Deadlock Avoidance for Free Choice Multi- Reentrant Flow lines: Critical Siphons & Critical Subsystems 1

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1 Deadlo Avodane fo Fee Choe Mul- Reenan Flow lnes: Cal Shons & Cal Subsysems P. Ballal, F. Lews, Fellow IEEE, J. Meles, Membe IEEE, J., K. Seenah Auomaon & Robos Reseah Insue, Unvesy of exas a Alngon, 73 Ja Newell Blvd. S., Fo Woh, X , USA. Insuo de Ingeneía y enología de la Unvesdad Auónoma de Cudad Juáez, Ave. del Chao 6 Ne. Juáez, Mexo. Absa hs ae has wo onbuons. Fs, we esen an analyss of deadlo avodane fo a genealzed ase of Mul Reenan Flow Lne sysems (MRF) alled he Fee Choe Mul Reenan Flow Lne sysems (FMRF). In FMRF, some ass have mulle esoue hoes; hene oung desons have o be made and uen esuls n deadlo avodane do no hold. hs analyss s based on he so-alled Cula Was (CW) of he esoues n he sysem. Fo FMRF, he well nown noons of Cal Shons and Cal Subsysems mus be genealzed and we edefne hese obes fo suh sysems. Ou seond onbuon ovdes a max fomulaon ha effenly omues he obes equed fo deadlo avodane. A MAXWIP dsahng oly s fomulaed fo deadlo avodane n FMRF sysems. Aodng o hs oly, deadlo n FMRF s avoded by lmng he wo n ogess (WIP) n he al subsysems of eah CW. A man onbuon of hs ae s a max fomulaon fo de and effen omuaon of Pe Ne obes. Index ems Deadlo Avodane, Pe nes, Dsee Even Sysems, MRF, FMRF, Inellgen Conol. Noaons DES dsee even sysems MRF mul-eenan flow lnes FMRF fee hoe mul-eenan flow lnes CW ula wa CB ula blong PN Pe Nes WIP wo n ogess I. INRODUCION R esoue assgnmen and as sequenng lay moan oles n alaons nvolvng moble weless senso newos, manufaung sysems, and ohe deson esoue sysems. Bu, he use of shaed esoues n hese dsee even sysems (DES) eaes mao oblems whle sequenng ass. If he assgnmen of he esoues s no oely made, seous oblems mgh ase. Suh oblems Suoed by ARO gan DAAD ; ARO gan ARO W9NF-5--34; NSF gan IIS-32655; NSF gan CNS-42282, Sngaoe SERC SRP gan 42228, NI Lead Use gan, exas ARP gan nlude blong and sysem deadlo [3, 9,, 7], whh ae dangeous suaons ha evenually so all he avy n he flow lne nvolved. Seveal exsng aoahes n he leaue ae desgned fo he ase of Mul Reenan Flow Lne sysems (MRF) [3, 7,, 7], whee esoues ae shaed and an efom moe han one as. Analyss of deadlo avodane s well undesood fo MRF. An moan ase n deadlo avodane saeges n MRF s he one of a Cula Wa (CW) [6] among he esoues. I has been shown n [3,, 6, 7] ha deadlo ous n MRF when blong develos n a CW. he analyss of shaed esoues beomes even hade when hee ae mulle hoes of esoues fo a gven as. hs means oung desons have o be made [4]. hese sysems ae a genealzed ase of MRF sysems alled he Fee Choe Mul-Reenan Flow Lnes (FMRF). In FMRF, moe han one esoue an efom same as n he sysem. In ohe wods, some ass n FMRF may no have edeemned esoues assgned. hs s he ase, fo nsane, n weless senso newos []. In he ase of FMRF, he nown mehods of deadlo avodane ovded fo MRF do no wo. Fo analyss, modelng and onol of DES, Pe Nes (PN) [3, 5] have been exensvely used. hough he PN famewo offes goous gound fo heoeal analyss, s vey nonvenen fo aual omuaonal analyss of he aal DES, ausng oblems of omuaonal omlexy n shedulng and deadlo avodane. PN heoy does no ovde effen omuaonal means fo omung obes suh as ula was, shons, e. Max ehnques ha exlo he PN suue an be used o allevae hs. Reenly, a max-based dsee even onolle has been oosed, ovng o be vey effen n omung PN obes and n sequenng ass n manufaung envonmens as well as senso newos [, 4, 5, 6,,, 2]. PN obes suh as Cula Was, Cal Shons, and Cal Subsysems have o be omued o avod deadlo n MRF. Due o he exsene n FMRF of deson laes, whh ae followed by wo o moe deson banhes, hese obes anno be used fo deadlo avodane hee whou edefnon, In hs ae we show how hese obes an be omued fo FMRF. he ey ssue s ha s neessay o edefne, o genealze he defnon of, al subsysems fo FMRF.

2 2 Also, we ovde a max fomulaon whh an effenly omue hese obes. hs max fomulaon allows fas and effen numeal omuaon ehnques o be aled o PN analyss. We show how a MAXWIP dsahng oly an be fomulaed fo FMRF o avod blong henomena. Unde hs oly, deadlo n FMRF an be avoded by lmng he wo n ogess (WIP) n he Cal Subsysems of eah CW. he es of he ae s oganzed as follows. In Seon II we desbe he oees ha haaeze FMRF sysems usng Pe Nes. In Seon III we show he oelaon beween ula was and suues efeed o as Cal Shons and Cal Subsysems. We genealze he noon of al shon o he ase of FMRF. Seon IV esens a max fomulaon ove an o/and algeba ha maes effen and de o omue he Pe Ne obes needed fo deadlo avodane. In Seon V we llusae he new noons by omung hem fo wo FMRF examles, one of whh adms deadlo, and one of whh does no. II. PERI NE ANALYSIS A. Pe Nes A Pe ne (PN) a bae dgah (P,, I, O), whee P s he se of laes, s a se of ansons, I s he se of nu as fom laes o ansons, and O s he se of ouus as fom ansons o laes. One an eesen I as an nu ndene max whh has I(,) f hee s an nu a fom lae o anson and O as an ouu ndene max whh has O(,) f hee s an ouu a fom anson o lae. he ndene max s defned as WO-I () Gven a node v (ehe anson o lae), we defne v as he e-se of v (se of nodes wh as o v) and v as he os-se of v (se of nodes wh as fom v). Smlaly, fo a se of nodes S {v }, defne S { v } and S { v }. he followng assumons allow one o eesen a dsee even sysem by a Pe Ne:. hee ae no mahne falues. 2. No e-emon. A esoue anno be emoved fom a ob unl s omlee. 3. Muual exluson. A sngle esoue an be used fo only one ob a a me. 4. Hold whle wang. A oess holds he esoues aleady alloaed o unl has all he esoues equed o efom a ob. B. Reenan Flow Lnes A seal ase of PN s he mulle eenan flow-lne sysem (MRF), see Fgue. Fo MRF sysems, we aon he se of laes, P J R PI PO, wh he laes n J, R, PI, PO eesenng esevely, he obs efomed, he avalably of esoues, nu of as, and ouu of odus. Eah a ah sas wh a PI-lae and emnaes wh a POlae. We denoe he se of ob laes J fo a ye as J so ha J J. Le he se of ansons along a ah be x, x 2,,x L, wh x and x L beng he nal and emnal ansons esevely. R() s he se of esoues needed by ob. Fo any esoue R defne he obs efomed by as J (). We aon he esoue se R as R s and R ns, wh R s beng he se of shaed esoues,.e. hose needed fo moe han one ob, and R ns beng he se of non-shaed esoues. hen, J ( ) f Rns and J ( ) > f Rs, wh S denong he adnaly of a se S (.e. he numbe of elemens). Moe geneal han MRF ae he fee-hoe mulleeenan flow-lne sysems (FMRF), whh have no edeemned esoue alloaon fo he obs. ha s, seveal dffeen esoues may be aable and avalable o efom a sef ob. hen, oung desons may be equed along he a ah abou whh esoues o use fo he nex ob, see Fgue 2. We fomally defne FMRF sysems as a lass of sysems sasfyng he followng oees (φ beng he emy se): Poees of FMRF. P, φ 2. on a ah, x P \ J φ and x L P \ J φ 3. J, R R : R( ) wh R ( ) 4. J, 5., J,, 6. J, J, l, φ l 7. R s φ hs means ha hee ae: () no self loos, (2) eah a ah has a well-defned begnnng and an, (3) evey ob eques only one esoue wh no wo onseuve obs usng he same esoue, (4) hee may be some obs ha an be done by dffeen esoues (.e. oung desons may have o be made), (5) hee ae no a ah loos, (6) fo any wo dsn obs on dffeen a ahs hee s no assembly,.e. wo a ahs anno mege no one, (7) hee ae shaed esoues. Aodng o oey 3, R( ) R R, wh he adnaly R ( ) ; Unde he foegong assumon, one has J ( ) J J. Poey 4 dsngushes MRF fom FMRF sysems. A FMRF sysem an have > fo some J, so ha oung desons ae needed. We all suh ob laes deson laes. In MRF, one has J. ha s, MRF ae a seal lass of FMRF. A anson x s sad o be a oseo anson of. A deson lae has mulle oseo ansons,.e. >. he esoues used by deson laes ae alled deson esoues. Fgue shows a samle MRF, whle Fgue 2 shows a samle FMRF. In Fgue he a ahs ae ndeen and nehe sl no eombne. R s a shaed esoue along a sngle a ah, and R2 s a shaed esoue beween wo a l

3 3 ahs. In Fgue 2 he a ahs sl a deson laes. he deson laes ae B and B2, eah followed by wo ansons. A hoe s equed hee o dede whh esoue (R o R2, esevely R3 o R4) o use fo he nex ob. P RA 2 R2A 3 RB 4 R3A 5 Po R4 R2 R R6 R3 P2 6 R4A 7 R2B 8 R6A 9 Po2 Fgue. MRF sysem B R B2 R3 B3 P B 2 RA 4 B2 6 R3A 8 B3 R5A Po gah n he dgah G w and an be obaned by ang unons of non-dson smle CW. Gven a CW C { }, one an aon he se of ansons as o +, whee o { x x C φ}, he se of nu ansons of wh nu as fom some ohe C, and + { x x C φ}, he se of nu ansons of wh no nu as fom any ohe C. We loosely say ha he ansons o ae n he CW C. he ob se of CW C { } s gven by J(C) U J ( ). n Paon hs as J(C) J ( C) + J ( C), whee J ( C) { J ( C), C} and o + { J ( C) +, }. J ( C) C Noe ha fo FMRF, one may have J ( C) J ( C) φ. In fa, f J (C) + s a deson lae wh C a smle CW,.e. > so ha has moe han one oseo anson, hen, J (C) + and J (C). hs s esely wha dsngushes deadlo avodane n MRF fom deadlo avodane n FMRF sysems. 3 R2A 5 7 R4A 9 R R2 R4 R5 Fgue 2. FMRF sysem M B B2 B3 a m b C. Cula Wa (CW) he followng bagound s aen fom [3,, 6]. We say esoue was fo esoue (denoed ) f he avalably of s an mmedae eequse fo he elease of,.e., φ. A wa elaon dgah s defned as G w (R, A) whee R s he se of nodes and A {a } s he se of edges wh a dawn f (.e. eah a eesens a anson n ). In G w, defne an R-ah beween and as a se of R-laes suh ha. hen s sad o wa ove an R-ah fo, denoed a, f hee s an R-ah beween and. A ula wa (CW) s a se of esoues C R, wh C >, suh ha fo any odeed a{, } C, a. A CW always onans a leas one shaed esoue. he smles CW s a se of esoues C R, suh ha fo some aoae e-labelng, one has 2... q, wh fo,, q. hs wll be efeed o as a smle ula wa. A smle ula wa s a smle u n he gah and s a CW no onanng any ohe CW. Consde a wa elaon gah G w and a CW C Gw. hen fo evey C, hee exss a leas one smle ula wa σ C suh ha σ. A CW s a songly onneed sub Pn Fgue 3. FMRF sysem Fgue 3 shows a FMRF sysem. Hee, R-B2-M-B3 s a CW C. Hee, a, m, b2 and b3 ae n J(C) and b s n J(C) +. b2 and b3 ae deson laes and B2 and B3 ae he eseve deson esoues. In FMRF sysems, due o he onsuon, he deson laes (b2 and b3 n hs examle) ae also a a of J(C) +. D. Mang, Plae Veo A lae P J R PI PO s sad o be maed when onans a oen, whh deng on he lae onanng ndaes an ongong ob, he exsene of an avalable esoue, a a n, o a odu ou. In a FMRF, he nal mang veo denoed as m assgns oens only o R and PI-laes. I s assumed ha he PO-laes ae always emy. Gven b b b3 3 2a 5 7 P, m() denoes he mang of,.e. he numbe of oens n. Gven a se of laes S, m(s) denoes he m2 M2 R2 9 2b 2 3 Pou

4 4 numbe of oens n S. A se of laes s sad o be unmaed o emy f none of s laes has any oens. Gven he se of laes P, he PN lae veo, o -veo, has dmenson of P, and one elemen oesondng o eah lae. Le he se of all laes be P,,..., }. { 2 Q hen he lae veo has Q elemens. Any se of m laes { P,2,..., m} an be eesened as a P -veo havng m enes and zeo enes ohewse. he P PN mang veo m( ) [ m( ) m( 2 ) L] N, wh he naual numbes N {,, 2...}, gves he numbe of oens n eah lae. Defne he PN anson veo, o x-veo, x o have dmenson of, and one elemen x oesondng o eah anson. Le he se of all ansons be { x, x2,... xl}. hen he anson veo has L elemens. A se of m ansons {,2,..., m} an be eesened as an x L-veo x havng m enes x and zeo enes ohewse. he well-nown PN mang anson equaon [5], m + ( ) m( ) + W x (2) + ( gves he new mang veo m ) n ems of he evous mang veo m ( ) and he ansons ha have fed aeang as s n x. A -nvaan s defned as a se of esoues and laes ha s n he null sae of W..e. W. (3) Noe ha f s a -nvaan, hen + m ( ) m( ) + W x m( ) so ha he numbe of oens n a -nvaan s onsan. One ye of -nvaan s gven by any esoue lus all of s obs, J (). E. Cula Blong A ula blong CB s a ula wa ha s emy and wll always eman so [3,, 6]. ha s, fo a CW C { }:. m(c), and 2. no oens wll eve be added o C. In hs suaon, one s sad o have deadlo, whee he esoues n he CW ae wang fo eah ohe and wll neve agan beome avalable. If a esoue on a gven a ah s nvolved n a CB, hen all downseam avy along ha a ah wll evenually. ha s, afe some me, he downseam obs on ha a ah wll neve agan be efomed. Le C {C } be a se of dson CW. hen, C s sad o be n CB f eah CW C s n CB. F. Shons he analyss of CB and deadlo an be aed ou fomally usng he noon of shon. A shon s a se of laes havng he oey ha s nu anson se s onaned n s ouu anson se.e. S S (4) A shon has he ey oey ha, one s unmaed, emans so. A mnmal shon of a CW C s he smalles shon onanng he CW. Defne a al shon fo a CW C as a smalles shon whh has he oey ha a CW s a CB f and only f he al shon s emy. I s shown n [, 2] ha a mnmal shon n MRF s a al shon. Fo MRF sysems, fo evey CW C, he se of laes S defned by S C J (C) (5) + s a mnmal shon as well as a al shon, whee J(C) + was defned eale [, 6]. In he ase of FMRF sysems, hs obe s sll a mnmal shon, bu anno be used n deadlo analyss due o deson laes. Hene s no a al shon as s now shown. Lemma : Fo FMRF sysems, S s a mnmal shon fo he CW C. Poof: o ove ha show S n S S s a shon, we need o,.e., evey anson havng an ouu lae S has an nu lae n S. I s nown ha, fo evey C, hee exss C,, suh ha φ. So, f, C R ns _ B wh B beng he deson esoues, hen, and hee exss some C,, suh ha { }. On he ohe hand, C R B ), hen fo ( s evey x }, ehe x { }, fo some C,, o { x, fo some J (C) +. Moeove, J C), { } fo some C by Poey 3. hs ( + oves ha shon. he mnmaly of Sˆ Sˆ, heefoe by onsuon, S s a S s obvous fom he way s onsued n FMRF sysems. If any J (C) + s lef ou, hen hee wll be some C Rs wh { } S. Smlaly f one of he esoues n C s no onsdeed, C s lef ou, hen hee wll be a leas one C wh { }. Boh he ases volae he defnon of shon. S Lemma 2: Le C be a smle CW and S onan a deson lae whose assoaed deson esoue s no a shaed esoue, hen emy. S onans a -nvaan and an neve be

5 5 Poof: Le C be a smle CW and S be defned by (5). Suose S onans a deson lae wh an assoaed esoue lae ha s no shaed,.e. J( ). hen C and >, so ha J(C) and J (C) +. heefoe, S. Moeove, S so ha he -nvaan J ( ) s n S. III. CRIICAL SIPHONS AND DEADLOCK ANALYSIS FOR FMRF SYSEMS Lemma shows ha S gven by (5) s ndeed a mnmal shon, even fo FMRF, bu Lemma 2 shows ha n some suaons fo FMRF, S an neve be emy. Sne deadlo an sll ou n suh suaons, s neessay o ovde an alenave fomula fo al shons fo FMRF. A. FMRF Cal Shons Le C s be he se of all smle CWs n a FMRF sysem. Le be a deson lae wh esoue lae R(). Defne he se of all he smle CWs n C s onanng he deson esoue as C { C C C} (6) s Gven a CW C, le he se of s deson laes be J de ( C) { J ( C) > } (7) Defne, C ( C) { C J ( C)} (8) de whh s he ey o deadlo analyss n FMRF sysems. We need o eusvely omue C (C), as eah ndvdual smle CW n C (C) defned n (8) may onan moe deson esoues, and so on as shown n Fgue 4. Hene we need an algohm o alulae C (C). Algohm shows a modfed veson of he bnay ee algohm [2]. C C2 C3 Fgue 4. ee eesenaon of smle CW deene hough deson laes C5 C6 Algohm : Comue C (C) fo a CW C Gven a CW C,. Calulae C (C) usng Equaon 8 2. x 3. y C (C) C4 4. If x y If any ( C C ( C)) C ( C) \ C x C (C) [ C (C) C (C x ) ] x x + y C (C) Go o 4 else x x + Go o 4 else 5. Reun C (C) Noe ha hs algohm omues he se C (C) fo a sngle CW. Also, when hs algohm emnaes, C (C)C (C (C)). hs algohm emnaes n one of he wo ways: a) J ( C ( C)) J ( C ( C)) φ o + b) φ +. In he dsussons ahead, we wll ove ha n ase (b), C (C) an neve be n CB and hene no CW n C (C) an be n CB. Defne a elaon C C, f hee exss a esoue C C. Noe ha sasfes he followng oees:. C C (Reflexvy) 2. If C C hen C C ( Symmey) 3. If C C and _ C C (No ansve) _ C C hen s no guaaneed ha heefoe, " " s no an equvalene elaon. Defne a seond elaon C C f hee exss a se of CW C suh ha C2... C C C n C. Noe ha,. C C (Reflexvy) 2. If C C hen C C 3. If C C hen (Symmey) C C (ansve) heefoe he elaon " " s an equvalene elaon [8] and aons he se of all he CW no dson equvalene _ lasses K(C) { C : C C },.e. K(C)K( C ). Coollay : Fo any smle CW C C (C), C (C) C ( C) _. Poof: he algohm meely omues he se C (C)K(C). heefoe, C (C) C ( C _ ). Lemma 3: C (C) s a CB f and only f:. φ + 2. m(c (C)) and 3. fo eah C ( C), J ( ) wh m ( ), x

6 6 J ( C ( C)) Poof: Neessy: Le C (C) be n CB.e s emy. hs means ha m(c (C)). Le and suose J ( C ( C)) +. hen ehe φ o hee exss a esoue C (C) suh ha φ. hs means ha a anson may fe and u a oen n R() so ha C (C) does no eman emy. heefoe, all he maed obs ae n J(C (C)) and no n J(C (C)) + and ondons and 3 hold. Suffeny: Condons and 3 of Lemma 3 mly ha any C (C) an ge a oen f and only f some C (C) wh, an ge a oen. Howeve by ondon 2 of Lemma 3, all he R-laes n C (C) ae emy, and hene none of hem an eve ge any oen. In ohe wods, C (C) s n CB. Lemma 4: Fo MRF, C s n CB f and only f: Suose S s no emy. hen hee s a lae suh ha. m(c (C)) and ' J ( C ( C)) + and m ( ' ). hs s a onadon. 2. fo eah C ( C), J ( ) wh m ( ), J ( C ( C)) Suffeny: S s a shon and by defnon, one s emy, wll eman so. hus, when m(c Poof: In MRF, C (C) C and ondon of Lemma 3 always (C)), wh all he oens los o some J ( holds. ), wh m() means ha Gven a CW C defne he obe, J( C ( C)) + and hene,. Sne S ( ) ( ( C)) C C J C + (9) he nex esuls show ha S s a al shon fo C fo FMRF sysems unde a ean ondon. Lemma 5: Fo any CW C, S shon fo C (C). Poof: One needs o show ha defned n (9) s a mnmal S S,.e., evey anson havng an ouu lae n S has an nu lae n S. By onsuon, S C (C),.e., he ouu laes fo hese ansons ae esoues n C (C). I s nown ha, fo evey C (C), hee exss C ( C),, suh ha φ. So, f C ( C) Rns, hee exss some C ( C),, suh ha { }. On he ohe hand, s C ( C) R, hen fo evey x }, ehe x { }, { fo some C ( C),, o x, fo some J ( C ( C)) +. Moeove, J C ( C)), { } fo ( + some C (C) by Poey 3. hs oves ha S S, Now, le hee be lae suh ha ' J ( C) and heefoe by onsuon, S s a shon. Mnmaly follows ' J ( C ( C)). In hs ase, ( ' ) C φ and by he way S s onsued. ( ' ) C ( C) φ. hs means ha C C (C), whh s a Noe, fo MRF, C (C) C, eoveed. Lemma 6: If ^ S and Lemma s S + a -nvaan and an neve be emy. φ, hen S onans Poof: If + φ, means ha hee exss a deson lae suh ha J ( C ( C)) + and J ( C ( C)). Also, by defnon of S, S. If s he oesondng esoue of, S, so he -nvaan J ( ) s also n S. Hene an neve be emy. he nex esul shows ha S s a al shon fo C (C). heoem : Gven C (C) and le φ. hen C (C) s n CB f and + only f S s emy. Poof. Neessy: If C (C) s n CB, Lemma 3 shows ha m(c (C)) and m() only fo. heoem 2: Le + φ, by Lemma 3 C (C) s n CB. + φ. hen C (C) an neve be n CB. Poof: Fo C (C) o be n CB, m(c (C)) and m() only wh J ( C ( C)). Suose also J( C ( C)) +. hs mles ( ) R C ( C). hus, hs oen an be fed. Hene, deadlo does no ou. hee exss a elaonsh beween Lemma 6 and heoem 2. When J ( C ( C)) + J ( C ( C)) φ, means S has a - nvaan and hene, C (C) an neve be n CB. Lemma 7: Gven a CW C{ }. hen () J ( C) ()If + ( C) + \ J ( C) + J φ,hen Poof: () Gven C{ }. hen by defnon C C (C). onadon. Hene, J ( C) { J ( C), C} o { J ( C ( C), C ( C)} J ( C ( C)) () In FMRF, f C s a CW wh deson lae hen, >, J (C) and also ossbly J (C) + as saed eale. Hene he se J ( C) + \ J ( C) } sly onsss of ob {

7 7 laes belongng o J(C) + whou deson laes. Le J ( C) + \ J ( C) } be denoed by J(C) s+. Now by defnon, { J(C) J ( C) + J ( C) and J(C (C))J(C (C)) + J(C (C)). By (), J ( C) J ( C ( C)). Sne C C (C), J(C) J(C (C)). heefoe, J(C)\J(C) J(C (C))\J(C (C)). he em J(C)\J(C) s equal o J(C) + \J(C).e. J(C) s+. Also, he em J(C (C))\J(C (C)) s J(C (C)) + as J ( C ( C)) + J ( C ( C)) φ. hus, J ( C) + \ J ( C) J ( C ( C)) +. he nex esuls show ha S s a al shon fo C. heoem 3: If J ( C ( C)) + J ( C ( C)) φ hen CW C s n CB f and only f C (C) s n CB. Poof: Neessy: Suose C be n CB and C (C) s no n CB. Case : If C does no onan deson laes hen CC (C) and he esul follows. Case 2: If C s n CB, hen m(c). Consde a esoue C and s ob lae suh ha s dead.e. m() fo all fuue fngs. Sne s dead, and hene, ( ) R R mus all be dead fo C o be n CB. C (C) and C ae elaed o eah ohe by he deson esoues. So, eang hs oess n a PN fom bawads o a ah of ansons and R-laes whh nlude all esoues of C (C), mus be dead. Fo C o eman n CB, no oen should be added o C and hene no oen should be added o C (C). hus, all he obs n C (C) mus be n J(C (C)). he ondon φ + ensues ha when C (C) s n CB, all he oens ae n J(C (C)) and no J(C(C)) +. Hene C s n CB. Suffeny: Le C (C) be n CB. hen m(c (C)) and hene m(c). Also by Lemma 3, all maed obs ae n J(C (C)). and no n J(C (C)) + and J ( C ( C)) J ( C ( C)) + φ. By Lemma 7 J ( C) and J(C) s+ J(C (C)) + whee J(C) s+ J ( C) + \ J ( C) }. Now, J(C) s+ J(C (C)) φ. { Hene fo all esoues n C and J ) havng m ( ), J(C). heefoe, C an ge a oen f and only f some ohe C (C) has a oen. Hene C s n CB. Coollay 2: Gven a CW C, and suose φ, hen C s n CB f and only f + S s emy. Poof: By heoem 3, C s n CB f and only f C (C) s n CB. hen heoem omlees he oof. Coollay 3: Gven ( ( C)) J C + ( φ, C (C) s n CB f and only f all he smle CWs n C (C) ae n CB. Poof: Neessy: Suose C (C) s n CB. Le hee be a smle CW C C ( ) suh ha s no n CB. I means C ehe m(c ) o oen an be added o C. Sne C C ( ), m( C ( C)) o oens an be added o C C (C). Hene, C (C) s no longe n CB. Suffeny: If all he smle CWs n C (C) ae n CB, means m(c ) and no oens wll eve be added o C. Sne C (C) U C, m(c (C)) and m(j(c (C) )) φ. Hene by defnon of CB, C (C) s n CB, ovded J ( C ( C)) J ( C ( C)) + φ. Coollay 4: Gven a CW C, and suose J ( C ( C)) J ( C ( C)) + φ, hen any CW C C (C) s n CB f C s n CB. Poof: By heoem 3, C s n CB f and only f C (C) s n CB. hen Coollay 3 omlees he oof. B. FMRF Cal Subsysems Fo bee eseve on ahevng dsahng oles wh deadlo avodane, he one of al subsysems [] s nodued. o avod deadlo, wo n ogess (WIP) mus be lmed whn ean al subsysems ha ae onsued by usng al shons, alled MAXWIP. A al subsysem fo a CW C n MRF s defned as he se of J-laes J(C). In ase of FMRF, Cal Subsysem fo a CW C an be defned f J ( C ( C)) J ( C ( C)) + φ and n hs ase s defned as he se of J-laes J(C (C)). Defne he se S S alled he suo of he bnay -nvaan ha mnmally oves S. Lemma 8: In FMRF, a smle CW C s no n deadlo f and only f one has m(j(c (C)) <m( S ). Poof: hee s CB n a sysem f and only f S s emy (by heoem ). Sne S S J ( C ( C)) s he suo of he bnay -nvaan, he oal numbe of s oens s onseved. hus, S wll be emy f and only f all he oens ae n J(C (C)).e. he al subsysem. he ondon m( S ) an be gven n ems of he nal mang of C (C), sne fo FMRF, he nal mang assgns oens only o R and PI laes. Noe ha one an now avod deadlo of a smle CW C by ensung ha he mang of s al subsysem S s less han he oal mang of he nal esoues n C (C). o be able o analyze FMRF sysems (and fo MRF sysems) and all s ossble deadlo suues, we need o denfy he se of all CWs.e smle CWs and CWs omosed of unon of nondson smle CWs (unons hough shaed esoues among smle CWs) [9, ]. Le us denoe hs se as CW. Lemma 9: hee s no CB n he FMRF sysem f fo evey C CW, he se C (C) s no n CB.

8 Poof: By heoem, when C (C) s no n CB, mles ha S s no emy. hen Lemma 8 omlees he oof. b. v s eesened by he veo v v Lemma : In FMRF, hee s no CB n he sysem f and only f fo evey C CW, one has m(j(c (C)) less han he nal 2. Le eesen a se of esoue laes. hen a. s eesened by he veo F mang of C (C). b. s eesened by he veo S hese esuls show ha we an avod deadlo by MAXWIP,.e. lmng WIP n all he al subsysems. Poof:. a: Any se of ob laes { J,2,..., m} s IV. MARIX COMPUAION OF PERI NE OBJECS PN ovde gea oal nsgh and mahemaal ehnques fo analyss, bu hey have had he defeny of no ovdng an effen omuaonal famewo fo smle omue-based analyss. We now nodue a max famewo fo omung suual obes of a PN ha oes hs defeny. Commensuae wh he aonng P J R PI PO wh J, R, PI, and PO beng he ses of ob laes, esoue laes, nu laes, and ouu laes esevely, aon he lae veo as v o wh v J, R, PI, o PO. Smlaly, aon he nu ndene max as I F F F F ]. Noe ha hee ae no nu as o [ v o ansons fom he laes n PO, so ha F o, a max of zeos. Paon he ouu ndene max as O [ S v S S S o ]. Noe ha hee ae no ouu as fom ansons o he laes n PI, so ha S. hese sub-maes ae Boolean maes havng enes of o. A. O/And Algeba Fo Comung PN Obes Gven Boolean maes A [ a ] and B [ b ], defne a logal o/and max algeba wheen addon oeaons ae elaed by logal o and mullaon oeaons by logal and. ha s, he max odu s defned by C A B wh ( a b ) ( a2 b2 ) ( a3 b3 ) L wh denong logal and and denong logal o. he max sum s defned by C A B wh a b. Noe ha hese max odus ae easly efomed usng sandad sofwae ogams nludng MALAB, e. hen one has omuaonal mehods fo omung PN obes based on he followng esul. Lemma : Max omuaon of PN e and os ses fo laes.. Le v eesen a se of ob laes. hen a. v s eesened by he veo F v v S eesened as a veo v havng m enes v and zeo enes ohewse. hen, by defnon v { } { xl I( l, ), some } x F ), some }. Moeove, { l v F v v { x l } wh xl [ Fv ) v( )] [ Fv ) v( )] x whh s equal o f and only f F v ) fo some.. b: Smlaly, by defnon, v { } {xl O(l, ),some } { l v x S ), some }. Also, v x S v x } wh { l xl [ Sv ) v( )] [ Sv ) v( )] whh s equal o f and only f S v ) fo some. 2. a: Any se of esoues laes { R,2,..., m} s eesened as a veo havng m enes and zeo enes ohewse. hen, by defnon { } { xl I( l, ), some } { l x F { x l } wh xl [ F ) ( )] [ F ) ( )] 2. b: Smlaly, by defnon, { } {xl O(l, ),some } x F ), some }. Moeove, { l x S ), some }. Also, x S x } wh { l xl [ S ) ( )] [ S ) ( )] whh s equal o f and only f S ) fo some. Lemma 2: Max omuaon of PN e and os ses fo ansons:. Le x eesen a se of ansons. hen a. x J s eesened by he veo S v x b. x R s eesened by he veo S x 8

9 9. x J s eesened by he veo x F v d. x R s eesened by he veo x F Poof: Pos ses of ansons ae ses of ob laes and esoues,.e. x { J, R}.. a. Any se of ansons { l l,2,..., n} s eesened as a veo x havng n enes x l and zeo enes ohewse. hen, by defnon x J { l } J { J O(, some l} { J S (, some l}. Moeove, S v x { } whee v S ( x( )} { S ( x } { v v l l whh s equal o f and only f Sv ( fo some l.. b. Any se of ansons { l l,2,..., n} s eesened as a veo x havng n enes x l and zeo enes ohewse. hen, by defnon x R { l } R { R O(, some l} { R S (, some l}. Moeove, S x { } whee S ( x( )} { S ( x } { l l whh s equal o f and only f S ( fo some l... Any se of ansons { l l,2,..., n} s eesened as a veo x havng n enes x l and zeo enes ohewse. hen, by defnon x J { l } J { J I(, some l} v { J F (, some l}. Moeove, F v x { } whee { v v l l F ( F ( x( )} { F ( x } whh s equal o f and only f v fo some l.. d. Any se of ansons { l l,2,..., n} s eesened as a veo x havng n enes x l and zeo enes ohewse. hen, by defnon x R { l } R { R I(, some l} { R F (, some l}. Moeove, x { } whee F ( x( )} { F ( x } F { l l F ( whh s equal o f and only f fo some l. B. Cula Was In Max Fom A wa elaon dgah fo MRF [2] s defned as W (S F ) () Eah one n he elemens w of W, eesens ha he dgah has an a fom esoue o esoue. CWs aea as loos n hs dgah. Smle CW aea as smle loos e.g. no onanng any smalle loos. W s used o omue all he smle CWs n MRF sysems usng a bnay sng algeba aoah used by [2] whh gves an ouu max CW. Eah CW s eesened as a ow n he max CW. In hs max CW eah eny of one n oson (,) means ha eah esoue s nluded n he h smle CW. Howeve, due o he omlexy of he Fee-Choe exenson of he MRF sysems, and due o he dvesy of loo ahs ha a se of esoues onaned n a smle CW mgh have, we need o denfy no only he esoues ha omose eah smle CW, bu also he ansons ha ln hem. hs wll gve us sef nfomaon needed o loae shons needed fo onsuons of ou deadlo oly fo FMRF sysems. We defne W (by dualy of W ) as W (F S ) () hs s a dgah of ansons. ha s, W s a dgah havng as fom anson o anson (by byassng a esoue.). Eah one n he elemens w of W, eesens ha he dgah has an a fom anson o anson. hen, one an denfy loos among ansons by usng sng algeba as above whh gves he ouu max CW. We loosely say s n CW f CW. Bu, even f we alulae wo ouomes fom he sng algeba, anson loos and esoue loos, we wll no be able o denfy whh se of anson loos oesond o whh se of esoue loos (due o he behavo of he algohm.). o do hs, defne, O S W (2) F O whee O s a zeo-max havng nxn elemens, O s a zeomax havng mxm elemens, n be he numbe of esoues o ows (olumn) of S (F ), and m be he numbe of ansons o ows (olumns) of F (S ). hs s a dgah of ansons and esoues R. Fo examle, hs dgah an be obaned by easng all he obs laes fom Fgue 3, and eeng he esoues, he ansons, and all he lns beween hem, as shown n Fgue 5. P n Fgue 5. Gahal eesenaon of dgah max W 6 7 R M B B 2 B 3 M 2 R P ou

10 Noe ha, usng hs dgah max W wh he bnay sng algeba algohm gven n [2], we ge he se C w [CW CW ], whee CW s he se of smle CW of esoues and CW s a se of smle CW of ansons. In ode o fnd he omlee se of smle CWs and he unon of non-dson smle CWs, we use he Guel algohm [] whh gves a max G. G ovdes he se of omosed CWs (ows) fom unons of smle CWs (olumns).e. an eny of n evey (, ) oson mles h smle CW s nluded n he h omosed CW. hen, we an alulae he se of loo esoues CW and loo ansons CW usng he followng onsuons, CW CW G CW (3) G CW (4) Noe ha denoes a smle CW, whle CW and CW efe o all he CWs (.e. smle and he unons). C. Max Algohm fo Comung C (C) Poey 4 of FMRF,.e J,, s wha dffeenaes FMRF fom MRF sysems. We aly hs oey of FMRF o omue C (C) usng maes. o do so, we mus fs omue C and J de (C) fo eah CW as equed o un Algohm. he nex algohm shows a oune whh an easly be mlemened usng ools suh as MALAB, o omue C and J de (C). Algohm 2: Calulang C usng maes fo : (CW ) f max(cw (,:)S F v )> [(,:) mles ene h ow] C (,:)CW (,:) Calulang J de (C) fo : (CW) f max(cw (,:)S F v )> J de (,:)CW (,:)S F v fo : J de fo : J de f J de (,) hen J de (,) f J de (,)> hen J de (,) Fndng nal C (C) DC S F v m fo : (CW ) f max(and(d(,:),j de (,:))) C (C)((m,:))CW (,:) mm+ Afe omung C, J de (C) and he nal C (C) usng Algohm 2, we use Algohm o omue C (C) eusvely. Noe ha Algohm only omues he se C (C) fo a CW C unde onsdeaon. hen, hs se s onveed o a sngle ow by usng unon of all he ows of C (C). he same oedue s followed fo all he ohe CWs n he sysem usng a smle fo loo o ge he omlee se C (C) fo all he CWs n he sysem. D. Cal Shons And Cal Subsysems In Max Fom In hs seon we use maes o omue he PN obes.e. Cal Shons and Cal Subsysems equed n Seon III fo deadlo analyss n FMRF sysems. C and C ae he se of nu and ouu ansons fom a CW C. One CW s onsued, we use Algohm 2 and hen Algohm o onsu C (C) fo eah C CW. Le (C) and C (C) be he se of nu and ouu C ansons fom C (C). In max fomulaon, s denoed as dc (C) and C (C) d esevely. I s omued as: dc (C) C (C) S (5) C (C) d C (C) F (6) o fnd he se of ansons CW (C (C)) beween esoues n he se C (C), we use, CW ( C ( C)) d C ( C) C ( C) d (7) he al subsysems J(C (C)) fo a gven CW ae gven by, CW ( C ( C)) F (8) he shon ob ses J(C (C)) + ae gven by, + d C ( C) ( d C ( C) C ( C) d ) F (9) Now we have all he mahney o omue usng maes he obes equed fo deadlo avodane usng heoem 3 and Lemma. E. MAXWIP Dsahng Poly Fo Deadlo Avodane In FMRF Lemmas 8, 9, and fomulae he MAXWIP dsahng oly fo deadlo avodane n FMRF sysems. Aodng o hs oly, CB n FMRF s avoded by lmng WIP n he al subsysems of eah CW C. hs oly an be mlemened as a eal-me deadlo avodane onol sheme by onollng he fng of he eedene ansons of he al subsysems. V. EXAMPLES We now llusae he new noons develoed n hs ae, as well as he owe of he max fomulaon, by omung hem fo wo examle FMRF. In he fs examle, deadlo an ou, whle a small hange n he suue yelds Examle 2, whee deadlo an neve ou. Examle : Consde Fgue 3. hs aula sysem onans hee deson esoues B, B2 and B3. Aodng o Seon IV, he elevan maes fo alulang he shon v v

11 obs and al subsysems ae F, S, F v and S v. F v s a max of obs equed o fe ansons. S v s a max of obs saed on fng of ansons. F s a max of esoues equed o fe ansons and S s a max of esoues eleased when ansons ae fed. hese maes fo he sysem n Fgue 3 ae gven by, R R2 M M2 B B2 B3 R R2 M M2 B B2 B F ' S b a 2a b2 m m2 b3 b 2b b a 2a b2 m m2 b3 b 2b F v ' S v he se of CWs n hs sysem usng he bnay sng algeba [] s gven by: R R2 M M2 B B2 B3 CW Eah ow n hs se eesens a CW e.g. he fs ow of hs se eesens he CW R-M-B-B3. Usng Algohm 2 and, he se C (C) fo hs sysem s omued as: R R2 M M2 B B2 B3 C (C) We obseve ha n hs ase, eah ow s he same, whh ndaes ha eah CW n CW s elaed by he deson esoues B2 and B3. hs s why we u wh he same esoues n eah ow of C (C). In fa, all he CWs n CW ae n he same equvalene lass C (C). he shon obs and al subsysems usng Equaons (8) and (9) ae gven by b a 2a b2 m m2 b3 b 2b J(C (C)) + b a 2a b2 m m2 b3 b 2b J(C (C)) In hs ase, φ + )) ( ( )) ( ( C C J C C J. Hene, f all he obs n J(C (C)).e. a, 2a, b2, m, m2, and b3 ae maed, hen aodng o heoem 3, C (C) s n CB and deadlo wll ou. Examle 2: Fgue 6 shows a small hange n he PN suue of Fgue 3. A new esoue R3 s added o he sysem whh efoms he as 3a,.e. R2 s no used fo 2a.

12 2 Pn Fgue 6. Case whee C J ( C ( )) + φ Hee, only he F and S maes hange. he maes F v and S v eman he same. hs s beause a new esoue s added o he sysem, bu he ass and he ansons eman he same. he addon of a new esoue esuls n an addonal olumn n he F and S maes. F R R2 R3 M M2 B B2 B S ' R R2 R3 M M2 B B2 B he oesondng CW and C (C) maes ae gven by: CW R R2 R3 M M2 B B2 B3 R R2 R3 M M2 B B2 B3 C (C) he shon obs and al subsysem usng Equaons (8) and (9) ae gven by: b a 2a b2 m m2 b3 b 2b J ( C ( C)) + b a 2a b2 m m2 b3 b 2b J ( C ( C)) J ( C ( C)) + We see ha φ. Hee, R3 C ( C) and heefoe, he as b3 J ( C ( C)) and B B2 B3 b b b3 3 a 2a 5 7 R M m m2 M2 9 R2 b 2a R3 2 3 Pou b3 J ( C ( C)) +. Hene, aodng o heoem 2, deadlo an neve ou. VI. CONCLUSIONS A heoy fo deadlo avodane n Fee Choe Mul- Reenan Flow Lnes (FMRF) s ovded ha exs nown esuls fo MRF. I s shown ha he ouene of deson laes, whh ae followed by wo o moe esoue ahs, means ha he usual noon of al shon used n MRF does no aly fo FMRF. heefoe, we defne a new noon of al shon fo FMRF ha s ed o deadlo avodane. A MAXWIP dsahng oly s fomulaed fo deadlo avodane n FMRF sysems. Aodng o hs oly, has been shown ha he wo n ogess (WIP) mus be lmed n he al subsysems o avod deadlo. A max fomulaon ha effenly omues he vaous PN obes equed fo deadlo avodane n FMRF has been ovded. hs max fomulaon allows.fas and effen numeal omuaon ehnques o be aled o PN analyss. REFERENCES [] Ballal P., Godano V., Lews F., "Deadlo fee dynam esoue assgnmen n mul-obo sysems wh mulle mssons: a max-based aoah," Po. Medeanean Conf. Conol & Auomaon, Anona, Ialy, June 26. [2] Coman,., Lesenson, C., and Rves R., Inoduon o Algohms, Pene Hall of Inda, 2. [3] Ezelea, J.; Colom, J.M.; Manez, J., A Pe ne based deadlo evenon oly fo flexble manufaung sysems, IEEE ansaon on Robos and Auomaon, Volume, Issue 2, Al 995 Page(s): [4] Godano V., Lews F., Meles J., uhano B., Coodnaon onol oly fo moble senso newos wh shaed heeogeneous esoues, Poeedngs of he IEEE Inenaonal Confeene on Conol and Auomaon, Budaes, June 25. [5] Godano V., Ballal P., Lews F., uhano B., Zhang J. B., Suevsoy onol of moble senso newos: mah fomulaon, smulaon, mlemenaon, IEEE ansaons on Sysems, Man and Cybenes, Pa B, Volume 36, Issue 4, Aug. 26 Page(s): [6] Godano V., Lews F., uhano B., Ballal P., Yeshala V., Max omuaonal famewo fo dsee even onol of weless senso newos wh some moble agens, Po. Medeanean Conf. Conol & Auomaon, Lmassol, Cyus, June 25. [7] Guel, A., Bogdan S., Lews F., Max Aoah o Deadlo- Fee Dsahng n Mul-Class Fne Buffe Flowlnes, IEEE ansaons on Auoma Conol, Volume 45, Issue, Nov 2 Page(s): [8] I. N. Hesen., os n Algeba, Blasdell Publshng Co., 964. [9] Lawley, M. Inegang Roung Flexbly and Algeba Deadlo Avodane Poles n Auomaed Manufaung Sysems, Inenaonal Jounal of Poduon Reseah, 38(3); , 2. [] Lews F., Guel A., Bogdan S., Doanal A. Pasavanu O., Analyss of deadlo and ula was usng a max model fo flexble manufaung sysems, Auomaa, vol.34, no. 9, Seembe 998.

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