Overview. Worst-case response time analysis of real-time tasks under FPDS. Motivation for FPDS. Scheduling model for FPDS. Scheduling model for FPDS

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1 Reder J. Brl, TU/e Iformatca, System Archtecture ad Networg Reder J. Brl, TU/e Iformatca, System Archtecture ad Networg Overvew Worst-case respose tme aalyss of real-tme tass uder FS Reder J. Brl Techsche Uverstet Edhove (TU/e) Mathematcs ad Computer Scece System Archtecture ad Networg Motvato Schedulg model Refuted aalyss pessmstc: W ECRTS 004 optmstc: W ECRTS 006 Revsed aalyss TU/e, CS-report 07-, 007 ECRTS 007 Remars Cocluso Mälardale Uversty, Swede, Aprl 008 Mälardale Uversty, Swede, Aprl 008 Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Motvato for FS Schedulg model for FS Fxed-rorty re-emptve Schedulg (FS): rawbacs of arbtrary pre-empto: cost, e.g. cache flushes ad re-loads; o-trval resource access protocols; Fxed-rorty No-pre-emptve Schedulg (FNS): Resolves drawbacs of arbtrary pre-empto, but at the cost of reduced schedulablty. Fxed-rorty Schedulg wth eferred pre-empto (FS): Betwee the extremes of FS ad FNS, although FNS s a specal case of FS. Based o model for FS Sgle processor; Set Γ of depedet perodc tass τ, τ,, τ ; Uque prortes, tass orderg: decreasg prortes; Cotuous schedulg; Characterstcs of tas τ : (release) perod T ; computato tme C ; (relatve) deadle ( T ); phasg ϕ. Basc assumptos: smlar to [Lu ad Laylad 7], but Mälardale Uversty, Swede, Aprl 008 Mälardale Uversty, Swede, Aprl Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Schedulg model for FS Schedulg model for FS Refemet for FS: A tas τ s a AG of m() o-pre-emptable subobs subob has computato tme C, C, C, C,6 C,7 Cosequeces for aalyss of FS: blocg: all tass, except for the lowest prorty tas; C, C,5 B = max max C, > m( ) Specalzato: a tas τ s a sequece of m() subobs we ow get m( ) C = C, C,4 C,8 C,9 = Whe m() = for all, we get FNS. Mälardale Uversty, Swede, Aprl potetal early completo of fal sub-ob C,m() : all tass, except for the hghest prorty tas Remar: C,m() wll be deoted as F Mälardale Uversty, Swede, Aprl 008 6

2 Reder J. Brl, TU/e Iformatca, System Archtecture ad Networg Reder J. Brl, TU/e Iformatca, System Archtecture ad Networg Refuted aalyss for FS Refuted aalyss for FS Based o aalyss for FS (recap) exted model wth resource sharg blocg B of τ ; crtcal stat of τ : smultaeous release wth all hgher prorty tass; max. blocg B by lower prorty tass. worst-case respose tme of tas s the smallest x R + satsfyg x = B + C + x C T < Mälardale Uversty, Swede, Aprl Ital pessmstc aalyss for FS corporated cosequece of blocg oly, smlar to FS wth resource sharg,.e. crtcal stat of τ : smultaeous release wth all hgher prorty tass; max. blocg B by lower prorty tass. worst-case respose tme of tas s the smallest x R + satsfyg x x B C C < T = + + hece, = (B + C ) Mälardale Uversty, Swede, Aprl Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Refuted aalyss for FS Refuted aalyss for FS Ital pessmstc aalyss for FS example tas T = C B (B + C ) = 4.6 whereas R,0 =.6 τ.4 τ tas τ tas τ Mälardale Uversty, Swede, Aprl tme 9 Ehaced aalyss for FS (99) also corporated early completo of fal sub-ob, fal sub-ob has started whe τ executed C (F ) where s a arbtrary small postve umber; ( ) = (B + C (F )) + (F ) we ow fd:.6 ( ) tas τ = (.4 + ) + (. ) = (. ).6 =.6 tas τ for 0.6 Mälardale Uversty, Swede, Aprl 008 tme 0 Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Refuted aalyss for FS Refuted aalyss for FS Ehaced aalyss for FS s pessmstc example 4 Ehaced aalyss for FS s optmstc a ext ob may mss ts deadle tas T C B τ 5 4 τ tas τ tas τ 7 tas τ tas τ Leged: deadle mss τ tas τ tme 5 7 aalyss yelds ( ) = 9 pessmstc, because blocg s actually (B ) + 0 tme cause: the fal sub-ob defers hgher prorty tass, causg hgher terferece for ext obs of τ Mälardale Uversty, Swede, Aprl 008 Mälardale Uversty, Swede, Aprl 008

3 Reder J. Brl, TU/e Iformatca, System Archtecture ad Networg Reder J. Brl, TU/e Iformatca, System Archtecture ad Networg Refuted aalyss for FS Revsed aalyss for FS Aalyss for deadles at most equal to perods: thy shall loo beyod the frst ob for FS ad tass wth varyg prortes: M. Gozález Harbour et al., RTSS 99; for preempto threshold schedulg: J. Regehr, RTSS 00; for FS: R.J. Brl, W ECRTS 006; for FNS ad CAN: R.J. Brl et al., RTN 006; for EF: M. Spur, Tech. Rep. INRIA 996. Ma amedmets resolvg the pessmsm: approach: rephrase crtcal stat; use (B ) + rather tha B, ad let ote: problem does ot occur for lowest prorty tas τ, because τ s ot bloced by other tass; resolvg the optmsm: approach: cosder all obs a actve perod; ote: problem does ot occur for hghest prorty tas τ, because τ does ot bloc other tass; Mälardale Uversty, Swede, Aprl 008 Mälardale Uversty, Swede, Aprl Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Revsed aalyss for FS Revsed aalyss for FS Resolvg the pessmsm rephrase crtcal stat of τ : smultaeous release wth all hgher prorty tass; sub-ob wth max. blocg B starts earler; use (B ) + rather tha B, ad let 0 lm = ( ) - where ( B + C F ) + F for < ( ) = ( C ( F )) + F for = Resolvg the pessmsm ef.: worst-case occuped tme WO (C) WO ( C) = lm ( C + ) worst-case occuped tme WO of tas s the smallest x R + satsfyg x x = C + + C T < ca be solved by meas of a teratve procedure Mälardale Uversty, Swede, Aprl Mälardale Uversty, Swede, Aprl Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Revsed aalyss for FS Revsed aalyss for FS Resolvg the pessmsm ef.: worst-case occuped tme WO (C) WO ( C) = lm ( C + ) we ow get: lm = lm ( ( B + C F ) + F ) 0 ( ( C ( F )) + F ) ( B + C F ) + F = WO ( C F ) + F for < for = for < for = Resolvg the optmsm oto of level- actve perod: (cumulatve) pedg load (t): amout of processg at tme t that stll eeds to be performed for the obs of tass τ wth that are released before tme t. level- actve perod s a terval [t s, t e ), where: (t s ) = 0; (t e ) = 0; (t) > 0 for t (t s, t e ). level- actve perod eds whe U <, or U = ad lcm of perods exst Mälardale Uversty, Swede, Aprl Mälardale Uversty, Swede, Aprl 008 8

4 Reder J. Brl, TU/e Iformatca, System Archtecture ad Networg Revsed aalyss for FS Resolvg the optmsm example (FS) tas T C τ τ 4 tas τ tas τ 0 level- actve perod tme Leged: preempto executo release Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Revsed aalyss for FS Resolvg the optmsm crtcal stat of τ the worst-case respose tme s assumed a level- actve perod, that starts wth a smultaeous release wth all hgher prorty tass, where sub-ob wth max. blocg B starts earler; worst-case legth WL of a level- actve perod s the smallest x R + satsfyg x = B + x C T level- busy perod Mälardale Uversty, Swede, Aprl Mälardale Uversty, Swede, Aprl Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Revsed aalyss for FS Revsed aalyss for FS Resolvg the optmsm worst-case respose tme = max, 0 < wl where WL wl = T, ( B + ( + ) C F ) + F T = WO (( + ) C F ) + F T for < for = Resolvg the optmsm teratve procedure (0) =,0 ( l+ ) ( l ) = max(,, l+ ) l=0,, procedure stops whe exceeds,, or the level- actve perod s over,.e. ( B + ( + ) C ) ( + ) T Note: smlar codto as for FS ad > T Kle et al., RMA Hadboo, KA, 99. Mälardale Uversty, Swede, Aprl 008 Mälardale Uversty, Swede, Aprl 008 Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Remars Coclusos Cotuous schedulg suprema, rather tha maxma, for < : blocg tme B worst-case legth WL of actve perod worst-case respose tme crtcal stat o-uform aalyss (τ caot be bloced) Aalyss of Cotroller Area Networ (CAN) same problem (FNS); see RTN 006; evolutoary mprovemet of flawed aalyss: uform for all tass ( ) see Real Tme Systems Joural of Aprl 007 Exstg aalyss for FS s both pessmstc ad optmstc Revsed aalyss resolved pessmsm: rephrased crtcal stat; used (B ) + rather tha B ; ad let resolved optmsm cosdered all obs actve perod; o-uform, because τ caot be bloced Ehaced tas-model based o a AG Mälardale Uversty, Swede, Aprl 008 Mälardale Uversty, Swede, Aprl

5 Reder J. Brl, TU/e Iformatca, System Archtecture ad Networg Reder J. Brl, TU/e Iformatca, System Archtecture ad Networg Refereces Acowledgemets R.J. Brl, J.J. Lue ad W.F.J. Verhaegh, Worstcase respose tme aalyss of real-tme tass uder fxed-prorty schedulg wth deferred preempto revsted wth extesos for ECRTS 07, CS-report 07-, Techsche Uverstet Edhove (TU/e), The Netherlads, Aprl 007. R.J. Brl, J.J. Lue, ad W.F.J. Verhaegh, Worstcase respose tme aalyss of real-tme tass uder fxed-prorty schedulg wth deferred preempto revsted, I: roc. 9th Euromcro Coferece o Real-Tme Systems (ECRTS 07), pp , July 007. Joha J. Lue ad Wm F.J. Verhaegh Ala Burs ad Robert I. avs (Uversty of Yor) IST fuded ARTIST Networ of Excellece o Embedded Systems esg Ramo A.W. Clout ad Ja H.M. Korst (hlps Research Laboratores) aoymous referees of the ECRTS 007 Mälardale Uversty, Swede, Aprl Mälardale Uversty, Swede, Aprl Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Reder J. Brl, r..brl@tue.l TU/e Iformatca, System Archtecture ad Networg Uform (pessmstc) aalyss I Uform (pessmstc) aalyss II Remember By defto, (C) WO (C), hece a uform (pessmstc) aalyss I:,,, ( B + ( + ) C F ) + F T for < = WO (( + ) C F ) + F T for =, = WO ( B + ( + ) C F ) + F T By defto, (C) WO (C), hece a uform (pessmstc) aalyss I: Moreover, WO (C) (C+ ), hece a uform (pessmstc) aalyss II:,,,,, = WO ( B, + ( + ) C F ) + F T, = ( B + ( + ) C ( F )) + ( F ) T Mälardale Uversty, Swede, Aprl Mälardale Uversty, Swede, Aprl