim Sym Mixd Job

36 T 3 T 3 Simpl SS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T S = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. T S Budg T S = r r d = p Simpl SS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T S = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. T S Budg J a2 arriv a = 7 r d = p T S r Ral-Tim Sym Mixd Job - 71 Ral-Tim Sym Mixd Job

37 T 3 T 3 Simpl SS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T S = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. T S Budg r d = p T S r Simpl SS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T S = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. R3b T S Budg J a3 arriv a = 15.5 r d = p T S r Ral-Tim Sym Mixd Job - 73 Ral-Tim Sym Mixd Job

38 Simpl SS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and T S = (5,1.5). Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 7 wih a2 = 2, and J a3 arriv a 3 = 15.5 wih a3 = 2. R3b R3b R3b T S Budg r d T S r T 3 Enhancing h Simpl SS Undr EDF W can improv rpon im of apriodic job by combining h Bacground Srvr wih h Simpl SS o cra a Sporadic/Bacground Srvr (SBS) ha xcu undr EDF.» Try o wri prcily h conumpion and rplnihmn rul for hi rvr! W can alo nhanc h Simpl SS by rplnihing h budg in chun a h SpSL rvr did.» Such a rvr wa propod by Ghazali and Bar.» I rul ar dfind in Prob 7.13 of Liu x. Ral-Tim Sym Mixd Job - 75 Ral-Tim Sym Mixd Job

39 Apriodic Job Srvr for Dadlin-Drivn Sym Th Toal Bandwidh Srvr (TBS) wa crad by Spuri and Buazzo (RTSS 94) o chdul» apriodic a who arrival im wa unnown bu» who wor-ca xcuion im (wc) wa nown.» A rivial admiion conrol algorihm ud wih h TBS can alo chdul poradic job (apriodic job who wc and dadlin i nown). Th conan uilizaion rvr wa crad by Dng, Liu, and Sun (Euromicro Worhop 97) o chdul» apriodic a who arrival im wa unnown bu» who wc and dadlin wa nown.» Howvr, hy alo wand o chdul apriodic job wih no dadlin and who wc wa unnown.» Thi rvr i almo h am a h TBS. Apriodic Job Srvr for Dadlin-Drivn Sym Liu prnaion of h conan uilizaion rvr and h TBS i poorly moivad and narly uninlligibl. You hould rad h papr, hy ma mor n han h marial in h boo. W will fir covr h TBS and hn h conan uilizaion rvr inc i may b air o undrand h valu of h conan uilizaion rvr whn hy ar prnd in hi ordr. Ral-Tim Sym Mixd Job - 77 Ral-Tim Sym Mixd Job

40 Toal Bandwidh Srvr (TBS) On way o rduc h rpon im of apriodic job who wc i nown in a dadlin-drivn ym i o» alloca a fixd (maximum) prcnag, U S, of h procor o h rv apriodic job, and» ma ur h apriodic load nvr xcd hi maximum uilizaion valu.» Whn an apriodic job com in, aign i a dadlin uch ha h dmand crad by all of h apriodic job in any faibl inrval nvr xcd h maximum uilizaion allocad o apriodic job. Thi approach i h main ida bhind h TBS. No: W u U S o dno h rvr iz (a Spuri and Buazzo do) rahr han a Liu do. u ~ TBS in Dadlin-Drivn Sym L TB b a TBS of iz U S in a a ym T chduld wih EDF. Thu, h rvr i allocad U S prcn of h oal procor bandwidh. Th rvr i rady for xcuion only whn i i bacloggd.» In ohr word: h rvr i only upndd whn i i idl. Conumpion Rul: Whn xcuing, TB conum i budg a h ra of on im uni pr uni of xcuion unil h budg i xhaud. Rplnihmn Rul: R1 Iniially, h xcuion budg = 0 and h rvr dadlin d = 0. R2 A im whn apriodic job J i wih xcuion im i arriv and h rvr i idl, d = max(d, ) i /U S and = i. If h rvr i bacloggd, no nohing. R3 Whn h rvr compl h currnly xcuing apriodic job, J i, (a) If h rvr i bacloggd, upda h rvr dadlin and budg: d = d i1 /U S and = i1. (b) If h rvr i idl, do nohing. Ral-Tim Sym Mixd Job - 79 Ral-Tim Sym Mixd Job

41 T 3 T 3 J a1 arriv a = 3 TB TBS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), bu U S = Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 6.9 wih a2 = 2, and J a3 arriv a 3 = 14 wih a3 = TB Budg = 1 d = max(0,3) 1/0.25 = 7 J a2 arriv a = 6.9 TB TBS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), bu U S = Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 6.9 wih a2 = 2, and J a3 arriv a 3 = 14 wih a3 = TB Budg = 2 d = max(7,6.9) 2/0.25 = 15 Ral-Tim Sym Mixd Job - 81 Ral-Tim Sym Mixd Job

42 T 3 T 3 J a3 arriv a = 14 TB TBS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), bu U S = Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 6.9 wih a2 = 2, and J a3 arriv a 3 = 14 wih a3 = TB Budg = 2 d = max(15,14)8=23 TB TBS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), bu U S = Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 6.9 wih a2 = 2, and J a3 arriv a 3 = 14 wih a3 = TB Budg Ral-Tim Sym Mixd Job - 83 Ral-Tim Sym Mixd Job

43 TBS Procor Dmand Obrv ha h oal bandwidh of h rvr i allocad o on apriodic job a a im. Morovr, if J i xcuion im i l han i pcifid wc, i, and job J i1 arriv bfor h dadlin for job J i, h TBS ffcivly prform bacground xcuion by rplnihing h rvr budg a oon a job J i i don. Howvr, h dadlin valu d = max(d,) i1 /U S for job J i1 i i no arlir han i1 /U S im uni from h dadlin of job J i. Thu, h dmand crad by any n job in an inrval ( 1, 2 ] i a mo n n i i = US = nus ( 2 1) ( US) i 1 i i 1 = = i U S Schdulabiliy wih a TBS Combining hi obrvaion wih Thorm 6.1, w g h following horm from Spuri and Buazzo: Thorm: A ym T of n indpndn, prmpabl, priodic a wih rlaiv dadlin qual o hir priod i chdulabl wih a TBS if and only if UT US 1 n whr U T = i h procor uilizaion of h = 1 p priodic a and U S i h procor uilizaion of h TBS. Ral-Tim Sym Mixd Job - 85 Ral-Tim Sym Mixd Job

44 Schdulabiliy wih a TBS Combining hi rul wih Thorm 6.2, w g h following horm: Thorm: A ym T of n indpndn, prmpabl, priodic a i chdulabl wih a TBS if n = 1 US 1 min(d, p ) whr U S i h procor uilizaion of h TBS. Sporadic job and h TBS Rcall ha a poradic job i li an apriodic job xcp i ha a hard dadlin.» Aum ach poradic job J i ha a rla im r i, wc i, and a rlaiv dadlin D i : J i = (r i, i, D i ) Th TBS aign dadlin uch ha h (abolu) dadlin d i aignd any job J i i d i = max(r i,d i-1 ) i /U S whr d 0 =1. Thu h TBS can guaran h dadlin of all accpd poradic job if i only accp job J i1 if r i1 D i1 d i1 and rjc i ohrwi. Ral-Tim Sym Mixd Job - 87 Ral-Tim Sym Mixd Job

45 Priodic, Apriodic, and Sporadic Job In om ym, w may wan o uppor all hr yp of job: priodic, apriodic, and poradic. Th TBS can do hi by accping all apriodic job and only accping a poradic job if i dadlin can b m. Bu hr i no valu in compling poradic job bfor hir dadlin, which h TBS will do. Thu Dng, Liu, and Sun crad h conan uilizaion rvr (CUS).» W can chdul poradic job wih h CUS and apriodic job wih h TBS in h am ym. CUS in Dadlin-Drivn Sym L CU b a CUS of iz U S in a a ym T chduld wih EDF. A wih h TBS, h rvr i rady for xcuion only whn i i bacloggd.» Thu, h rvr i only upndd whn h rvr i idl. Conumpion Rul: Whn xcuing, CU conum i budg a h ra of on im uni pr uni of xcuion unil h budg i xhaud. Rplnihmn Rul: R1 Iniially, h xcuion budg = 0 and h rvr dadlin d = 0. R2 A im whn apriodic job J i wih xcuion im i arriv and h rvr i idl, (a) If < d, do nohing; (b) If d, d = i /U S and = i. R3 A h dadlin d of h rvr, (a) If h rvr i bacloggd, upda h rvr dadlin and budg: d = d i1 /U S and = i1. (b) If h rvr i idl, do nohing. Ral-Tim Sym Mixd Job - 89 Ral-Tim Sym Mixd Job

46 T 3 T 3 J a1 arriv a = 3 CU CUS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and U S = Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 6.9 wih a2 = 2, and J a3 arriv a 3 = 14 wih a3 = CU Budg = 1 d = 3 1/0.25 = 7 J a2 arriv a = 6.9 CU CUS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and U S = Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 6.9 wih a2 = 2, and J a3 arriv a 3 = 14 wih a3 = CU Budg = 6.9 < d = 7 do nohing! Ral-Tim Sym Mixd Job - 91 Ral-Tim Sym Mixd Job

47 T 3 T 3 J a2 arriv a = 6.9 CU CUS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and U S = Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 6.9 wih a2 = 2, and J a3 arriv a 3 = 14 wih a3 = CU Budg = 2 d = 7 2/0.25 = 15 J a3 arriv a = 14 CU CUS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and U S = Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 6.9 wih a2 = 2, and J a3 arriv a 3 = 14 wih a3 = CU Budg = 14 < d = 15 do nohing! Ral-Tim Sym Mixd Job - 93 Ral-Tim Sym Mixd Job

48 T 3 T 3 J a3 arriv a = 14 CU CUS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and U S = Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 6.9 wih a2 = 2, and J a3 arriv a 3 = 14 wih a3 = CU Budg = 2 d = 158 = 23 CU CUS wih EDF Schduling Sam Ta S: = (3,0.5), = (4,1), T 3 = (19,4.5), and U S = Aum apriodic job J a1 arriv a 1 = 3 wih a1 = 1, J a2 arriv a 2 = 6.9 wih a2 = 2, and J a3 arriv a 3 = 14 wih a3 = CU Budg Ral-Tim Sym Mixd Job - 95 Ral-Tim Sym Mixd Job

49 Commn on h CUS Dadlin and rplnih amoun ar h am in boh rvr. Th main diffrnc bwn h CUS and h TBS i ha h CUS nvr rplnih h rvr budg arly. Thu, h TBS acually yild br avrag rpon im for apriodic job (and i wa crad bfor h CUS ) Morovr, i would appar ha h TBS may b abl o accp mor poradic job han h CUS. Th valu of h CUS i no clar, and Liu do a rribl job arguing for i! Ral-Tim Sym Mixd Job - 97 TBS and CUS Summary Boh rvr can b modifid o uppor h ca whn h wc of apriodic job i unown:» fix h xcuion budg a om valu and aum h rvr ha a priod of /U S. Boh rvr can b modifid o rclaim unud rourc whn h acual xcuion im i l han h wc ha w aumd:» rduc h currn dadlin of h rvr by ( -)/U S uni bfor rplnihing h budg. W can hav mulipl TBS/CUS rvr a long a h oal procor uilizaion/dniy i no grar han 1. Ral-Tim Sym Mixd Job