Multicore mixed-criticality with a hierarchical real-time scheduler and resource servers

IT 15026

Examensarbete 30 hp

April 2015

Multicore mixed-criticality

with a hierarchical real-time

scheduler and resource servers

Emanuel Berg

Masterprogram i datavetenskap

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Multicore mixed-criticality with a hierarchical

real-time scheduler and resource servers

This is a real-time mixed-criticality system on a dual-core Linux desktop. The hardware/software architecture employs memory throttling so that modules can be isolated in execution as well as in analysis/certification.

There are two asymmetrically dedicated cores. A critical core runs a hierarchical scheduler (hs) with critical software: hard-coded, or Linux processes with associated metadata. The best-effort core runs arbitrary software.

The cores share DRAM and the memory bus. Best-effort activity can delay critical software.

Interference is throttled/bounded to retain real-time computability.

When necessary, hs freezes/thaws best-effort core software. Task schedulers have budgets to bound memory interference. If depleted, the best-effort core is frozen whenever tasks from such a scheduler execute. Budgets are reset periodically. The scheduling algorithm is a polled-preemptive EDF, with critical-group CPU budgets. Task model: sporadic.

Two resource servers uphold isolation in the face of shared resources: memory, and critical-core CPU time.

A memory experiment shows that best-effort core activity is throttled dynamically. Alas, the

submission mechanism lacks precision in worst-case scenarios: hs is itself suspectible to best-effort interference, so the throttling mechanism can brake before it can effectuate.

Ämnesgranskare: Philipp Rümmer Handledare: Pontus Ekberg

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h?2 `2bQM iQ /BpB/2 i?2 +`BiB+H ibFb BMiQ ;`QmTb Bb iQ #2 #H2 iQ 2tT`2bb M/ 2KTHQv  KQ`2 ~2tB#H2 M/ }M2@;`BM2/ `2H@iBK2 bvbi2KX AM i?Bb T`QD2+i-i?2`2 `2 Qi?2` T`K2i2`b T`i 7`QK i?2 +`BiB+HBiv H2p2H ii+?2/ iQ 2+? ;`QmTX

h?2 ibF +HbbB}+iBQMb `2 iQ #2 mb2/ #v i?2 `2H@iBK2 QT2`iBM; bvbi2K iQ ;mB/2 Bib +iBQMbX _2H@iBK2 MHvbBb Bb iQ #2 TTHB2/ iQ i?2 ;`QmTb b BbQHi2/ 2MiBiB2b- M/ 7Q` M 2MiB`2 bvbi2K QM2 ;`QmT i  iBK2- b B7 i?2 Qi?2` ;`QmTb /B/ MQi 2tBbiX

_2H@iBK2 PT2`iBM; avbi2K

h?2 `2H@iBK2 QT2`iBM; bvbi2K U_hPaV Bb i?2 bQ7ir`2 i?i KF2b bm`2 i?i i?2 `2H@iBK2 ibFb `2 2t2+mi2/ BM bm+?  rv i?i i?2v `2 HH ;m`Mi22/ iQ +QKTH2i2 #27Q`2 i?2B` /2/HBM2b 2tTB`2X AM T`BM+BTH2- i?2`2 b?QmH/ #2 b2p2`H rvb iQ BKTH2K2Mi bm+?  bvbi2KX AM T`+iB+2- i?2 KQbi +QKKQM bQHmiBQM Bb iQ ?p2  `2H@iBK2 b+?2/mH2` BMbi2/ Q7 i?2 Q`/B@ M`v b+?2/mH2`- M/ i?2M iQ MMQii2 2+? ibF rBi? `2H@iBK2 K2i/i- bm+? b rQ`bi@+b2 2t2+miBQM iBK2 M/ /2/HBM2X h?2 `2H@iBK2 b+?2/mH2` rBHH T`Q+2bb i?2 K2i/i- b r2HH b i?2 Qp2`HH bvbi2K bii2 UBM+Hm/BM; i?2 iBK2V- bQ i?i ibFb `2 2M[m2m2/ M/ b2H2+i2/ 7Q` 2t2+miBQM BM  KMM2` i?i rBHH 2Mbm`2 +QKTHBM+2 rBi? i?2 `2H@iBK2 +QMbi`BMibX

h?2 `2bQM iQ +`2i2  ibF@#b2/ bvbi2K Bb i?i `2H@iBK2 bvbi2Kb BMi2`+i rBi? i?2 QmibB/2- T?vbB+H rQ`H/X "2+mb2 i?2`2 `2 bQ KMv bT2+ib Q7 i?2 T?vbB+H rQ`H/ i?i +M #2 K2bm`2/- M/ BM bQ KMv rvb Ĝ M/ HH ?b iQ #2 /QM2 +QMiBMmQmbHv- pB`imHHv BM T`HH2H Ĝ  TQr2`7mH bQHmiBQM Bb iQ 2KTHQv  b2i Q7 ibFb- r?2`2BM 2+? ibF Bb `2bTQMbB#H2 7Q` +QHH2+iBM; M/ T`Q+2bbBM; /i Q7 QM2 T`iB+mH` bm+? bT2+iX 6Q` 2tKTH2- QM2 ibF `2TQ`ib i2KT2`im`2- MQi?2` pQHmK2- v2i  i?B`/ bvbi2K T`QtBKBiv iQ Qi?2` Q#D2+ib-M/ bQ QMX h?Bb KQ/2H KBKB+b i?2 ;2M2`H@Tm`TQb2 T`Q+2bb@#b2/ +QKTmiBM; KQ/2H- r?B+? BM Bib2H7 ?b KMv /pMi;2bX

h?2`2 `2 b2p2`H /pMi;2b iQ ?pBM; b2p2`H +`BiB+HBiv ;`QmTb,

Ç h?2 bvbi2K Bb KQ`2 KQ/mH`, ibFb +M #2 #`Qm;?i BM M/ Qmi b ;`QmTb- M/ 2+? ;`QmT +M #2 MHvx2/ 7Q` `2H@iBK2 b+?2/mH#BHBiv BM/2T2M/2MiHv Q7 i?2 Qi?2` ;`QmTbX

Ç h?2 bvbi2K Bb KQ`2 r2HH@/2}M2/- M/ Q`/2`2/X 6Q` 2tKTH2- #v bbB;MBM; /Bz2`2Mi *Sl@#m/;2ib iQ /Bz2`2Mi ;`QmTb- i?2 `2bT2+iBp2 BK@ TQ`iM+2 Q7 i?2 ;`QmTb b bm#bvbi2Kb `2 K/2 2tTHB+Bi, BM im`M- i?Bb rBHH z2+i i?2 #2?pBQ` Q7 i?2 bvbi2K BM 2t2+miBQMX

Ç h?2 bvbi2K Bb KQ`2 ~2tB#H2, i?2 /Bz2`2Mi *Sl@#m/;2ib bbB;M2/ iQ /Bz2`2Mi ;`QmTb +M #2 mb2/ iQ bbB;M  #m/;2i i?i Bb /QTi2/ iQ i?2 ivTB+H 2t2+miBQM Tii2`Mb Q7 i?2 ibFb Q7 i?i ;`QmTX

a+?2/mH#BHBiv

6Q` Mv ibF b2i Ur?2`2 i?2 ibFb /?2`2 iQ  bT2+B}+ ibF KQ/2HV- M/  T`iB+mH` b+?2/mHBM; H;Q`Bi?K- i?2`2 Bb  i2bi i?M +M /2i2`KBM2 b+?2/mH#BHBivX h?i i2bi M@ br2`b i?2 [m2biBQM, dz*M i?2 H;Q`Bi?K KF2  b+?2/mH2-i?i BMpQHp2b HH i?2 ibFb BM i?2 b2i- bQ b+?2/mH2-i?i- B7 i?2v `2 2t2+mi2/ ++Q`/BM;Hv- HH ibFb rBHH +QKTH2i2 #27Q`2 i?2B` /2/HBM2b\Ǵ A7 bQ- i?2 ibF b2i ?b Tbb2/ i?2 b+?2/mH@ #BHBiv i2bi 7Q` i?i H;Q`Bi?K, i?2 ibF b2i Bb b+?2/mH#H2X

RXj `2H@iBK2 BbQHiBQM

h?2 +`BiB+H ;`QmTb Kmbi #2 BbQHi2/ 7`QK 2+? Qi?2` BM `2H@iBK2 MHvbBbX A7 Mv QM2 ;`QmT Bb /22K2/ b+?2/mH#H2- i?Bb Kmbi ?QH/ 7Q` Mv `mM@iBK2 b+2M`BQ r?B+? +M BMpQHp2 i?2 ibFb Q7 i?i ;`QmT- b r2HH b i?Qb2 Q7 HH Qi?2`bX h?i Bb- BM Q`/2` iQ T`QpB/2 ;m`Mi22b 7Q`  T`2/B+i#H2 2t2+miBQM Q7 i?2 ibFb Q7 Mv QM2 +`BiB+H ;`QmT- Bi Bb M2+2bb`v iQ T`Q+2bb i?2 `mMiBK2 #2?pBQ` Q7 i?2 ibFb Q7 HH Qi?2` ;`QmTb b r2HHX

S`2/B+i#H2

*QKTmi2` bvbi2Kb `2 /2i2`KBMBbiB+ #v /2}MBiBQMX A7 i?2v `2

T`2/B+i#H2- i?i /2i2`KBMBbK Bb HB7i2/ iQ  ?mKM H2p2H, i?2

mb2` Bb #H2 iQ KF2  bii2K2Mi Ĝ i?i bQK2i?BM; T`iB+mH` rBHH ?TT2M- M/ MQi Hi2` i?M i  +2`iBM iBK2X h?2 bii2@ K2Mi Bb i?2M T`Qp2/ iQ ?QH/ #v mb2 Q7 7Q`KH K2i?Q/bX A7 i?2 bii2K2Mi ?QH/b BM i?2Q`v b r2HH b T`+iB+2- i?2 bvbi2K Bb T`2/B+i#H2- i H2bi rBi? `2bT2+i iQ i?i T`iB+mH` bii2K2MiX  T`2/B+i#H2 bvbi2K +QMbBbib Q7  b2i Q7 HBF2rBb2 T`2/B+i#H2 bii2K2Mibf#2?pBQ` Tii2`MbX

AM T`+iB+2- i?2 `2bQm`+2 b2`p2` bQHmiBQM Bb M ii2KTi iQ 7Q`KHBx2 M/ bBK@ THB7v bm+? +QMbB/2`iBQMb iQ i?2 TQBMi r?2`2 M BM/BpB/mH ;`QmT /Q2b MQi ?p2 iQ #2 MHvx2/ BM  rv i?i +iBp2Hv ?b iQ +QMbB/2` i?2 T2+mHB`BiB2b Q7 i?2 Qi?2` ;`QmTb- i?2B` ibFb- M/ biimb2bX

q?2M2p2` M22/2/- i?2 bvbi2K Kmbi BMi2`p2M2 iQ T`2p2Mi mM`mHv #2?pBQ`-`B;?i #27Q`2 Bi ?TT2MbX A7 i?2 ibFb Q7  +2`iBM ;`QmT +QHH2+iBp2Hv ?p2 /2TH2i2/ i?2B` HHQ+i2/ `2bQm`+2b- i?2 ;`QmT Kmbi #2 ?BM/2`2/ 7`QK 7m`i?2` +iBpBiv- H2bb Bi rBHH D2QT`/Bx2 i?2 T`2/B+i#BHBiv Q7 ibFb i?i #2HQM; iQ Qi?2` ;`QmTbX

_2bQm`+2b

h?2 `2bQm`+2b Q7  +QKTmi2` +M #2 2Bi?2` BM ?`/r`2 Q` bQ7ir`2X >`/r`2 `2bQm`+2b `2 i?2 K2KQ`v- i?2 *Sl- i?2 /BbF- 2i+X am+? `2bQm`+2b +M #2 2Bi?2` biiB+ Ĝ 2X;X- i?2 /BbF bBx2 Ĝ Q` /vMKB+ Ĝ 2X;X- i?2 pBH#H2 K2KQ`v i i?2 iBK2 Q7  +2`iBM rQ`FHQ/X _2bQm`+2b +M HbQ #2 BM bQ7ir`2X A7  T`Q;`K ?b #22M bbB;M2/  +2`iBM MmK#2` Q7 K2KQ`v +@ +2bb2b- i?Bb #m/;2i Bb  `2bQm`+2, B7 y- Bi Bb  /2TH2i2/ `2bQm`+2 mMiBH Bi ;2ib `2bmTTHB2/X

RX9 /Bz2`2Mi FBM/b Q7 +QKTmi2` bvbi2Kb

h?2 ;2M2HQ;v Q7 +QKTmi2` bvbi2Kb Ĝ 7`QK  /2bFiQT Pa iQ  7mHHv@~2/;2/ `2H@iBK2 bvbi2K BM i?2 T?vbB+H rQ`H/ Ĝ +M #2 bF2i+?2/ HBF2 i?Bb,

Ç BMi2`+iBp2 MQM@iBK2b?`BM; Ĝ h?Bb Bb  ivTB+H +QKTmi2` bvbi2K 7Q`  mb2` iQ mb2 7Q` p`BQmb- KmM/M2 Tm`TQb2bX h?2 bQ7ir`2 i?i `mMb QM i?Bb bvbi2K /Q2b MQi ?p2 Mv bbQ+Bi2/ /2/HBM2b, 2p2`vi?BM; Bb bBKTHv 2t2+mi2/ rBi?Qmi Mv ;m`Mi22b- #mi HbQ rBi?Qmi Mv +mK@ #2`bQK2 `2H@iBK2 Hv2` Q` K2i/i ii+?2/ iQ 2t2+miBM; T`Q;`KbX aiBHH- BM bQK2 b2Mb2- 2p2M bm+? bvbi2Kb `2 dz`2H@iBK2Ǵ- #2+mb2 ?mKM@ +QKTmi2` BMi2`+iBQM rQmH/ #2 mM#2`#H2 B7 7Q` 2tKTH2  F2vbi`QF2 bQK2iBK2b ?/ iQ T`Q+2bb 7Q` b2p2`H b2+QM/b #27Q`2 i?2 +?`+i2` T@ T2`2/ QM i?2 b+`22MX

Ç iBK2b?`BM; Ĝ  iBK2b?`BM; bvbi2K Bb M BMi2`+iBp2 bvbi2K- #mi Bi Bb HbQ  KmHiBT`Q;`KK2/ M/ KmHiBibF2/ bvbi2K i?i bmTTQ`ib b2p2`H +iBp2 mb2`b iQ #2 HQ;;2/ BM i i?2 bK2 iBK2X h?2 BKTH2K2MiiBQMb Q7 bm+? bvbi2Kb Q7i2M 2KTHQvb i?2 T`Q+2bb@#b2/ `+?Bi2+im`2- r?B+? Bb HbQ r?i Bb 7QmM/ BM KMv `2H@iBK2 bvbi2KbX h?Bb b?Qrb i?2 T`QtBKBiv #2ir22M iBK2b?`BM; M/ `2H@iBK2 bvbi2Kb- i?2 /Bz2`2M+2 #2BM; i?2

#b2M+2 Q7 /2/HBM2b BM  iBK2b?`BM; bvbi2KX AM ;2M2`H-  iBK2b?`BM; bvbi2K Bb Q`B2Mi2/ iQr`/ +iBp2 ?mKMb- r?Q T`QpB/2 BMbi`m+iBQMb r?i i?2 +QKTmi2` b?QmH/ /Q- r?BH2  `2H@iBK2 bvbi2K Bb Q`B2Mi2/ iQr`/ i?2 QmibB/2 T?vbB+H rQ`H/- M/ r?i2p2` ;Q2b QM i?2`2 Bb bKTH2/ M/ T`Q+2bb2/ b BMTmi iQ i`B;;2` p`BQmb `2bTQMb2bX (kj- TX j)

Ç bQ7i `2H@iBK2 Ĝ >2`2- bQ7ir`2 Bb bbQ+Bi2/ rBi? /2/HBM2bX A7  /2/@ HBM2 +MMQi #2 ?2H/- ?Qr2p2`- i?i /Q2b MQi BKTHv bvbi2K@rB/2 /Bb7mM+@ iBQMX _i?2` Bi Bb mT iQ i?2 bvbi2K /2bB;M2`b UM/ bQK2iBK2b i?2 mb2`V iQ };m`2 Qmi r?i iQ /Q BM bm+? M 2p2MiX

Ç }`K `2H@iBK2 Ĝ >2`2- /2/HBM2b `2 bQ7i BM i?2 b2Mb2 i?i #`2+?2b `2 MQi 2M+Qm`;2/- #mi HHQr2/c QM i?2 Qi?2` ?M/- /2/HBM2b `2 ?`/ BM i?2 b2Mb2 i?i +QKTmiiBQM r?Qb2 `2bmHi ``Bp2b Hi2 ?b x2`Q pHm2- M/ Bb /Bb+`/2/X UAM bQK2 b2iiBM;b- /2Hv2/@+QKTmiiBQM /i +M 2p2M #2 /M;2`Qmb B7 mb2/ b B7 Bi ?/ ``Bp2/ BM iBK2XV

Ç ?`/ `2H@iBK2 Ĝ >2`2-  /2/HBM2 #`2+? BKTHB2b  bvbi2K 7mHiX Ç +`BiB+H `2H@iBK2 Ĝ >2`2-  /2/HBM2 #`2+? BKTHB2b  ?x`/ iQ T2QTH2Ƕb

?2Hi?X *`BiB+H `2H@iBK2 Q#pBQmbHv BKTHB2b ?`/ `2H@iBK2X q?i Bb MQi b Q#pBQmb Bb i?i ?`/ `2H@iBK2 /Q2b MQi BKTHv +`BiB+H `2H@iBK2X 6Q` 2tKTH2- +QMbB/2`  +QKTmi2` ;K2,  7bi@T+2- BMi2`+iBp2 ;K2 Bb 7Q` HH T`+iB+H Tm`TQb2b ?`/ `2H@iBK2- 2p2M i?Qm;? i?2 BKTH2K2M@ iiBQM T`Q##Hv H+Fb 2tTHB+Bi /2/HBM2b Ĝ biBHH- B7 i?2 ;K2 /Q2b MQi `2+i pB`imHHv BMbiMiHv iQ BMTmib M/ BM@;K2 2p2Mib- Bi Bb mMTHv#H2X >Qr2p2`- M mMTHv#H2 ;K2 Bb MQi  ?x`/ iQ i?2 ?2Hi? Q7 Mv HBpBM; i?BM;- bQ 7Q` i?Bb `2bQM Bi Bb MQi +QMbB/2`2/ +`BiB+H `2H@iBK2X

RX9XR ?v#`B/b M/ ;`v xQM2b

h?2 `2H@iBK2 bvbi2K +HbbB}+iBQMb `2 MQi +mi BM biQM2X JMv bvbi2K +QM@ bBbi Q7 bm#bvbi2Kb Q7 /Bz2`2Mi Mim`2bX +imHHv- i?2`2 +M #2 +QK#BMiBQMb rBi?BM  bm#bvbi2K Bib2H7, +QMbB/2`  `2H@iBK2 bvbi2K r?2`2 bQ7ir`2 Bb b@ bQ+Bi2/ rBi? #Qi?  bQ7i M/  ?`/ /2/HBM2X (e- TTX k@j) A7 i?2 T`Q;`K /Q2b MQi +QKTH2i2 #27Q`2 i?2 bQ7i /2/HBM2- i?2 bvbi2K `2HHQ+i2b `2bQm`+2b bQ i?2 T`Q;`K rBHH #2 KQ`2 HBF2Hv iQ +QKTH2i2- [mB+F2`X PMHv r?2M i?2 ibF biBHH ?b MQi +QKTH2i2/ i i?2 iBK2 Q7 i?2 ?`/ /2/HBM2- bvbi2K 7mHi Bb ~;;2/X

RX9Xk r?2`2 /Q2b i?Bb T`QD2+i }i BM\

h?2 ?B2``+?B+H b+?2/mH2` b+?2/mH2b TB2+2b Q7 bQ7ir`2 r?B+? HH ?p2 T`K@ 2i2` pHm2b Ĝ i?2 `2H@iBK2 K2i/iX h?mb- Mv bvbi2K i?i `mMb QM iQT Q7 ?b rBHH #2 i H2bi  bQ7i `2H@iBK2 bvbi2KX "mi Bi +QmH/ HbQ #2  }`K Q` ?`/ `2H@iBK2 bvbi2K- Q` bQK2i?BM; BM #2ir22M- rBi? MQM2 Q` KBMBKH +?M;2b iQ i?2 +Q/2 `2[mB`2/X lHiBKi2Hv- Bi Bb i?2 mb2` r?Q /2+B/2b r?i bQ7ir`2 rBHH `mM M/ ?Qr i?i bQ7ir`2 rBHH #2?p2- M/ r?2MX h?Qb2 /2+BbBQMb rBHH b?T2 i?2 2MiB`2 bvbi2K- BM+Hm/BM; Bib TH+2 BM i?2 `2H@iBK2 H//2`X

b 7Q` MQr- r?i Bb ?`/@+Q/2/ Bb i?2 bTQ`/B+ ibF KQ/2H, bQ7ir`2 ibFb `2 MMQii2/ rBi? i?2 T`K2i2`b q*1h- /2/HBM2- M/ T2`BQ/X

q*1h

q*1h UqQ`bi *b2 1t2+miBQM hBK2V Bb i?2 rHH@+HQ+F iBK2-Q` `2bTQMb2 iBK2- Q7  T`Q;`K, i?2 KQmMi Q7 iBK2 i?i Tbb2b 7`QK BMpQ+iBQM iQ +QKTH2iBQM- BM i?2 rQ`bi TQbbB#H2 +b2 UBX2X- Bi Bb BKTQbbB#H2 i?i Bi +QmH/ 2p2` iF2 Mv HQM;2` iQ bm++2bb7mHHv i2`KBMi2 i?2 ibFVX h?2 q*1h T`K2i2` Bb ?2HT7mH BM KQ/2Hb- iQ /Q +QKTmiiBQM- M/ Bi +M HbQ bi22` bQ7ir`2 #2?pBQ`X LQM2i?2H2bb- Bi Bb KQbi Q7i2M MQi TQbbB#H2 iQ /2`Bp2 i?2 q*1h- BM bTBi2 Q7 Km+? *QKTmi2` a+B2M+2 `2b2`+? /2pQi2/ iQ K2i?Q/b iQ i?i 2M/X (RN- TX RRy)

h?2`2 `2 b2p2`H `2bQMb q*1h Kv #2 BM+QKTmi#H2, ibF@2t2+miBQM +QM/BiBQMb Kv p`v Ĝ 2ti2`MHHv Ui?2 T?vbB+H +QMi2tiV- #mi HbQ BMi2`MHHv U2X;X- bvbi2K HQ/Vc HbQ- ibFb Kv /2T2M/ QM Qi?2` ibFb iQ +QKTH2i2 #27Q`2 i?2K- Q` iQ `2H2b2 `2bQm`+2b i?2v ?p2 KQMQTQHBx2/c M/- i?2 bvbi2K Kv #2  KmHiB+Q`2- Q` 2p2M /Bbi`B#mi2/ rBi? bvM+?`QMQmb K2bb;2 TbbBM; BM #2ir22MX (9- TX j)

AM i?Bb T`QD2+i- #Qi? BM bQ7ir`2 M/ BM MHvbBb- Bi Bb bbmK2/ i?2 q*1h Bb +Q``2+i- r?2`2p2` Bi TT2`bX +imHHv- q*1hb `2 bBKTHv bbB;M2/ #v i?2 mb2` rBi? MQ [m2biBQMb bF2/ B7 Bi Bb `2HBbiB+ Q` MQiX q?2`2p2` pBH#H2- BM/i p2`B}+iBQM Bb /QM2 BM QMHv i2`Kb Q7 Qi?2` T`K2i2` BM/i, i?2 bbQ+Bi2/ bQ7ir`2 Bb MQi 2tKBM2/X

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a+?2/mH2`

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रतततततततततततततततततततऴ द JOREDO VFKHGXOHU द द O OLIHWLPH द द S SHULRG द द U UDWH द द 6 VFKHGXOHUV द द D DOJRULWKP द सतततततततॐततततततततततत़ द रततततक़ततततततततततततततततततततततततततऴ द द रततक़तततततततततततततततततऴ रतततततक़तततततततऴ

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Ç h?2 ibF ?b M2p2` 2t2+mi2/- M/ ?b MQi v2i #22M TH+2/ BM i?2 `2/v [m2m2 Ur?2`2 Bi rQmH/ ?p2 bii2 _1.uVX

Ç h?2 ibF ?b ?/ i H2bi QM2 BMbiM+2 Q7 Bi@ b2H7 2t2+miBM; Ui?2 _lLLAL: bii2Vc Bi +QKTH2i2/ Bib +QKTmiiBQM- M/ ?b #22M Tmi QM ?QH/ Ui?2 >PG. bii2VX >Qr2p2`- i?2 iBK2 Q7 M2+2bb`v ?QH/BM; U/m2 iQ i?2 bTQ`/B+ ibF KQ/2HV ?b i`MbTB`2/- bQ MQi?2`- M2r BMbiM+2 Q7 i?2 ibF +M BM/22/ #2 TH+2/ BM i?2 `2/v [m2m2 Ĝ QMHv i?Bb ?b MQi ?TT2M2/- v2iX

q?B+? QM2 Q7 i?2b2 bBimiBQMb Bb i ?M/ Bb MQi Q7 T`+iB+H Q` MHviB+H BKTQ`iM+2- #mi i?2v BHHmbi`i2 i?2 +v+HB+ Mim`2 Q7 i?2 HB72 Q7  ibF BM i?Bb bvbi2KX h?2 `2bQM 7Q` i?Bb bii2 Bb iQ ;2i i?2 bTQ`/B+- b2KB@ `M/QK #2?pBQ` Q7 ibF ``BpHbX

LQi2 i?i M mM2KTHQv2/ ibF ?b MQi ``Bp2/ bQ Bi /Q2b MQi ?p2  T2M/BM; /2/HBM2- M/ Bi +MMQi #2 b2H2+i2/ 7Q` 2t2+miBQMX

_1.u h?2 ibF Bb [m2mBM;X Ai ?b #22M /2THQv2/ Ĝ Bi ?b ``Bp2/ Ĝ M/ Kv BM T`i ?p2 #22M 2t2+mi2/- QMHv-B7 bQ- Bi ?b #22M T`22KTi2/ Ĝ 2Bi?2` rv- Bi Bb MQi +QKTH2i2/ bQ Bi rBib- [m2mBM;X HH iBK2 bT2Mi BM i?Bb bii2 Bb //2/ iQ i?2 `2bTQMb2 iBK2 Q7 i?2 ibFX

_lLLAL: h?2 ibF Bb 2t2+miBM;, i?2 b+?2/mH2` i?i ?QH/b i?2

ibF ?b MQi /2TH2i2/ Bib *Sl@#m/;2i- M/ i?2 ibF ?b i?2 KQbi BKK2/Bi2 /2/HBM2 7Q` HH ibFb i?i `mM QM MQM@/2TH2i2/ b+?2/mH2`bX q?2M 2t2+miBM;- i?2 ibF Bb +HQbBM; BM QM i?2 #bQHmi2 /2/HBM2 b Km+? b r?2M Bi B/H2b BM i?2 `2/v [m2m2X >Qr2p2`- +QMi``v iQ r?2M B/HBM;- i?2 ibF Bb HbQ +HQbBM; BM Ui i?2 bK2 bT22/V QM Bib QrM i2`KBMiBQM- i r?i TQBMi Bi ?b `2+2Bp2/ 2MQm;? *Sl iBK2 iQ +QKTH2i2 Bib +QKTmiiBQM M/ i2`KBMi2 bm++2bb7mHHvX

>PG. h?2 T`2pBQmb ibF BMbiM+2 ?b +QKTH2i2/- M/ #27Q`2

MQi?2` BMbiM+2 Q7 i?2 bK2 ibF +M ``Bp2- Bi Kmbi ?QH/ mMiBH Bib T2`BQ/ Bb Qp2`- BM ++Q`/M+2 rBi? i?2 KBMBKH@BMi2```BpH- QM2@BMbiM+2@i@@iBK2 TQHB+v Q7 i?2 bTQ`/B+ ibF KQ/2HX

.1Gu1. h?2 ibF ?b KBbb2/ Bib /2/HBM2, i?2 #bQHmi2 /2/@

HBM2 Bb BM i?2 Tbi- M/ i?2 ibF ?b MQi #22M bbB;M2/ i?2 *Sl Ubii2 _lLLAL:V 7Q`  iBK2 i?i 2[mHb Q` 2t+22/b i?2 q*1h Q7 i?2 ibFX AM i?Bb bvbi2K-  /2@ Hv2/ ibF rBHH MQi 2p2` #2 +QKTH2i2/- MQ` rBHH Mv 7m`i?2` BMbiM+2b Q7 i?i ibF #2 `2H2b2/X

3 avbi2K /2b+`BTiBQM

h2`KBMQHQ;v, h?Bb b2+iBQM Bb  /2b+`BTiBQM Q7 i?2 ?B2``+?B+H b+?2/mH2` b

 TB2+2 Q7 bQ7ir`2X h?2 T`K2i2`b +Q``2bTQM/ /B`2+iHv iQ *YY /i ivT2b, BM bQK2 +b2b +Hbb2b- BM bQK2 +b2b T`BKBiBp2bX JQbi T`K2i2`b `2 T`i Q7 i?2 BMi2`7+2 M/ +M i?mb #2 bT2+B}2/ #v i?2 mb2` rBi?Qmi Mv +?M;2b iQ i?2 bQ7ir`2 `2[mB`2/X PMHv i?2 T`K2i2` MK2b `2 bBKTHB}2/- KQbi Q7i2M iQ bBM;H2 H2ii2`bc M/ PP `2HiBQMb?BTb `2 2tT`2bb2/ b T`2Mi?2bBx2/ HBbib, i?Bb Bb iQ KF2 i?2 /2b+`BTiBQM KQ`2 ++2bbB#H2- M/- Hi2`- iQ 7+BHBii2 i?2

mb2 Q7 i?2 T`K2i2`b BM 7Q`KmHbX LQM2i?2H2bb i?Bb `2KBMb  /2b+`BTiBQM Q7 +imH bQ7ir`2 M/ Bb MQi  KQ/2H iQ #2 p2`B}2/ #v 7Q`KH K2i?Q/b Ĝ r?B+?-Hb- i #2bi rQmH/ p2`B7v QMHv i?2 KQ/2H Bib2H7X

3XR i?2 iQT@b+?2/mH2`

g Bb i?2 bBM;mH`- Q` `QQi- iQT@b+?2/mH2`,

g = (l, p, r, S, a) 2[X URV- iQTnb+?2/mH2`

T`K2i2` MK2 UmMBiV /2b+`BTiBQM

l HB72iBK2 Kb h?2 bvbi2K HB72iBK2X h?Bb Bb i?2

KQmMi Q7 iBK2 i?i i?2 iQT@b+?2/mH2` rBHH 2t2+mi2- 7`QK i?2 iBK2 Q7 BMpQ+@ iBQMX b i?2 iQT@b+?2/mH2` Bb i?2 iQT@ KQbi #Q/v Q7 i?2 bvbi2K- l 2[mHb i?2 bvbi2K HB72iBK2X

p T2`BQ/ Kb h?2 ;HQ#H T2`BQ/X i i?2 #2;BMMBM;

Q7 2+? T2`BQ/ i?2 ibF@b+?2/mH2`b ?p2 i?2B` #m/;2ib U`2VbmTTHB2/ iQ i?2B` BM@ /BpB/mH- T`2@bbB;M2/ H2p2HbX

r `i2 Kb h?2 b+?2/mHBM; `i2 #v r?B+? g rBHH

b+?2/mH2 SX i 2p2`v r- 2t2+miBQM +QK2b iQ  ?Hi, b+?2/mHBM; Bb /QM2-M/ 2t2+miBQM `2bmK2bX

S b+?2/mH2`b h?2 b2i Q7 ibF b+?2/mH2`b i?i i?2 iQT@

b+?2/mH2` b+?2/mH2bX

a H;Q`Bi?K h?2 b+?2/mHBM; H;Q`Bi?K i?i g T@ THB2b iQ S- BM Q`/2` iQ b2H2+i r?i ibF iQ 2t2+mi2 U7Q` HH s ∈ SVX h?2 H;Q@ `Bi?K mT/i2b M/ KBMiBMb i?2 bi@ imb2b Q7 HH Qi?2` ibFb M/ b+?2/mH2`b b r2HHX

3Xk ?Qr i?2 iBK2 T`K2i2`b `2Hi2

0 < r < p < l 2[X UkV- `2Hi2

A7 MQi 0 < r i?2`2 rBHH #2 +QMbiMi T`22KTiBQM M/ MQi Mv ibF 2t2+miBQM i HHX

A7 MQi r < p- i?2M i 2+? iB+F- i?2 #m/;2ib Q7 HH s ∈ S rBHH ?p2 #22M `2bmTTHB2/- M/ r?i ?TT2M2/ #27Q`2 i?2 iB+F rBHH MQi BM~m2M+2 i?2 b2H2+iBQM Q7 i?2 M2ti s$<sub>- b i?2M UM/ HrvbV- 2p2`v s</sub>$ <sub>∈ S UBM+Hm/BM; sV Bb 2[mHHv</sub> `2/vX A7 bQ- i?2`2 rBHH MQi #2 Mv BbQHiBQM #2ir22M i?2 +`BiB+H ;`QmTb b i?i Bb 2tT`2bb2/ M/ BKTH2K2Mi2/ b i?2 *Sl@#m/;2ib Q7 i?2 b+?2/mH2`bX A7 i?2 #m/;2ib +MMQi #2 /2TH2i2/- i?2v +MMQi THv  `QH2 BM i?Bb bvbi2KX A7 MQi p < l- i?2`2 rBHH MQi #2 Mv `2bmTTHB2b Q7 #m/;2ibX

3Xj ibF b+?2/mH2`

h?2 ibF@b+?2/mH2`b `2 i?2 BKTH2K2MiiBQM Q7 i?2 +`BiB+H ;`QmTbX

S %= ∅ b- mMH2bb bQ- i?2`2 Bb MQi Mv s ∈ S 7Q` g iQ b+?2/mH2 UQ` Mv ibFb iQ 2t2+mi2VX

h?2 b2i Q7 ibF b+?2/mH2`b Bb,

LQiiBQM, h?2 bm#b+`BTib `2 QMHv ;Bp2M iQ BM/B+i2 i?i i?2 T`QT2`iB2b

#2HQM; iQ  +2`iBM ibF b+?2/mH2`X  bBM;H2 ibF@b+?2/mH2` s rQmH/ #2 7mHHv /2b+`B#2/ b s = (m, b, P, α)X

3XjXR mi- i?2 KtBKmK #2bi@2zQ`i +Q`2 K2KQ`v ++2bb2b

6Q` 2+? ;HQ#H T2`BQ/ UQ7 iBK2 pV- i?2 bQ7ir`2 i?i `mMb QM i?2 #2bi@2zQ`i +Q`2 Kmbi MQi 2t+22/  +2`iBM MmK#2` Q7 K2KQ`v ++2bb2b Ĝ miĜ /m`BM; i?2 2t2+miBQM Q7 i?2 +`BiB+H ibFb Q7 siX

h?i rv BMi2`72`2M+2 7`QK i?2 #2bi@2zQ`i +Q`2 pBb@@pBb  bT2+B}+ +`BiB+H ;`QmT +M #2 [mMiB}2/- M/ i?mb #QmM/2/X h?2 #QmM/ Bb mb2/ BM `2bTQMb2@ iBK2 MHvbBb Q7 i?2 +`BiB+H ibFb QM  ibF@b+?2/mH2` #bBbX

3XjXk bi- i?2 *Sl@#m/;2i

bi Bb i?2 #m/;2i bbB;M2/ iQ siX h?2 iBK2 i?2 +QKTmi2` bT2M/b 2t2+miBM; 

t ∈ Pi rBHH `2/m+2 i?2 rQ`FBM; #m/;2i Q7 si #v i?2 bK2 KQmMi Q7 iBK2X

q?2M i?2 rQ`FBM; #m/;2i Bb /2TH2i2/ 7Q` s_i- g rBHH MQi +QMbB/2` s<sub>i</sub>c ?Qr2p2`-g rBHH `2biQ`2 i?2 rQ`FBM; #m/;2i Q7 si iQ bii i?2 i`MbBiBQMb #2ir22M ;HQ#H T2`BQ/bX

b_i > 0 b Qi?2`rBb2 g rBHH M2p2` +QMbB/2` s_iX

h?2 *Sl@#m/;2i 72im`2 KF2b Bi TQbbB#H2 iQ BbQHi2 i?2 +`BiB+H ;`QmTb 7`QK

2+? Qi?2`X 6Q` 2+? ;HQ#H T2`BQ/- i?2 ibFb Q7 si Ĝ Pi Ĝ +MMQi /2Hv

Mv ibF i?i #2HQM;b iQ MQi?2` ;`QmT #v Mv KQ`2 i?M bi- i?2 #m/;2i Q7 siX

3XjXj α_i- i?2 UibFV b+?2/mHBM; H;Q`Bi?K

α_iBb i?2 b+?2/mHBM; H;Q`Bi?K s<sub>i</sub>mb2b iQ T`QTQb2  t ∈ P_i7Q` 2t2+miBQMX

AM T`BM+BTH2- α<sub>i</sub> +M #2 M HiQ;2i?2` /Bz2`2Mi H;Q`Bi?K i?M i?2 ;HQ#H

b+?2/mHBM; H;Q`Bi?K aX αi +M HbQ #2 /Bz2`2Mi 7`QK QM2 ibF@b+?2/mH2` iQ MQi?2`X AM i?Bb T`QD2+i- HH ibF@b+?2/mH2`b /Q +QKKQMTH+2 1.6- r?BH2 i?2 iQT@b+?2/mH2` /Q  bHB;?iHv KQ/B}2/ 1.6- i?2 KQbi MQi#H2 //BiBQM #2BM; i?2 `mHBM;@Qmi Q7 b+?2/mH2`b i?i ?p2 /2TH2i2/ #m/;2ibX

3X9 P

- i?2 bTQ`/B+ ibF b2i

Pi = {tj} = {(Cj, Dj, Tj)} 2[X U9V- bTQ`/B+nibFnb2i

Pi %= ∅ b Qi?2`rBb2 i?2`2 `2 MQi Mv ibFb iQ b+?2/mH2X

3X8 i?2 bTQ`/B+ ibF KQ/2H

h?2 bTQ`/B+ ibF KQ/2H Bb  bKHH@#mi@BKTQ`iMi 2ti2MbBQM iQ GBm M/ Gv@ HM/Ƕb T2`BQ/B+ KQ/2H BM r?B+? 2+? ibF ?b QMHv irQ T`K2i2`b, q*1h-M/ T2`BQ/X (R9- TTX 9e@eR) h?2 bTQ`/B+ KQ/2H Bb /Bz2`2Mi 7`QK i?2 T2`BQ/B+ KQ/2H #2+mb2 Bi /2+QmTH2b i?2 /2/HBM2 7`QK i?2 T2`BQ/ (Re) r?2`2b i?2 T2`BQ/ `2KBMb b M mTT2` #QmM/ 7Q` i?2 /2/HBM2 M/ HbQ /2i2`KBM2b i?2 7`2[m2M+v Q7 ibF BMbiM+2 `2H2b2bX

h?2 bTQ`/B+ ibF KQ/2H Q7 i?Bb bvbi2K /Q2b MQi /Bz2` 7`QK i?2 bTQ`/B+ ibF KQ/2H +QKKQMHv mb2/ BM `2H@iBK2 b+?2/mHBM;,

tj = (Cj, Dj, Tj) 2[X U8V- bTQ`/B+nibF

3X8XR ibF T`K2i2`b

Ç C_j Bb i?2 q*1hX

Ç Dj Bb i?2 /2/HBM2- `2HiBp2 iQ i?2 ``BpH iBK2X

Ç T_j Bb i?2 KBMBKH BMi2`@`2H2b2 iBK2, BX2X- i?2`2 +M #2 i KQbi QM2 BMbiM+2 Q7 tj 7Q` 2+? TjX

3X8Xk ibF T`QT2`iB2b

Ç i?2 ibFb /Q MQi b?`2 Mv ibF@bT2+B}+ `2bQm`+2b Ç i?2`2 `2 MQi Mv ibF@T`2+2/2M+2 +QMbi`BMib Ç i?2`2 Bb MQi Mv +QKKmMB+iBQM #2ir22M ibFb

N a+?2/mH#BHBiv K2i?Q/, `2bTQMb2 iBK2

PM2 K2i?Q/ iQ 2Mbm`2 b+?2/mH#BHBiv Bb iQ bbmK2 i?2 rQ`bi@+b2 b+2M`BQ, BX2X- i?i i?2 #2bi@2zQ`i +Q`2 bQ7ir`2 T`Q/m+2b  KtBKmK MmK#2` Q7 K2K@ Q`v ++2bb2b- i HH iBK2b- M/ i i?2 rQ`bi iBK2b rBi? `2bT2+i iQ i?2 +`BiB+H ibFb +QKTH2iBM; #27Q`2 i?2B` /2/HBM2b Ĝ r?BH2- bBKmHiM2QmbHv- +`BiB+H ibFb `2 `2H2b2/ BM i?2 KQbi mM7pQ`#H2 rv Ĝ i?2 rv i?i +mb2b KtBKmK [m2mBM;X

NXR r

- i?2 rQ`bi@+b2 `2bTQMb2@iBK2 7Q` t

rj Bb i?2 rQ`bi@+b2 `2bTQMb2@iBK2 7Q` i?2 ibF tj, Bi Bb i?2 KtBKmK@H2M;i? iBK2 bTM 7`QK i?2 ``BpH Q7 tj- iQ Bib +QKTH2iBQMX

h?2 7QHHQrBM; i2`Kb // iQ i?2 /2Hv Q7 t<sub>j</sub>X h?2 bmK Q7 i?Qb2- THmb C<sub>j</sub>- i?2 q*1h Q7 tj- Bb 2[mH iQ i?2 rQ`bi@+b2 `2bTQMb2 iBK2,

rj = Qj+ oj+ ij+ Cj 2[X UeV- `2bTQMb2niBK2

NXRXR Q_j- [m2mBM;

Q_j Bb i?2 rQ`bi@+b2 iBK2 bT2Mi [m2mBM; 7Q` tjX q?2M  ibF Bb [m2mBM;- Bi Bb BM i?2 _1.u bii2X h?2 /2}MBiBQM Q7 [m2mBM; Bb r?2M2p2` tj +MMQi 2t2+mi2-r?BH2 bQK2 Qi?2` ibF MQi QMHv +M- #mi BbX Ub  +QMb2[m2M+2- Bi Bb MQi +QMbB/2`2/ [m2mBM; r?2M HH ibFb `2 BM+iBp2- b Bb i?2 +b2 /m`BM;  ;HQ#H b+?2/mHBM; BMi2``mTiXV h?2`2 `2 irQ UQ` i?`22V bBimiBQMb r?2M tj +M #2 [m2m2/- r?B+? KF2b mT 7Q`  iQiH [m2mBM; /2Hv 7Q`KmH Q7,

NXRXk z- i?2 `2bTQMb2 iBK2 BM ;HQ#H T2`BQ/b

h?2 rQ`bi@+b2 UH`;2biV MmK#2` Q7 ;HQ#H T2`BQ/b i?i +M i`MbTB`2 /m`BM; i?2 2tBbi2M+2 Q7  ibF Bb- rBi?Qmi `QmM/BM;,

z = r_pj 2[X U3V- T2`BQ/b

NXRXj Qg_j- ;`QmT /2/HBM2 [m2mBM;

>2`2- t<sub>j</sub>Bb [m2mBM; #2+mb2 Bi ?b  H2bb BKK2/Bi2 /2/HBM2 i?M MQi?2` ibF BM i?2 bK2 ;`QmT Ĝ  ibF i?i #2HQM;b iQ i?2 bK2 b+?2/mH2` b tjX

LQi2 i?i Dmbi #Qmi Mv ibF +M ?p2 t_jT`22KTi2/ M/fQ` [m2m2/ i bQK2 TQBMi,  ?B;?2` T`BQ`Biv Bb 2[mBpH2Mi iQ  KQ`2 BKK2/Bi2 #bQHmi2 /2/HBM2X h?Bb Bb  /vMKB+ T`QT2`iv i?i /Q2b MQi BKTHv Di ≤ Dj 7Q`  ?B;?2` T`BQ`Biv ibF ti, B7 ti rb `2H2b2/ #27Q`2 tj- i?2 #bQHmi2 /2/HBM2 Q7 ti Kv biBHH #2 KQ`2 BKK2/Bi2 i?M i?i Q7 tjX h?2 b2i hp(tj) Bb HH ibFb i?i i bQK2 TQBMi +M ?p2  ?B;?2` T`BQ`Biv i?M tjX

h?2 rQ`bi@+b2 bvbi2K [m2mBM; Bb, Qg<sub>j</sub> = ! {tj,tk}⊆Pi,tk∈hp(tj) " rj Tk # C<sub>k</sub> 2[X UNV- ;`QmTn[m2mBM;

LQi2, hp(tj) Bb 2z2+imHHv HH ibFb 2t+2Ti 7Q` tj Bib2H75 "2+mb2 Q7 i?i-i?Bb b+?2/mH#BHBiv K2i?Q/ Bb `2/m+2/ iQ i?i Q7  }t2/@T`BQ`Biv bvbi2K- QMHv rQ`b2- b ?2`2- HH ibFb rQmH/ ?p2 2[mH T`BQ`BivX q?BH2 i?2`2 2tBbi K2i?Q/b iQ +QKTmi2 i?2 `2bTQMb2 iBK2 7Q` BM/BpB/mH ibFb mM/2` 1.6- i?Qb2 `2 +QKTHB+i2/ M/ ivTB+HHv MQi rQ`i? i?2 2zQ`i b HQM; b i?2 Tm`TQb2 Bb iQ +QKTmi2 ;2M2`H b+?2/mH#BHBivX aiBHH- i?2 7Q`KmHb ?2`2 `2 BM+Hm/2/ iQ BHHmKBMi2 i?2 K2+?MB+b Q7 i?Bb BKTH2K2MiiBQMX

NXRX9 Qs

j- +`Qbb@;`QmT Ubvbi2KV /2/HBM2 [m2mBM;

h?2 ibF t_j Q7 i?2 b+?2/mH2` s<sub>i</sub> +M #2 /2Hv2/ #v  ?B;?2`@T`BQ`Biv ibF i?i #2HQM;b iQ MQi?2` b+?2/mH2`X AM i?2 rQ`bi +b2- HH Qi?2` b+?2/mH2`b rBHH