Non-preemptive scheduling algorithms and schedulability conditions for real-time systems with precedence and latency constraints

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(1)Non-preemptive scheduling algorithms and schedulability conditions for real-time systems with precedence and latency constraints Liliana Cucu, Yves Sorel. To cite this version: Liliana Cucu, Yves Sorel. Non-preemptive scheduling algorithms and schedulability conditions for real-time systems with precedence and latency constraints. [Research Report] RR-5403, INRIA. 2004, pp.33. �inria-00070600�. HAL Id: inria-00070600 https://hal.inria.fr/inria-00070600 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Non-preemptive scheduling algorithms and schedulability conditions for real-time systems with precedence and latency constraints Liliana Cucu — Yves Sorel. N° 5403 Decembre 2004. N 0249-6399. ISRN INRIA/RR--5403--FR+ENG. Thème COM. apport de recherche.

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(35) Z\D[-`;`_s¢¤r3pN±rrvs¤slnr`;6x2l2` pX`!l2ln[s¤ln` spvirp{u1lnln[ln`;` x2r³u{lml2l2p ` x2ArXp{`¢lnpvA[r`;vsNrx2B6uvln[p sr3q-(io!`_+k uv1)r's¤r3±-rsl2`%xn`_-` lns¤lnspvr'p{

(36) u{¢¢Sln[`%pN-`_x2u{lnspvrk_§ º ¾ !#" $&%('

(37) < #/*)

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(52) # 0O.- # , /) S A# #L

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(75) # KO /O , /H E ,A / + (# . .#/ ( ( O

(76) #/O

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(81) .

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(88) # / O , . ) O K%-=% , K # *) & E& % #) O ' 0/ ( , #) .O

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(90) . LO,/

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(158) x2k¡pNr ¢i%[`lnx2p `_AuNku{l2r-[` l2x2p ` B¢tu1l2lns¤pNp%+r X` 8:8ykno&3[pj`_`_3k?qr¢`_p{ l H I \ Z [ ` 2 k & o [ Y ` 3 q kn£`!l&k¦xnp1o!©Npv`_x2kLx2`_lnkn[--u1pNlr3¢tu{pNsrx®s¢¤s¤l2l|uvp£is¤x&kmk6¢tu1lnql2sr` rs¤`Yxno `_ks¤`Y¢u{kmk `Yln§¶u{spvx&Z\r o&[ [8:`'8.uv^L¢¤uv`_rpN^L·r^6¦s¤ur u{¢¢55'x2` -¢tu{u1p~¢l2¢¤knp1s¤knpNsr k¡¢¤`%= lnp£k2o&ln[±[`_rp~km q`³¢¤ln`Y^¦[k:pNslnrkn[`s¤^6p~k2kmuvo&`;¢d[¢`_`_knuN`!ql&3k¢¤s¤`Yru{k6?x2u{`³l2rp6s ^¦` ¢¤s¤r~`_rln^¦sst^6o ^¦uvu{¢Cu ¢§ l2oa.p pvpNrrln[kns` `®l2~s¤pNqk2r³o&` [r~`Yl2pv3¢¤xiNq°{ln¢t`[u{©N`?` sl|¢¤rLs¤.l|sivpS°.xnxn`_ln`_[¢¢u{u{` lni·lnss©vpv`_u{r¢¤x2i;k_`v§§l2p;Z\Z\ln[[[stk¯` stk6opvsuv^Lrxml2ssuv¢¤l2s¢3s¤`_pNpvk¦r®x&l23s[`k¯u1xvll2s[©v:`\`_`³r®l|.^6jp%iu²o_i l2uv[knN` `_s¤pNkH©Nxn`'u{`_x2^fuv`:r¶r6p{q lyr[sts 3~stò&q[®r~`l2`skno_r-o&u{[¢D`Yk°{3xnl2`_q[¢¢tu{stu{k¯lnskn©vsq¢`_sl|3¢¤ii ¥ kn`_o!lnspvr5§ . . L(A,C) A. B. C L(D,I). D. F. H. K. L. L(J,L) J. svqx2` 'F u1ln`_ro!i o!pvr-k|l2x2uvs¤r~l2k ÃÃ ËMON P,Q%R. I.

(159) 5². LO,/

(160) . .º%O O# A, B ∈ V (* / /.O< ,

(161) *#/- 0/ *. )#/"! " ! @#@ %O-= % ,& #! = . ,* - # !/ .#. G = (V, E) - ,& / O# s +PC(A,<B)s ∈+PC1 - / @O# Ss ∈+SC < s + C 1 ) S 0 ∀H &6∈ M(A, , ,) B) O .O ) , H %O.> &#/ H∃H 6∈AM(A,AB) /H& s0 + CB < sB0 + CH < sH0 + C -L& δ (S) ≤ δ (S 0 ) S0 ∀S 0 ∈ S 0. 1. A. A. H. H. B. B. AB. AB. :`<[u²©N` su{x2`;+Ckno&[`Y<3qs¢`_+CxnpN^ ° Np x\ .¾5l2p ¾`dd[.u²©v`Yu{` o ruvs qkm.`+`;SC[stu²k©Nu¯<` kno&s[`_3+qC¢`.knq°3o&pv[%rl2¢i6[u1pvl X∀H` x&u1l26∈s¤pNrM(A, k:-`_¢¤pNB)rv°{sr`_?sl2[lnp` x

(162) M(A, B) s s X ?C B C δ (S) = pvx6knpv^¦`k2o&[`Y3q¢` S §H∈¨S`%[-°.u²¢¤©v` `l N = {H ∈ V knqo&[ l2[u1l H 6∈ M(A, B) u{r s + C < s +C <s +C } X X ? + B δ (S ) = C + C , ∀S ∈ S :i knq3l2x2uN§ ol2s¤s¤rruv¢¤ln¢i[`³:` `;~[qu²u1©vl2`s¤pNr$k ?C+ B¦u{r ? B!°\.`° pv3l&u{sr δ§yZ\[(S`;¢)` ^¦−^6δ u¦sk\(S)x2p1=©v`Y P C ≥ 0,∀S ∈ S δ (S) ≤ δ (S ) ∀S ∈ S .º& / #/= / ),/ T

(163) # 0O A, B, C # D ∈ V - ) (A, B) 8:8 (C, D) ) (A, B) = (C, D) M(A, B) M(C, D) = ∅ 1 ¾5 ¾ ¦¨£`%xnp1©N`¯jio pvr~lnx&uvso!lnspvrl2[u1ls¤ (A, B) = (C, D) ln[`_r M(A, B) T M(C, D) = ∅ § ¨£`Lknq°.X`%pNkn[-`?u²ln©v[` u{l ∃F ∈ M(A, B) T M(C, D) §:`Yo u{q-km` F ∈ M(A, B) °«x2`_knX`_ol2s¤©N` ¢i F ∈ M(C, D) ? " B P (A, F ) ∈ P uvr ?D B P (F, B) ∈ P x2`_knX`_ol2s¤©N` ¢i ? +B P (C, F ) ∈ P uvr ? /B P (F, D) ∈ P §PpNrH`LZ\xn[u1pNstl2^ k:[®sk.?" x2spvB:r ^fuvo!rpNur~ln-?x&u{uvBs3x¯°stoxnl2l2`Yps¤kmpNXl2r[`_o!`L lnsp{s©vl2ln[` [¢ìvl2x;[°~`dx2pvu{^s`!x;±u{r?Drs¤lnB:®spvu{lnrr[ p{`_xn5?`L "x2` Bs¢tkd°3u1:l2rs¤p`¯pNrEpN3u{='l2lnuv[® s¤r [x2sPpvo&[^¸(A,knl2u²[i3D)`LkHl2¢uN∈[k|u1l;Pl:lnu{[u{rs` xdx2`¯lnPp stk\(C,lnu1[l.`?B)¢±`_uNx&k|k|∈ll r-p{u{l\sx_ln§:x2¨q``¡u{[r-u²©v`:`;u6[o pvu²©vr~`lnx&uv3stol2s¤pNr³kmTp¦l2[`?knqXpNknsl2s¤pN§ rln[u{lln[`_xn`¡sk F ∈ M(A, B) T M(C, D) stk Z\[`dx2pjp{5pNx<l2[`%o M(A, uvkn`pv>B)x2` ¢tu1lnsM(C, pvCr 8:8~stk\D)kms^¦=s¤¢t∅u{x.lnp¦o uNkm`¯p{>x2` ¢tu1l2s¤pNr A§Z\[`d¢¤`_^¦^¦u?stk.xnp1©N`_ H. B. A. B. H. H. A. A. H. B. I. AB. I∈M(A,B). 0. 0 H. H. 0 B. 0. 0 A. AB. 0. I. H. AB. AB. AB. 0. 0. H∈N. I∈M(A,B). 0. A. B. 0. 0. 0. 0. AB. H∈N. H. ÁÂÃ5ÁÅÄ.

(164) 5Y.

(165) "!$# !" %&'#( )#+*, -$./ /0/. l:u¡¢u{ln` r-o!3i¦pj`Yo!k pvr-rk|p{l2ldx2uvo!s¤pvr~r~l l&L(A, srX` x2x&` u1¢tu1l2s¤lnpNspvrBr X8:` 8~¢pvpvrx.vs¤rsrxn¦`_¢lnu{p¦lnsl2pv['r sk= ¢u{ lml2s¤ln` [ x B) vuo pvrrp{Z\kml2ln[[x&` u{`¯x sr~¢¤¢t`_u1l ^¦l2`? ^¦rouLi;o pvx2rp1km©vlnT`Yx&u{k

(166) slnr~[l u{L(C, { u s ' r v u j r i v p D) M(A, B) M(C, D) l2o_p6uvknu¡`_k_¢t°~u1lnl2[` r`%o ^¦iso!rpNs¤^6rk|uvl2¢>x2uvkms¤` r~ll o!o!pNpNxnrNx2l&`_u{kn-srpNk:rpv3r=sr¢i6∅pvl2Bp6X§'` `_Z\x&uNu1[o&l2[ss¤k¡pN¢trs¤u1^¦k:ln`_-r`_¢o!s¢¤`_pNi k%ro!vl2pv[sr-ru1k|?l?l2x2ln`_uvpLuvs¤o&r~ln[ [l ssk^¦k\o sstrpv3rs¤`_^6rNkmlnl2uvx&s¢.o_u{u{sknr~¢D`!§l_l?°jo kmpvpLx2xnpN`Yx.kmXl2pv[r`;l|s¤:rp ºj¾ º , G = (V, E) *. #4!

(167) #.O K"! # %-= # # / #/ .-( / '0 # ,,- %

(168) #0 #

(169) O= - #/

(170) O , :

(171) )# 0 8:8 - ,& '= ' # , # & @)#/< -E /% .#O ) #/ E -A ) X C ≤ L -),/ #/ A1 # B #/C'! #A)# ,,H AB H∈M(A,B) /% .#O 1 ¾5¾ 6Z\[`;¢` ^¦^6u$;5 kn[p1k.ln[-u1l_°jpNx\l|.p¦pN-`_x2u{lnspvrk A u{r- B xn`_¢u{ln`YjiuLu{ln[5°k2o&[`Y3q¢s¤r. pNZ\r[¢si?kpN^¦-`_`_x2uvu{rlnkspvl2r[k<u1 l'[s¤Lso&[u -kn`_o&¢¤[pN`Yr3%qln¢`vp °¯M(A, -pv` rl|.¢i·`_` pvr XA` x&u1lnl2p s¤pNBrk s¤¢¢S[3so&`_[ o xn-`Yuv`_kn¢¤pN`\rln[`dl2p ©1u{¢q`p{ δ (.)uvxn` § B) ¤ s r. [ s & o [ k 2o&s[¢¢C`Y§3Z\q¢[`_` x2`!-` pvl|x2.`¡`_uv` rr 'A-l2`Ypo uvBqkm°` 3pLrp{lk2u1l2sTkmiuL¢u{ln`_ro!io pvr°-kmlnpvx&xdu{su{r~¢l¢-Lu{sx2k °j l2[[` sr'o&[µruvp¦xn`¡p{M(A, l2s¤[r®` x x2` k2¢to&u1B)[l2`_s¤pNrq¢¤8:` 8 pl2N[x;stk\s¤r l|ijx2-` ¢t`Nu1§ lnspvr = ? ¢¤`_^¦^¦u 56u{r-M(A, µ¢` ^¦^6B)u³ BM(C, °>xnpN^ D)rp1= ∅pNr5°5:`¦ s¤¢¢yo!pNrkmst3`_x%pvr¢i®kno&[`_3q¢`_k;pv } [u²k2©jo&s¤[r`_Jqu ¢¤` ¢tu1Sl2` rknou{ilnstk|o!±-pN`_rk¦k|l2u{x2¢uv¢\s¤r~ln[l4`³? ¢tu1`!l2±` rrs¤o lnispvr o!pN+rBk|§l2x2uv:s¤r~`_l&o_k?u{qs¤%kn` u{r¶pvr¢is¤ δ ≤° ` Ls¤ln[` x° :∀A`'[u{u²r-©v ` ∀B °5pvxd.`L[u²©N` s X °Sln[∀H`_r£u6∈ knM(A, o&[`_3q¢B)` S k2u1lnstkm±`_k;u{¢¢ln[` s +C < s +C +C < s +C ¢tu1l2` ro iµo!pNrkmlnx&u{srNl&k;suvrµpNr¢iµs¤ °pvxuv¢¤¢ A u{r B [u²©jsr³u ¢tu1l2` ro i C ≤L o pvrkmlnx&u{sr~l_§

(172) Z\[`;ln[` pvx2` ^0sk\x2p1©v`Y. º 3 O , & >)#/< - /% .#O # . $ . ,)#/0/+8:8 . ,)#/0/ = -@& >EO # @ %

(173) , / /"! <4#:)# ,,-4 O%

(174) #/O .+,O#/H<:O ( , #) .O

(175) # 0O * )'!/'!4 > AB. AB. AB. H. H. A. A. B. B. H. H. AB. H. AB. H∈M(A,B). # ,,- O%

(176) #/O 1 O % O# % ,)A "#/) '%

(177) # , &' / #) 1 %( º ' '#/) '%

(178) # , & # * - / 0 ), #E%-=%< D#)

(179) # 0O /H&J# ,,- /% .#O(

(180) )# 0 8:8 . ,)#/0/ = O !

(181) #.O ! , !

(182) / /%0 & )#/< - /% .#O L(A, C) - L(D, F ) - L(H,1 I) #/ - . % O . = - - L(J, L) , /E#/ (A, C) - (D, F ) - (H, I) # -K

(183) , , - - (J, L) 1 '@ )# ,,- O%

(184) #/O #/

(185) A. +O# # = 1, ∀A ∈ V 1 E @ /0 &#/ # #/ # . ,& @>

(186) )# 0B 8:8 -L & @>

(187) )# 0' = 1 3 O @, C A . / ) # * . / . 0 $ . 0 ,. @ & / # L ) # ,. , 0 ,. E ' % L . *. !

(188). # ,. . & / # & @. % # ). ' . , 0 % , ) .O

(189) # 0O 1 ( :% O= &#/E&' / 0 :#/ &

(190) #() # /. * / 0.4# , = %- O%

(191) #/O # )& %-= 1 !/

(192) ! %%O 4#( 1 .

(193) . I r6pNx2` x

(194) lnp%xn`Ykm`_rNlHl2[`x2`_knq¢¤l2kyo pvro ` x2rs¤rkmi3kmln` ^6kH s¤ln[6¢tu1l2` ro i?o pvrkmlnx&u{sr~l2k

(195) s^LXpNkn`_¦pNr¦u{sx2k. sr'xn`_¢u{lnspvr"A°.`;sr~lnx2pjqo!`;l|:p?r` rp{l&u1lnspvr-k § ÃÃ ËMON P,Q%R.

(196) 5_ . LO,/

(197) . L(A,C) A. B. C. L(D,F) D. E. F L(H,I). G. H. I. L(J,L) J. K. L. svqx2` F<gji3kmln`_^¦k\pvpN-`_x2u{lnspvrk:¢tu1l2` ro io!pNrk|l2x2uvs¤r~l&k.sr'xn`_¢u{lnspvr98:8pvx\sr'xn`_¢u{lnspvr. t=0. t=1. A. t=2. B. =. t=3. C. D. E. F. G. J. K. L. H. I. svqx2` Fyg3o&[`Y3q¢`%knu{lnstk|ijsr6u{¢¢>o!pNrk|l2x2uvs¤r~l\p{ln[`kni3k|l2` ^. ÁÂÃ5ÁÅÄ.

(198) 5².

(199) "!$# !" %&'#( )#+*, -$./ /0/. ->u{s¤ªNsx:`.>[-s`!r?luvk\ln(A,[pv`\XB), `x&x2u1` l2©j(C,s¤spNpvrqk:D)k