P U B L I C A T I O N I N T E R N E 1800 PARTIAL ORDER TECHNIQUES FOR DISTRIBUTED DISCRETE EVENT SYSTEMS: WHY YOU CAN T AVOID USING THEM

Transcription

1 I R I P U B L I C A T I O N I N T E R N E 1800 N o S INSTITUT DE RECHERCHE EN INFORMATIQUE ET SYSTÈMES ALÉATOIRES A PARTIAL ORDER TECHNIQUES FOR DISTRIBUTED DISCRETE EVENT SYSTEMS: WHY YOU CAN T AVOID USING THEM ERIC FABRE, ALBERT BENVENISTE ISSN I R I S A CAMPUS UNIVERSITAIRE DE BEAULIEU RENNES CEDEX - FRANCE

2

3 INSTITUT DE RECHERCHE EN INFORMATIQUE ET SYSTÈMES ALÉATOIRES Cmpus Buliu Rnns Cx Frn Tél. : (33) Fx : (33) Prtil Orr Thniqus for Distriut Disrt Evnt Systms: why you n t voi using thm * Eri Fr, Alrt Bnvnist ** Systèms ommunints Projt DistriCom Pulition intrn n 1800 My pgs Astrt: Monitoring or ignosis of lrg sl istriut Disrt Evnt Systms with synhronous ommunition is mning tsk. Ensuring tht th mthos vlop for Disrt Evnt Systms proprly sl up to suh systms is hllng. In this ppr w xplin why th us of prtil orrs nnot voi in orr to hiv this ojtiv. To support this lim, w try to push lssil thniqus (prlll omposition of utomt n lngugs) to thir limits. W fous on on-lin thniqus, whr ky iffiulty is th hoi of propr t struturs to rprsnt th st of ll runs of istriut systm. W isuss th us of prviously known struturs suh s xution trs n unfolings. W propos n ltrntiv n mor ompt t strutur ll trllis. W stuy th pprtus n to xtn th us of ths t struturs to rprsnt istriut xutions. An w show how suh t struturs n us in prforming istriut monitoring n ignosis. Th thniqus rport hr wr us in n inustril ontxt for fult mngmnt n lrm orrltion in tlommunitions ntworks. This rport srv s support for plnry rss tht ws givn y th son uthor t WODES Ky-wors: Disrt Evnt Systms, istriut systms, ignosis, prtil orrs, unfolings, fult mngmnt (Résumé :tsvp) * This rport hs n writtn s support to th plnry rss givn y th son uthor t WODES This work hs n support in prt y joint RNRT ontrts Mg n Mg2, with Frn Tlom R&D n Altl, fun y frnh Ministèr l Rhrh, n y irt ontrts with Altl. This ppr rports on xprin n joint work with Stfn Hr n Clu Jr, from IRISA. It is s on tight ooprtion n intrtion with Chistoph Dousson from Frn Tlom R&D n Armn Aghsryn from Altl. ** IRISA-INRIA, Cmpus Buliu, Rnns; surnm.nm@inri.fr Cntr Ntionl l Rhrh Sintifiqu (UMR 6074) Univrsité Rnn Ins Rnns Institut Ntionl Rhrh n Informtiqu t n Automtiqu unité rhrh Rnns

4 Thniqus orrs prtils pour ls systèms à événmnts isrts réprtis: pourquoi n put-on s y soustrir Résumé : C oumnt xpliqu pourquoi l on n put éhppr u rours ux moèls orrs prtils pour l lgorithmiqu s systèms àévénmnts isrts réprtis. Dns l présnt oumnt nous nous n tnons à l survilln t n tritons ps u ontrôl. Ls thniqus présntés ont été utilisés ns un ontxt inustril, pour l gstion réprti lrms ns ls résux téléommunitions. Mots lés : systèms àévénmnts isrts, systèms réprtis, ignosti, orrs prtils, épligs, gstion lrms

5 Prtil Orr Thniqus for Distriut DES 3 Contnts 1 Introution 4 2 Disussing rlt work Th pr-omput pproh How out istriut systms? Th lngug s pproh Unfoling s monitoring Unfolings to rprsnt sts of runs for utomt Unfoling s monitoring Bsi oprtors on ourrn nts n unfolings Ftorizingunfolings Unfoling s moulr monitoring Six si prolms Distriut monitoring Prolm n Distriut on-lin monitoring Prolm Trllis s monitoring Prolms 4 n Osrvtion ritri n trlliss Trllis s monitors Bsi oprtors on trlliss Prolmswithsomosrvtionritri Rvisiting osrvtion ritri n trlliss Ftorizing trlliss From trllis to prtil orr mol2 6 Extnsions n furthr rsrh issu5 6.1 Builing mols for lrg systms: slf-moling Proilisti tru onurrny mols Tim tru onurrny mols Dynmilly hnging systms Prolm Inomplt mols Conlusion 27 A Appnix, pplition ontxt: istriut fult mngmnt in tlommunitions ntwork9 PI n 1800

6 4 Eri Fr, Alrt Bnvnist 1 Introution Sin th pionring work y Rmg n Wonhm, th Disrt Evnt Systms (DES) ommunity hs vlop rih oy of frmworks, thniqus, n lgorithms for th suprvision of DES. Whil most uthors hv onsir suprvision of monolithi utomton or lngug, ntrliz frmworks hv n mor rntly onsir [7] [11] n [21, 32]. Whil iffrnt rhitturs hv n stui y ths uthors, th typ of sitution onsir is th following: Th systm onsir is osrv y finit st of gnts, inx y som finit inx st I. Agnti n osrv vnts ll y som sulpht L i L of th mssg lpht. Lol isions prform y th lol gnts r thn forwr to som ntrl suprvisor, whih tks th finl ision rgring osrvtion; isions mhnisms vill to th suprvisor r simpl poliis to omin th isions forwr y th lol gnts,.g., onjuntion, isjuntion, t [32]. Of ours, thr is no rson why suh ntrliz stting shoul quivlnt to th ntrliz on. Thrfor, vrious notions of ntrliz osrvility, ontrollility, n ignosility hv n propos for h prtiulr rhittur, s.g., [32]. Diing upon suh proprtis n thn om infsil [31]. Whrs ths r importnt rsults, thy fil to rss th issu of lrg systms, whr glol mol, glol stt, n somtims vn glol tim, shoul voi. Aoringly, in this work, w onsir istriut systm A with susystms A i,i I n st of snsing systms O i,i I tth to h susystm. Th gol is to prform th monitoring of A unr th following onstrints: suprvisord i is tth to h susystm; suprvisor D i os not know th glol systm mol A; it only knows lol viw of A, onsisting of A i plus som intrf informtion rlting A i to its nighors; suprvisor D i sss osrvtions m y O i ; th iffrnt suprvisors t s prs; thy n xhng mssgs with thir nighoring suprvisors; thy onur t prforming systm monitoring; no glol lok is vill, n th ommunition infrstrutur is synhronous. Fwr rsults r vill on DES monitoring tht omply with ths rquirmnts. In this ppr, w shll first try to solv this prolm in th lssil frmwork of utomt, lngugs, n thir prlll omposition; w rfr to this s th squntil frmwork sin tim n stts r glol, n runs r squns of vnts. To ount for our istriut stting, w voi mnipulting glol mols (w stik with prouts of utomt inst), n w voi mnipulting glol runs (w stik with sts of synhroniz lol runs inst). Suh typ of stuy hs, for xmpl, n prform y Su n Wonhm [28, 29, 30] y vloping istriut lgorithm involving suprvising prs tht xhng mssgs to hiv ithr lol onsistny (th prs gr on th rstritions of thir monitoring Iris

7 Prtil Orr Thniqus for Distriut DES 5 rsults to thir ommon intrfs) or glol onsistny (h pr tully omputs th lol projtion of th glol monitoring rsult). Thir lgorithm mnipults lngugs, i.., sts of runs. Similr work hs n prform inpnntly y [25] n [26]. To proprly sl up, th issu of ffiintly rprsnting ths sts of runs must rss, s [6], [14] [19], n [4]. This is th min fous of th prsnt ppr. W first invstigt th us of xution trs to rprsnt sts of runs of n utomton. W show how to rprsnt monitoring or ignosis, oth off-lin n on-lin, in trms of suh xution trs. W thn show how xution trs n ftoriz with propr notion of prout, whn th utomton itslf is prout. An w show how this prout of xution trs n omput in istriut wy, y using lif propgtion typ of lgorithm involving hoti n synhronous xhngs of mssgs twn th suprvising prs. W lso show how this n prform on-lin, whil osrvtions r ollt y th prs. Instrumntl in prforming this r som ky oprtors on xution trs, nmly: intrstion, projtion, n prout. Still, this is not ntirly stisftory: vn though xution trs r lol, thy grow xponntilly with osrvtion lngth. W thus propos to rus n ol i from ontrol, nmly trlliss of runs suh s us in th lssil ynmi progrmming or Vitri lgorithms. In trllis, th st of ll runs is rprsnt y 1/ suprimposing ommon prfixs, n 2/ mrging futurs of runs tht rh intil stts n hv intil lngth for thir pst (i.., hv xplin th sm numr of vnts in thir pst). Intrstions n prouts n sily fin for trlliss. Unfortuntly, whil prouts n intrstions n proprly fin for trlliss, projtions nnot. As onsqun, no istriut lgorithm n vlop with trlliss s ov. Th vry prolm is tht, whil fining trlliss in th lssil wy, w us glol ounting ritrion for mrging futurs of runs. Th solution is to rpl glol ountrs y multi-ountrs, i.., to hv on ountr for h pr. This is not nw i in ft, s multi-loks, lso ll vtor loks, hv n introu in th 80 s y omputr sintists [24, 20] to kp trk of onsistnt glol stts y istriut prs, in th ontxt of istriut systms. With multi-ountrs, ll th n pprtus works for trlliss (intrstion, projtion, n prout) n istriut on-lin monitoring lgorithms n vlop. Ths lgorithms us muh lss mmory thn thos using xution trs, not to spk out thos mnipulting lngugs irtly. In ft, othr vli ritri for mrging futurs n us s wll. Rquirmnts for suh ritri is tht thy projt wll onto th omponnts; this oviously hols for multi-ountrs, ut not for glol ountrs. Th ottom lin is tht th right pitur for glol run is rthr st of synhroniz lol runs, h lol run possssing its own lol ritrion for mrging futurs. This mns tht prtil orr viw of glol xutions is in rquir. Of ours, if intrnl onurrny lso xists within h iniviul susystm, thn using prtil orr stting lso within h lol trllis is rommn. W will isuss th iffiultis in oing this, s lso [17] n [4]. PI n 1800

8 6 Eri Fr, Alrt Bnvnist Th ppr is orgniz s follows. Rlt work is isuss in Stion 2; losst work to ours is tht of Su n Wonhm [28, 29, 30]. In Stion 3 w invstigt istriut ignosis in th lssil frmwork of utomt quipp with prlll omposition. Sts of runs s wll s ignoss r rprsnt s xution trs, lso ll unfolings. A istriut ignosis lgorithm is prsnt, whr th suprvisors t s prs y xhnging mssgs, synhronously. This lgorithm suffrs from n xssiv siz of t struturs whil prforming ignosis: unfolings r not ompt nough. Eri Fr propos using trllis inst, mor ompt t strutur lry onsir in th ontrol ommunity in th ontxt of ynmi progrmming n Vitri lgorithm. W invstigt th us of this mor ffiint t strutur in Stion 4. In prtiulr w xplin why propr us of it rquirs kin of prtil orr viw of istriut xutions. How to mov to full flg prtil orr viwpoint for ignosis is isuss in Stion 5. Rlt prolms tht must onsir ut r not isuss in this ppr r rifly list in Stion 6. Finlly, Appnix A rports on our pplition xprin in th ontxt of fult mngmnt in tlommunitions ntworks n srvis. No proofs r givn in this tutoril ppr, propr rfrns r givn for ths. 2 Disussing rlt work Following th lssil stting, w mol our systm for monitoring s n utomton A = l (S, L,, ), whr S is th st of stts, L is th st of lls, n s > s is th trnsition l rltion. Cll run squn of sussiv trnsitions: σ : 0 > l s1 2 > s2... n not y Σ A th st of ll runs of A. Prtition L s L = L o L u,whrl o n L u r th osrv n unosrv lls, rsptivly, n lt Proj o (σ) thvisil projtion of σ otin y rsing unosrv lls from σ, n rpling stts y ounting of osrv lls (s Fig. 2). Dnot y Σ A,o = {Proj o (σ) σ Σ A } th st of ll osrvtions of A. Thmonitor of A is n lgorithm tht omputs, for vry osrvtion O Σ A,o, th st of runs xplining O, nmly: 2.1 Th pr-omput pproh Proj 1 o (O) (1) This pproh rsss wkr vrsion of th monitor : only finl stts of runs in Proj 1 o (O) r of intrst, rthr thn th omplt runs xplining O. Th monitor tks th form of ompil lgorithm, mning tht strutur is sttilly omput (spifilly trministi utomton) tht, whn f with osrvtion O, livrs th sir solution. This is simpl n wll known. With nottions s ov, th lgorithm is: 1. Comput th invisil rh A /Lu y hiing, in A, lls longing to L u ; A /Lu is in gnrl nontrministi. Iris

9 Prtil Orr Thniqus for Distriut DES 7 2. Dtrminiz A /Lu. This is known in th DES litrtur s n osrvr n is simplifi vrsion of Lfortun s ignosr [11],.g., without oumnting th fults tht ourr. 2.2 How out istriut systms? Rll th prout of utomt (lso ll prlll omposition in DES litrtur): A 1 A 2 = (S 1 S 2,L 1 L 2,, (,1,,2 )) (2) l whr (, ) > (s 1,s 2 ) iff th utomt progrss ithr lolly (ss (i) n (iii)) or jointly (s (ii)): (i) l L 1 \ L 2 l > s 1 s 2 = (ii) l L 2 L 1 l > s 1 l > s 2 (iii) l L 2 \ L 1 s 1 = l > s 2 Nxt, following our rquirmnts, ssum tht th utomton for monitoring omposs s A = i I A i (3) whr A i = (S i,l i, i,s i,0 ), L i = L o,i L u,i, n L = L o L u, with L o = L o,i. i I Do w hv: A /Lu? = i I A i/lu,i Th nswr is ys if thr is no hin intrtion: i, j : L i L j L u = With this lst ssumption, omputing th monitor n on ntirly lolly. Howvr, this is too strong n ssumption, s th intrsting s is whn hin intrtions our twn omponnts,.g., fult fft propgtion. Without this unrlisti ssumption, w o not know how to omput th monitor in istriut wy. 2.3 Th lngug s pproh This pproh ws populriz y Su n Wonhm [29, 30]. Th i is th following : q.(1)nsolvproviσ A n O r of rsonl siz th ov uthors ll Σ A th lngug of th systm. Tking this omputtion s n tomi stp, on n rss PI n 1800

10 8 Eri Fr, Alrt Bnvnist lrg systms in th following mnnr. Assuming A omposs s in (3), w n ompos Σ A s Σ A = i I Σ Ai whr nots th prlll prout of lngugs. Givn n osrv squn O, or ttr, olltion of osrv squns (O i ) i I, on pr omponnt A i, th monitor is otin in two stps: 1. omput th st V i = Proj 1 o,i (O i) of ll runs of A i mthing lol osrvtions O i ; 2. omput th prlll prout of ths lol sts of runs V = i I V i. In prti, on is not so muh intrst in V, i.. runs of A xplining ll osrvtions, thn in th projtions of V on h omponnt A i. As mttr of ft, th lttr n omput mor ffiintly thn V y omintion of projtion, mrg n prout oprtors. This pproh os not rquir tht th intrtions twn omponnts r osrv. Th uthors istinguish lol onsistny whr th lol solutions gr on thir intrfs, n glol onsistny whr th lol solutions r projtions of th glol solution. W o not giv mor tils hr sin th pris sription of this pproh is til in th nxt stion, with mjor iffrn howvr : inst of ssuming tht (1) n fftivly solv (whih hols for smll systms only), w py ttntion to th ffiiny of th t strutur to no runs of A, n w solv (1) in istriut wy n rursivly s th numr of osrvtions inrss. 3 Unfoling s monitoring From now on, w shll onsir on-lin lgorithms, whih o not pr-omput monitors or osrvrs. Ky to this r t struturs n thniqus to mnipult sts of runs in n ffiint wy. Th simplst on is prsnt n stui first. 3.1 Unfolings to rprsnt sts of runs for utomt Th prinipl of unfolings is to rprsnt sts of runs y suprimposing ommon prfixs of thm, thus otining tr-shp t strutur. 1 Ourrn nts n Unfolings. An (S, L)-ourrn nt is tr whos rnhs n nos r ll y two finit lphts not y L n S, rsptivly. Lt A =(S, L,, ) n utomton. Its unfoling U A is th uniqu (S, L)-ourrn nt whos rnhs r ll th runs of A, h run ing rprsnt only on (U A is Iris

11 Prtil Orr Thniqus for Distriut DES 9 Figur 1: Automton A n prfix of its unfoling U A. uniqu up to n isomorphism of ll trs). Fig. 1 shows n utomton n prfix of its unfoling (unfolings r infinit s soon s utomt possss loops). By us, w lso ll runs th mximl rnhs of ny (S, L)-ourrn nt. If n utomton A =(S, L,, ) is suh tht (S, ) is tr, thn U A intifis with A; y us, w sy tht A is n unfoling. For xmpl, singl run is n unfoling. 3.2 Unfoling s monitoring Th monitor for A =(S, L,, ),L= L o L u is rfin in trms of unfolings s follows: D = f U A O (4) Th so onstrut D ontins ll runs of D tht xplin som prfix of O. W n rovr our originl finition (1) from D y pruning wy th runs of D tht o not xplin O ntirly. This is illustrt in Fig. 2. Th onstrution of D n prform inrmntlly n f on-lin, whil sussiv vnts of O r riv. Not tht th rnh (, 0) > (s3, 1) longing to D fils to xplin th postfix {r, f } of th osrvtion squn. It shoul thrfor prun to gt th sir form (1). Suh pruning n prform with ly xtly 1, i.., whn riving th vnt ll r in O. In ft, this finition lso works if O is not n osrvtion squn, ut rthr n utomton. For xmpl, O = O 1 O 2 n th utomton rprsnting th st of intrlvings of two onurrnt msurmnts y two inpnnt n non synhroniz snsors. This mns tht w n s wll onsir th s of istriut snsing. 1 Unfolings ssoit to utomt, i.., squntil mhins, r usully ll xution trs. Howvr, sin w shll onsir oth squntil n prtil orr thniqus, w prfr lling thm lry unfolings. PI n 1800

12 10 Eri Fr, Alrt Bnvnist 0 (, 0) f f r f f f f f s 3 s 3 r r 1 (, 1) r (s 3, 1) f s 3 f f 2 3 (, 2) f f (, 3) (s 3, 3) A U A U O = O U A O = U A U U O Figur 2: Computing D = U A O. 3.3 Bsi oprtors on ourrn nts n unfolings To vlop our istriut on-lin lgorithms, w will n th following oprtors on unfolings or ourrn nts: intrstions, projtions, n prouts. Ths r introu nxt. Intrstion. For V n V two (S, L)-ourrn nts, thir intrstion V V is th (S, L)-ourrn nt whos runs r th ommon runs of V n V. Projtion. Lt V n (S, L)-ourrn nt. For L L n π : S S totl surjtion from S onto som lpht S,lt Proj L,π(V) (5) th projtion of V on L, otin y pplying th following two ruls, whr th trm mximl rfrs to prtil orring y inlusion: 1. ny mximl rnh l 1 > s1 l 2 > s2 l 3 > s3...s n 1 l n > sn suh tht k =1,...,n 1,l k L n l n L, is rpl y π( ) l n > π(sn ); 2. ny mximl rnh l 1 > s1 l 2 > s2 l 3 > s3...s n 1 l n > sn Iris

13 Prtil Orr Thniqus for Distriut DES 11 suh tht k =1,...,n,l k L, is rpl y π( ). Stts tht r not onnt r rmov. If V is prfix of th unfoling U A of som prout utomton A = A A, thn w simply writ Proj A (V) inst of Proj L,π(V). Prout. Using th slf-rprouing proprty U V = V if V is n ourrn nt, w n fin notion of prout for ourrn nts s follows: V U V = f U V V whr th prout of utomt ws fin in (2). Suh prouts r ssoitiv n ommuttiv. Th runs of V U V r simply otin y synhronizing th runs of V n of V. 3.4 Ftorizing unfolings Th following rsult is instrumntl in gtting th istriut monitoring lgorithms: Thorm 1 ([14]) W r givn prout A = i I A i of utomt. 1. W hv U A = U i I U A i = U i I Proj A i (U A ) whr h Proj Ai (U A ) is (gnrlly strit) prfix of U Ai. 2. For h i I, ltv i prfix of th unfoling U Ai,nltV = f U i I V i thir prout. Thn V is prfix of U A, i.. vli st of runs for A, n s ov on hs V = U i I V i = U i I Proj A i (V) whr h Vi = f Proj Ai (V) is (gnrlly strit) prfix of V i. In ition, Vi is th miniml omposition of V oring to lphts L i in tht ny othr omposition V = U i I V i,whrv i hs lpht L i, is suh tht Vi is prfix of V i. This thorm sys tht th unfoling of A n omput s prout of unfolings, n point 2 xprsss tht ny st of runs fin y prout form tully mits miniml prout form, fin y its projtions. Thorm 1 is funmntl rsult to vlop istriut lgorithms s on unfolings. PI n 1800

14 12 Eri Fr, Alrt Bnvnist 3.5 Unfoling s moulr monitoring Now, w onsir istriut stting in whih oth th systm n its snsors r istriut: A = i I A i, L i = L i,o L i,u O = i I O i, with lpht L i,o W llow tht not ll intrtions r osrv, i.. L i L j L i,o L j,o is llow, n w lso llow tht pirs of intrting omponnts isgr on whih ll is osrv or unosrv, i.. UsingThorm1in(4)yils: L i,o L j L j,o L i is llow. D = U i I U A i O i = U i I ( UAi U O i ) This suggsts fining moulr monitoring y D mo = f (D i ) i I, whr D i = Proj Ai O i (D) (6) n th lttr stisfis U i I D i = D. 3.6 Six si prolms Th following si prolms must rss, w shll o this in th squl: Prolm 1 Comput D mo without omputing D. Prolm 2 Comput D mo y tthing suprvising pr to h sit. Prolm 3 Comput D mo on-lin n on th fly. Prolm 4 Arss synhronous istriut systms. Prolm 5 Avoi stt xplosion u to th onurrny twn n possily within th iffrnt omponnts. Prolm 6 Arss hngs in th systms ynmis. Iris

15 Prtil Orr Thniqus for Distriut DES Distriut monitoring Prolm n 2 Distriut monitoring rlis on th following funmntl rsult: Thorm 2 ([14, 15, 19] ) Lt (A i ) i=1,2,3 thr utomt suh tht (L 1 L 3 ) L 2 (A 2 sprts A 1 from A 3 ) n onsir prfix of U A1 A 2 A 3 fin y V 1 U V 2 U V 3,whrV i is som prfix of U Ai.WritProj i (.) for short inst of Proj Ai (.). Thn, th following formuls hol: Proj 2 (V 1 U V 2 U V 3 ) = Proj 2 (V 1 U V 2 ) } {{ } lol to (1,2) Proj2 (V 2 U V 3 ) } {{ } lol to (2,3) } {{ } lol to 2 (7) Proj 1 (V 1 U V 2 U V 3 ) = Proj 1 ( V 1 U Proj 2 (V 2 U V 3 ) ) } {{ } (8) } lol to (2,3) {{ } lol to (1,2) Proof: Esy if w rmmr tht th runs of V U runs of V n th runs of V. V r otin y synhronizing th Dfin th following oprtors, tth to th pir of sits (i, j) nsiti, rsptivly: Msg Vi V j = f Proj j (V j U V i ) Fus( V i, V i ) = f V i V i Noti tht th Fus oprtor gnrlizs to ny numr of mssgs. Using ths oprtors, ruls (7) n (8) rsptivly rwrit s Proj 2 (V 1 U V 2 U V 3 ) = Fus ( ) Msg V1 V 2, Msg V3 V 2 (9) Proj 1 (V 1 U V 2 U V 3 ) = Msg (Msg V3 V 2 ) V 1 (10) Lt (A i ) i I olltion of utomt. Dfin its intrtion grph s th following non irt grph: its vrtis r ll with th inis i I, n w rw rnh (i, j) iff no othr inx k I xist suh tht A k sprts A i from A j. Fig. 3 illustrts th rsulting lif propgtion lgorithm whn th intrtion grph of (A i ) i I is tr. This lgorithm rsults from sussiv pplitions of Thm. 2 with th shuling init y th inx from 1 (1st stp) to 6 (lst stp). Th rrows pit mssg propgtion, n fusion ours whn two or mor mssgs rh th sm no. At th n, fusion of ll inoming mssgs is prform t h no, whih yils th sir projtion of U i I V i on h no. PI n 1800

16 14 Eri Fr, Alrt Bnvnist Figur 3: Blif propgtion lgorithm whn th intrtion grph of (A i ) i I is tr. For th pplition to istriut monitoring, simply prform th sustitution: A i A i O i. Not tht intrtions twn omponnts my our through unosrv lls (this is in ft th intrsting s for fult ignosis). Th ov rigi n strongly synhroniz shuling is not ptl for istriut monitoring. Th following lmm hlps ovroming this: Lmm 1 ([17]) Th two mps r inrsing w.r.t. h omponnt. (V i, V j ) Msg Vi V j (V i, V i ) Fus( V i, V i ) As onsqun, hoti itrtions whr mssgs r snt synhronously to nighors, put into uffrs t rption, thn r n fus t ny tim to prpr nxt mssg, will onvrg to th sm rsult s th rigi shm of Fig. 3. Th lttr is just th shm minimizing th numr of ommunitions twn sits. Whn th intrtion grph of (A i ) i I posssss yls, thn this lgorithm n still us. At th quilirium, it yils lol onsistny in th sns of [28, 29, 30], mning tht lol monitors gr on thir intrfs. Howvr this lgorithm os not omput in gnrl lol projtions of th glol monitor Proj i (D), it only omputs som uppr pproximtion of thm, s [16]. So fr this rss Prolm n 2. Nxt, w onsir Prolm Distriut on-lin monitoring Prolm 3 W shll s tht solving th lttr n on gin y using Lmm 1. To riv on-th-fly lif propgtion, onsir th following itionl oprtor tth to sit i: Grow( O i,l i ) = f ppn to O i nw lol vnt ll l i (11) Iris

17 Prtil Orr Thniqus for Distriut DES 15 n onsir lso th following tomi oprtor otin s follows: pik nighoring no i 0 of i, notyi 1,...,i n th othr nighoring nos of i, nprform: V i := Fus( V i1, V i2,..., V in ); Msg Vi V i0 (12) Eh sit prforms on of th two oprtions (11) or (12), synhronously, in hoti wy. Thnks to Lmm 1, th rsulting hoti itrtions onvrgs to th sm vlu s for th shm shown in Fig. 3, n th lgorithm is inrmntl. S [17] for til nlysis in prtil orr ontxt. 4 Trllis s monitoring Prolms 4 n 5 So fr w sm to hv rh stisftory solution of Prolm 3. Di w rss Prolms 4 n 5? Not quit so: our solution is somhow hting. In gnrl, unfolings grow xponntilly in with with thir lngth, s Fig. 1. This oms prohiitiv whn onsiring on-th-fly lgorithms. W woul hppy with t struturs hving oun with long th prossing. Trlliss, whih hv n us for long tim in ynmi progrmming lgorithms, r goo nits for this. In this stion w isuss trllis s monitoring. Agin, w ply th sm gm y first insisting tht nothing fny shll introu. So w stik with th lssil squntil stting (utomt n thir prouts). At som point, howvr, w will s tht onsiring prtil orrs nnot voi. 4.1 Osrvtion ritri n trlliss Unfolings r simpl strutur to rprsnt sts of runs, for utomt. Howvr, whn pth of th unfoling rnhs, its snnts sprt for vr. Trlliss hv n us in ynmi progrmming (or in th populr Vitri lgorithm), y mrging, in th unfoling, futurs of iffrnt runs oring to pproprit ritri. For xmpl, mrg th finl nos of two finit runs σ n σ if: 1. Thy gin n trmint t intil stts (this first onition is mntory to nsur tht σ n σ hv intil futurs); 2. Thy r quivlnt oring to on of th following osrvtion ritri: () σ n σ possss intil lngth; 2 () σ n σ possss intil visil lngth (y not ounting silnt trnsitions); () Slt som L o L n rquir tht σ n σ stisfy Proj Lo (σ) =Proj Lo (σ ); () Assum A = i I A i n rquir tht σ n σ hv intil lngths whn rstrit to th iffrnt lol lphts L i. 2 This is th osrvtion ritrion us in ynmi progrmming or Vitri lgorithm. PI n 1800

18 16 Eri Fr, Alrt Bnvnist W now formliz th onpt of osrvtion ritrion: Dfinition 1 (osrvtion ritrion) An osrvtion ritrion θ : L L θ is prtil funtion rlting two finit lphts; θ xtns to wors s usul, n w tk th onvntion tht θ(w) =ɛ, th mpty wor, if no symol of w hs n img vi θ. Lt T grph whos nos r ll y S n rnhs r ll y L { } (whr spil symol mns th sn of ll). For θ n osrvtion ritrion, sy tht two rnhs n s init l 1 > s1 l 2 > s2 l 3 > s3...s n 1 l n > sn s init of T r θ-quivlnt iff l 1 > s 1 l 2 > s 2 l 3 > s 3...s m 1 l m > s n s init = s init,s n = s n n θ(l 1 l 2 l 3...l n )=θ(l 1l 2l 3...l m) Nottion. By us of nottion, w shll somtims writ θ(σ) inst of θ(l 1 l 2 l 3...l n ), whn l 1 l 2 l 3...l n is th wor prou y run σ, sov. Dfinition 2 Lt T irt grph whos nos r ll y S n rnhs r ll y L { }, nltθ : L L θ n osrvtion ritrion. T is n (S, L, θ)-trllis if it stisfis th following onition: ny two rnhs origint from th sm no of T n trmint t th sm no of T iff thy r θ-quivlnt. As onsqun, vry iruit of T must ll y wor whos img y θ is ɛ. Exmpls orrsponing to th ov ss () () r () L θ = {1}, Dom(θ) =L { }. () L θ = {1}, Dom(θ) =L. () L θ = L o,nθ(l) =l iff l L o, θ(l) ing othrwis unfin. () L θ = I, nθ(l) =i if l L i. For V n (S, L)-ourrn nt n θ : L L θ n osrvtion ritrion, th pir (V,θ)givs ris to trllis T (V,θ), otin y mrging xtrml stts of miniml (for inlusion) θ-quivlnt rnhs of V. For A =(S, L,, ) n utomton, n θ n osrvtion ritrion, fin T A,θ = f T (U A,θ) Trlliss r illustrt in Fig. 4, for th ov ss (), (), n (). isuss ltr. Cs () will Iris

19 Prtil Orr Thniqus for Distriut DES 17 Figur 4: Top. Lft: A; right: unfoling U A. Bottom. Lft: T () () () A ;mi: T A ;right:t A, with L o = {, }. Lls of trnsitions r omitt in th trlliss. Loops in trlliss r sh, thy orrspon to pths in th unfoling whos lls r unfin unr θ. 4.2 Trllis s monitors Th trllis s monitor for A =(S, L,, ),L= L o L u is fin s D = f T (A O),θ (13) whr th osrvtion ritrion θ is isuss nxt. Consir th following thr ltrntivs for θ: (i) Osrvtion ritrion θ : L {1} is th prtil funtion suh tht θ(l) =1ifl L o, n othrwis θ(l) is unfin osrvtion ritrion θ ounts th visil glol lngth; (ii) Osrvtion ritrion θ : L L o is th prtil funtion suh tht θ(l) =l if l L o, n othrwis θ(l) is unfin osrvtion ritrion θ rors th glol osrv squn; (iii) For A = i I A i n O = i I O i, w lso onsir th osrvtion ritrion θ : L I, whih is th prtil funtion suh tht θ(l) =i if l L o,i, n othrwis PI n 1800

20 18 Eri Fr, Alrt Bnvnist unfin osrvtion ritrion θ ounts th visil lol lngths. (This orrspons to s () of prvious stion.) Not tht osrvtion ritrion (i) is th lssil on, us in ynmi progrmming. It ws illustrt y igrm T () A of Fig. 4. Also, not tht D s fin in (13) n omput on-lin long with th roring of th osrvtion O. Compring th ov thr osrvtion ritri. Lt (O i ) i I tupl of lol osrvtion squns ollt y th iffrnt snsors. Thn, O = i I O i, thir prout, is in ft th st of ll possil intrlvings of th lol osrvtions O i. Thn, vry run of D xplins som prfix of on mong thos intrlvings. Two suh runs, σ n σ, will mrg oring to osrvtion ritrion (i) iff 1/ thy trmint t intil stts of A O n 2/ thy possss intil glol lngth. In ft, th trminl stt of σ (or σ ) ontins, s prt of its omponnts, th trminl stt of its O omponnt, whih is tupl (n i ) i I,whrn i is th lngth of osrvtion O i (s Fig. 2 for th oing of osrvtions). Thus hving intil trminl stts implis, for σ n σ, tht thy hv xplin osrvtions with qul lol lngths. Thus, lthough osrvtion ritri (i) n (ii) iffr for gnrl trlliss, thy oini for th prtiulr trlliss T (A O),θ fining monitors, us of th prsn n spil form of O. On th othr hn, sin h lol osrvtion onsists of singl squn, knowing th lngth of prfix of it ntirly trmins this prfix. Thrfor, osrvtion ritri (ii) n (iii) r gin quivlnt for us in monitoring. To summriz, osrvtion ritri (i), (ii), n (iii) iffr in gnrl, ut thy r quivlnt whn us in th ontxt of monitoring, i.., thy will rsult in intil mrgs. In th nxt stion w will s tht osrvtion ritri (ii) n (iii) yil vli luli involving intrstions, projtions, n prouts, whrs (i) won t. An w will xplin why. 4.3 Bsi oprtors on trlliss Bsi oprtors r fin nxt. Intrstion. For T n T two (S, L, θ)-trlliss, thir intrstion T T is th uniqu (S, L, θ)-trllis whos runs r th ommon runs of T n T. Prouts. Two osrvtion ritri θ : L L θ n θ : L L θ r ll omptil if θ n θ gr on L L ; in this s, fin thir join θ θ y Assuming θ n θ omptil, fin (θ θ )(l) = if l L thn θ(l) lsθ (l) S T S = f T US S (=T US U U S ) (14) Iris

21 Prtil Orr Thniqus for Distriut DES 19 whr th osrvtion ritrion us in fining T US S is θ θ. Suh prouts r ssoitiv n ommuttiv. Projtion. Projtions n fin in th sm wy for trlliss s for unfolings. Lt T n (S, L, θ)-trllis, n lt L L n π : S S totl surjtion from S onto som lpht S. Dfin th projtion Proj L,π(T ) s in (5) y pplying rul n 2 to th rnhs of T. 4.4 Prolms with som osrvtion ritri Th ov notions ris numr of iffiultis, pning on th osrvtion ritri us. Th following two prolms our whn using θ s in s (i) ov. s 0 (,s 0) s 1 (,s 1) (,s 1) s 2 (,s 2 ) (,s 2 ) (,s 1 ) s 3 s 3 s 3 (s 3,s 2 ) s 3 A A T A,θ T A A,θ θ Proj {,,,},π (T A A,θ θ ) Figur 5: Illustrting prolm with prouts n projtions of trllis. Osrvtion ritrion is y ounting th numr of non-silnt rnhs ling to th onsir vnt. Th projtion onsists in 1/ rsing th vnts not ll y,,,, n 2/ rmoving vi projtion π th prim omponnt of th stt. Th trllis strutur is not stl unr projtions if θ ounts th visil lngth, glolly (figur 5). Th lst igrm shown is otin y prforming projtion s xplin. It os not yil vli trllis, howvr, sin th two rnhs > s1 > s2 n > s2 shou not onflunt us thy hv iffrnt lngths. 3 3 W my insist living with this prolm n still us suh trlliss with thir prouts n projtions; unfortuntly, orrting this my rquir unoun ktrking of Proj {,,,},π (T A A,θ θ )inorr to rmov inorrt mrgs. PI n 1800

22 20 Eri Fr, Alrt Bnvnist s 0 (,s 0) s 1 (,s 1 ) (,s 1 ) s 2 (,s 2 ) (,s 2 ) s 3 (s 3,s 2 ) s 3 Figur 6: A prolm in pturing prfixs of runs. Digrm 1: prfix of T A,θ. Digrm 2: T A,θ. Digrm 3: tking th prout of igrm n 2. Digrm 4: projting this prout on th 1st omponnt yils fk run: > s2 > s3 whih is not prt of th 1st igrm. Projting prfixs of trllis yils fk itionl runs if θ ounts th visil lngth, glolly (figur 6). Usully, whn projting th lngug of prout utomton, prfixs of runs of th prout projt into th orrsponing prfixs of runs of th omponnts. This is not th s hr. Wht is th prolm? Th prolm with this glol osrvtion ritrion is tht it is not prsrv y projtions. This ls us to hrtriz whih r th vli osrvtion ritri to hnl istriut systms. 4.5 Rvisiting osrvtion ritri n trlliss Ronsir th sm prolm on th sm xmpl of Fig. 5, y using now osrvtion ritrion (iii) of Stion 4.2. Th rsult is shown on Fig. 7. Why is this th right solution? Th funmntl rson is tht Θ = f θ θ projts wll: if Θ(σ 1 )=Θ(σ 2 )forsompir (σ 1,σ 2 ) of runs, thn w must hv θ (Proj A (σ 1 )) = θ (Proj A (σ 2 )) n θ (Proj A (σ 1 )) = θ (Proj A (σ 2 )). W formliz this nxt y rvisiting Dfinition 1. Dfinition 3 (istriutl osrvtion ritrion) Lt L = i I L i omposition of lpht L, nlt(θ i ) i I fmily of pirwis omptil osrvtion ritri. St Θ= f i I θ i. Sy tht Θ is istriutl 4 if, for ny two wors w, w L (th Kln losur of L): Θ(w) =Θ(w ) θ i (π i (w)) = θ i (π i (w )) hols, for vry i I, whr π i : L L i is th mp onsisting in rsing th symols not longing to L i. 4 Distriutl osrvtion ritri r ll hight y E. Fr in [18]. Iris

23 Prtil Orr Thniqus for Distriut DES 21 (,s 0) (,s 0) (,s 1) (,s 1) (,s 1) (,s 1) (,s 2) (,s 2) (,s 2) (,s 1) (,s 2) (,s 2) (,s 2) (,s 1) (s 3,s 2) (s 3,s 2) (s 3,s 2) (s 3,s 2) (s 3,s 2) U A A T A A ;θ θ Figur 7: Som nw igrms r shown: U A A is th intrlving s unfoling of A A ; T A A ;θ θ is th intrlving s trllis of A A, uilt with osrvtion ritrion θ θ,whrθ n θ ount th numr of trnsitions prform y A n A, rsptivly. Not tht this osrvtion ritrion is m visil hr y simply ollting th pirs (i, j) of inis of th ompoun stts (s i,s j ) of th prout. Th prolm with osrvtion ritrion (i) of Stion 4.2 is tht it is not istriutl, whrs (ii) n (iii) r istriutl. Trlliss uilt with istriutl osrvtion ritri n ftoriz s shown nxt. 4.6 Ftorizing trlliss Thorm 3 ([19]) Lt A = i I A i prout utomton n Θ= f i I θ i orrsponing istriutl osrvtion ritrion. 1. W hv T A,Θ = α i I T Ai,θ i = α i IProj i (T A,Θ ) whr Proj i () nots th projtion on A i. 2. For h i I, ltt i prfix of th trllis unfoling T Ai,θ i,nltt = f α i I T i thir trllis prout. W hv In ition, T i T = α i I Proj i(t ) = f Proj i (T ) is th miniml omposition of T oring to lph- hs lpht L i,is ts L i in tht ny othr omposition T = α i IT i,whrt i suh tht L T i L T i. This thorm is illustrt on Fig. 8. Now, w hv ll th n pprtus for roing wht ws on for moulr unfoling s monitoring. W o not rpt this. PI n 1800

24 22 Eri Fr, Alrt Bnvnist Proj A (T A A ;θ θ ) Proj A (T A A ;θ θ ) s 0 s 0 s 1 s 1 s 1 s 2 s 2 s 2 s 2 s 3 s 3 pplying stp n 2 of projtion onto A Figur 8: Illustrting Thorm 3 on ftoriz forms. Th mi igrm shows th rsult of pplying rul n 2 fining (5), to th vli trllis T A A ;θ θ shown in Fig. 7. Applying th lst stp yils th finl rsult. Disussion. Th importnt proprty of istriutility for n osrvtion ritrion shoul not om s surpris to us. For xmpl, osrvtion ritrion (iii) is nothing ut th onpt of vtor lok introu for th nlysis of istriut systms n lgorithms in th 80 s y Mttrn [24] n Fig [20]. Using vtor loks mounts to rgring xutions of th ovrll istriut systm s tupls of synhroniz lol xutions. This is just prtil orr viw of istriut xutions, whr lol xutions r still onsir squntil. 5 From trllis to prtil orr mols In th pring stion, w hv sn tht runs of istriut systms shoul sn s prtil orrs, otin y synhronizing th squntil runs of omponnts. Now, if th omponnts of th istriut systm intrt synhronously, thn intrnl onurrny lso must xist within h omponnt. Hn, th runs of omponnt shoul thmslvs sn s prtil orrs. Thus it mks sns to onstrut vrint of unfolings or trlliss, whr runs ppr s prtil orrs. This is illustrt in Figur 9. Avntgs n iffiultis r isuss nxt. Avntgs: Prtil orr unfolings r ttr thn intrlving ons in tht thy rmov imons within th omponnt or systm onsir. This uss rution in siz. Furthrmor, whn long ut finit runs r onsir for th monitoring prolm, it my tht prtil orr unfolings prform nrly s wll s intrlving s Iris

25 Prtil Orr Thniqus for Distriut DES 23 (,s 0) (,s 2 ) (,s 1 ) (,s 2 ) (s 3,s 2 ) (,s 1 ) (,s 2 ) (s 3,s 2 ) (s 3,s 1 ) T A A ;θ θ s 0 s 0 s 1 s 1 s 1 s 2 s 2 s 3 s 2 s 2 s 3 U po A A s 3 T po A A ;θ θ s 3 Figur 9: Showing th prtil orr unfoling U po po A A n trllis T A A ;θ θ ; for omprison, w hv lft th squntil trllis T A A ;θ θ. Not tht th imon hs isppr in oth ss. trlliss; this is,.g., th s whn most mrg in th onsir trllis origint from imons in th intrlving smntis. Prtil orr trlliss r ttr thn intrlving ons in tht thy rmov imons within th omponnt or systm onsir. This uss rution in siz. Prtil orr unfolings n trlliss n quipp with notions of prout n intrstion. Diffiulty: th projtion of prtil orr unfoling or trllis n somtims not rprsnt s nothr prtil orr unfolings or trllis, s Figur 10. This figur shows PI n 1800

26 24 Eri Fr, Alrt Bnvnist A A A s 0 s 0 s 1 s 1 s 2 s 2 s 3 s 0 s 0 s 1 s 1 s 0 s 1 s 1 s 2 s 2 s s 1 s 3 1 s 2 s 1 s 2 s 2 s 3 s 2 s 3 s 2 s 3 s 2 U po A A A s 3 s 3 Proj A A (U po A A A ) Figur 10: Th figur shows istriut systm with two omponnts, writtn s (A A ) A. This mns tht th first omponnt is lry istriut systm n thrfor hs intrnl onurrny. W show on th right th prtil orr unfoling of this istriut systm. Som onflits r pit in in thik gry sh lins n som uslitis r pit in thik gry soli lins. Projting on th first omponnt shoul yil th lst igrm, hving th onflits n uslitis in it. Unfortuntly, ths nnot ptur y ourrn nt fturs, with th vill nos. An nrih strutur is n. th prolm with prtil orr unfolings, ut th sm iffiulty hols with prtil orr trlliss. Iris

27 Prtil Orr Thniqus for Distriut DES 25 Solutions whn using prtil orr unfolings. Whn using prtil orr unfolings, th iffiulty n irumvnt y on of th following mns: 1 st mtho: nhn ourrn nts with possil itionl uslitis n onflits, not rsulting from th grph strutur of th nt. This is th pproh tkn in [14, 15]. 2 n mtho: non ourrn nts n us vnt struturs inst. Evnt struturs r sts of vnts quipp irtly with uslity rltion n onflit rltion, with no us of onition nos to grphilly no onflit. This is th pproh tkn in [17]. 3 r mtho: kp ourrn nts s suh, ut voi th nhnmnt us in th 1 st mtho y xhnging mssgs in th form of so-ll intrlving struturs, s [4]. With ths moifitions, th pring thniqus for istriut monitoring with prtil orr unfolings pply. Th vlopmnt of similr thniqus for prtil orr trlliss is unr progrss. 6 Extnsions n furthr rsrh issus In this stion w rviw som furthr prolms rising from pplitions n w rw orrsponing rsrh irtions. 6.1 Builing mols for lrg systms: slf-moling As xplin in Appnix A, rlisti pplitions suh s fult mngmnt in tlommunition ntworks n srvis rquir mols of omplxity n siz fr yon wht n onstrut y hn. Thus, ny mol s lgorithm woul fil rssing suh typ of pplition unlss propr mns r foun to onstrut th mol. In som ontxts inluing th on rport in Appnix A, n utomti onstrution is possil. On pproh vlop in [1] is ll slf-moling. Its prinipl is illlustrt in Fig. 11. To onstrut mols, th following prior informtion is ssum vill: () A finit st of prototyp omponnts is vill, n ll systms onsir r otin y omposing instns of ths prototyp omponnts. In our pplition ontxt, ths prototyp omponnts r spifi y th iffrnt ntwork stnrs us (s list in th lft most ox of Fig. 11), in th form of Mng Clsss. In this ontxt, th numr of lsss for onsirtion is typilly smll ( ozn or so). In ontrst th numr of instntit omponnts in th systms my hug (from hunrs to thousns). () For h prototyp omponnt, hviorl mol is vill in on of th forms w isuss in this ppr. This is th mnul prt of th moling. It ws on,.g., y Altl, for th s of ll stnrs shown in th lft most ox of Fig. 11) [1]. PI n 1800

28 26 Eri Fr, Alrt Bnvnist stnrs: SDH, WDM OTN, GMPLS... hvior of gnri ntwork lmnts pturing rhittur (ntwork isovry) utomti lgorithm gnrtion n ploymnt utomti hviorl mol gnrtion Figur 11: Slf-moling. () Systm rhittur n utomtilly isovr. By systm rhittur w mn th strutur of th systm (list of instns n thir topology n intronntions). This ssums tht so-ll rflxiv rhitturs r us, i.., rhitturs rrying struturl mol of thmslvs. This is for xmpl th s in our ontxt, whr this trsk is rfrr to s ntwork isovry. Hving (), (), n () llows to onstrut utomtilly th systm mol (A i ) i I n vn gnrt n ploy th monitoring lgorithm utomtilly [1]. 6.2 Proilisti tru onurrny mols In rl-lif pplitions, monitoring n ignosis gnrlly yil miguous rsults. For xmpl, in rl-lif systms, multipl fults must onsir; s rsult, it is oftn possil to xplin th sm osrvtions y ithr on singl fult or two inpnnt fults. This motivts onsiring proilisti mols n vloping mximum liklihoo lgorithms. In oing this, w woul oviously lik tht nonintrting susystms r proilistilly inpnnt. Non of th lssil proilisti DES mols (Mrkov hins, Hin Mrkov Mols, Stohsti Ptri nts, stohsti utomt) hs this proprty. Smy As [2, 3] hs vlop th funmntls of tru onurrny proilisti mols. Iris

29 Prtil Orr Thniqus for Distriut DES Tim tru onurrny mols In prforming monitoring or ignosis, physil tim (vn impris) n us to filtr out som onfigurtions. Tim systms mols r n for this. Cnits r tim utomt n onurrnt or prtil orr vrsions throf [9]. 6.4 Dynmilly hnging systms Prolm 6 So fr w mntion this prolm ut i not rss it in this ppr. In ft, rssing it is th vry motivtion for onsiring run-s on-lin lgorithms in whih no ignosr is sttilly pr-omput. Mols of ynmilly hnging DES r not lssil. A vrity of thm hv n propos in th ontxt of istriut systms. Ptri nt systms [13] r systms of qutions rlting Ptri nts; ths mols llow for ynmi instntition of prfin nts. Vrints of suh mols xist in th Ptri nt littrtur. Grph Grmmrs [27] r mor powrful s thy us uniform frmwork to rprsnt oth th movmnt of tokns in nt n th rtion/ltion of trnsitions or sunts in ynmi nt. Grph Grmmrs hv n us y Hr t l. [22] for ignosis unr ynmi ronfigurtion. This sujt is still in its infny. 6.5 Inomplt mols For lrg, rl-lif systms, hving n xt mol (i.., pting ll osrv runs whil ing t th sm tim non trivil) n hrly xpt. Th kin of lgorithm th DES ommunity vlops gts stuk whn no xplntion is foun for n osrvtion. In ontrst, pttrn mthing thniqus suh s hronil rognition [12] vlop in th AI ommunity r lss pris thn th DES mol s thniqus ut o not suffr from this rwk. Lvrging th vntgs of DES mol s thniqus to pting inomplt mols is hllng tht must rss. 7 Conlusion W hv isuss ignosis of lrg ntwork systms. Our rsrh gn n rquirmnts stting wr motivt y th ontxt of our ongoing ooprtion with Altl, s rifly rport in th ppnix. Th fous of this ppr ws on on-lin istriut ignosis, whr ignosis is rport in th form of st of hin stt historis xplining th ror lrm squns. In this ontxt, ffiiny of t struturs to rprsnt sts of historis is ky issu. W hv tri to vit lst possil from th lssil stting, whr istriut systms r mol through th prlll omposition of utomt or lngugs. Our onlusion is tht, to rtin xtn, opting prtil orr viwpoint nnot voi. To th lst, istriut xutions must sn s prtil orr of intrting onurrnt squns of vnts. Of ours, opting truly onurrnt stting in whih xutions r systmtilly rprsnt s prtil orrs is lso possil. PI n 1800

30 28 Eri Fr, Alrt Bnvnist This htroox viwpoint riss numr of nonstnr rsrh issus, som of whih wr list in th prvious stion. Whil our group hs strt rssing som of ths, muh room rmins for furthr rsrh in this xiting r. Anothr importnt rmrk w lik to stt is th usfulnss of tgoril thniqus in nlysing th issus w isuss in this ppr. Not tht w hv onsir lrg vrity of t struturs to rprsnt sts of runs. For h of thm, w hv onsir th wish st of si oprtors. Gtting th sir ftoriztion proprtis n om rl nightmr if only pstrin thniqus r us s,.g., [17] for suh sitution. In ontrst, tking tgoril prsptiv [23] signifintly hlps struturing th rsrh prolms n fousing on th right proprtis for hking. It lso prvnts th rsrhr from roing vrints of hr proofs. S for instn [4, 18, 19]. Iris

31 Prtil Orr Thniqus for Distriut DES 29 A Appnix, pplition ontxt: istriut fult mngmnt in tlommunitions ntworks Th thniqus rport in this ppr wr vlop in th ontxt of ooprtion with th group of Armn Aghsryn t Altl Rsrh n Innovtion. A monstrtor hs n vlop for istriut fult ignosis n lrm orrltion within th ALMAP Altl MAngmnt Pltform. Mor rntly, n xplortory vlopmnt hs n prform y Armn Aghsryn n Eri Fr for th Optil Systms usinss ivision of Altl. Th systm onsir is shown in Fig. A.1. In this pplition, ignosis is still prform ntrlly, ut th systm for monitoring is lrly wily istriut. Dignosis ovrs oth th trnsmission systm (optil fir, optil omponnts) n th omputr quipmnt itslf. Fult propgtion ws not vry omplx ut slf-moling prov ssntil in this ontxt. Prformn of th lgorithms ws ssntil. A typil us s of istriut monitoring is illustrt if Figs. A.2 4. Fig. A.2 illustrts ross-omin mngmnt n impt nlysis. Th ntwork for monitoring is th optil ring of Pris r with its four suprvision ntrs. Whn fult is ignos, its possil impt on th srvis ploy ovr it is omput this is nothr kin of mol s lgorithm. As for th optil ring itslf, Fig. A.3 shows th systm for monitoring. It is ntwork of svrl hunrs of smll utomt ll mng ojts hving hnful of stts n intrting synhronously. Du to th ojt orint ntur of this softwr systm, h mng ojt posssss its own monitoring systm. This monitoring systm tts filurs to livr propr srvi; it rivs, from nighoring omponnts, mssgs initing filur to livr srvi n sns filur mssgs to nighors in s of inorrt funtioning. This ojt orint monitoring systm uss lrg numr of runnt lrms trvlling within th mngmnt systm n susquntly ror y th suprvisor(s). Fig. A.4 shows typil fult propgtion snrio involving oth horizontl (ross physil vis) n vrtil (ross mngmnt lyr hirrhy) propgtion. Th prolm of rognising uslly rlt lrms is ll lrm orrltion. Fig. A.5 shows how monitoring rsults r rturn to th oprtor, y proposing nit orrltions twn th thousns of lrms ror, i.., whih lrm uslly rsults from whih othr lrm. This shows y th wy tht ignosis is not nssrily formult, in rl lif pplitions, s tht of isolting spifi pr-fin fults. PI n 1800

32 1 Figur A.1: th sumrin optil tlommunition systm onsir for th tril with Altl Optil Systms usinss ivision n Altl Rsrh n Innovtion. 2 C2 LSP1.1 LSP1.2 2 C root us C1 1 LSP1 A1 LSP2.1 3 LSP2.2 MPLS Domin C 2 A LSP2 1 3 D 2 LSP3 MPLS Domin B B 3 1 LSP3.1 LSP3.2 4 impt srvis B3 A2 MPLS Domin A B1 B2 Figur A.2: filur impt nlysis.

33 3 SDH Ring Montroug Figur A.3: th SDH/SONET optil ring of th Pris r, with its four nos. Th igrm on th lft zooms on th strutur of th mngmnt softwr, n shows its Mng Ojts St Oun Aurvillirs 4 AU-AIS AU-AIS Montroug Gntilly AU-AIS AU-AIS isl isl AU-AIS AU-AIS MS-AIS isl MS-AIS TF LOS isl LOS TF Figur A.4: showing filur propgtion snrio, ross mngmnt lyrs (vrtilly) n ntwork nos (horizontlly).

34 5 Corrlt lrms Figur A.5: rturning lrm orrltion informtion to th oprtor.

35 30 Eri Fr, Alrt Bnvnist Rfrns [1] A. Aghsryn, C. Jr, J. Thoms. UML Spifition of Gnri Mol for Fult Dignosis of Tlommunition Ntworks. In Intrntionl Communition Confrn (ICT), LNCS 3124, Pgs , Fortlz, Brsil, August [2] S. As, A. Bnvnist. Brnhing Clls s Lol Stts for Evnt Struturs n Nts: Proilisti Applitions, in: FoSSCSV. Ssson (itor), 2005, vol. 3441, pp [3] S. As n A. Bnvnist. Tru-onurrny Proilisti Mols: Brnhing lls n Distriut Proilitis for Evnt Struturs. Informtion n Computtion, 204 (2), F [4] P. Bln, S. Hr, n B. König. Distriut Unfoling of Ptri Nts. Pro. of FOS- SACS 2006, LNCS 3921, pp , Springr [5] P. Broni, G. Lmprti, P. Poglino, M. Znll, Dignosis of Lrg Ativ Systms, Artifiil Intll. 110, pp , [6] A. Bnvnist, E. Fr, S. Hr, C. Jr, Dignosis of synhronous isrt vnt systms, nt unfoling pproh, IEEE Trns. on Automti Control, vol. 48, no. 5, pp , My [7] R.K. Bol, J.H. vn Shuppn, Dntrliz Filur Dignosis for Disrt Evnt Systms with Costly Communition twn Dignosrs, in Pro. 6th Int. Workshop on Disrt Evnt Systms, WODES 02, pp , [8] R.K. Bol, G. Jirovnu, Distriut Contxtul Dignosis for vry Lrg Systms, in Pro. of WODES 04, pp , [9] T. Chtin, C. Jr. Tim Suprvision of Conurrnt Systms using Symoli Unfolings of Tim Ptri Nts, in: 3r Intrntionl Confrn on Forml Molling n Anlysis of Tim Systms (FORMATS 2005), Springr Vrlg, Sptmr 2005, LNCS 3829, p [10] O. Contnt, S. Lfortun, Dignosis of Moulr Disrt Evnt Systms, in Pro. of WODES 04, pp , 2004 [11] R. Douk, S. Lfortun, D. Tnktzis, Coorint Dntrliz Protools for Filur Dignosis of Disrt Evnt Systms, J. Disrt Evnt Dynmi Systms, vol. 10(1/2), pp , [12] Christoph Dousson, Pul Gorit, Mlik Ghll: Sitution Rognition: Rprsnttion n Algorithms. IJCAI 1993: Iris

36 Prtil Orr Thniqus for Distriut DES 31 [13] R. Dvillrs n H. Klul. Solving Ptri Nt Rursions Through Finit Rprsnttion. Pro of IASTED 04. [14] E. Fr, Ftoriztion of Unfolings for Distriut Til Systms, Prt 1 : Limit Intrtion Cs, Inri rsrh rport no. 4829, April html [15] E. Fr, Ftoriztion of Unfolings for Distriut Til Systms, Prt 2 : Gnrl Cs, Inri rsrh rport no. 5186, My [16] E. Fr, Convrgn of th turo lgorithm for systms fin y lol onstrints, Iris rsrh rport no. PI 1510, [17] E. Fr, A. Bnvnist, S. Hr, C. Jr, Distriut Monitoring of Conurrnt n Asynhronous Systms, J. Disrt Evnt Dynmi Systms, spil issu, vol. 15 no. 1, pp , Mrh [18] E. Fr, Distriut ignosis s on trllis prosss, in Pro. Conf. on Dision n Control, Svill, D. 2005, pp [19] E. Fr, C. Hjiostis. A trllis notion for istriut systm ignosis with squntil smntis. In Pro. of Wo006, Ann Aror, USA, July 10-12, [20] C.J. Fig. Logil tim in istriut omputing systms. IEEE Computr 24(8), 28 33, [21] S. Gn, S. Lfortun, Distriut Dignosis Of Disrt-Evnt Systms Using Ptri Nts, in pro. 24th Int. Conf. on Applitions n Thory of Ptri Nts, LNCS 2679, pp , Jun, [22] S. Hr, A. Bnvnist, E. Fr, C. Jr. Fult Dignosis for Distriut Asynhronous Dynmilly Ronfigur Disrt Evnt Systms, in: IFAC Worl Congrss Prh 2005, [23] Sunrs M Ln. Ctgoris for th Working Mthmtiin. Springr Vrlg, [24] F. Mttrn. Virtul tim n glol stts of istriut systms, Pro. Int. Workshop on Prlll n Distriut Algorithms Bons, Frn, Ot. 1988, Cosnr, Quinton, Rynl, n Rort Es., North Holln, [25] Y. Pnol, M-O. Corir, L. Roz, A ntrliz mol-s ignosti tool for omplx systms. Int. J. on Artif. Intl. Tools, Worl Sintifi Pulishing Comp., vol. 11(3), pp , [26] W. Qiu n R. Kumr. A Nw Protool for Distriut Dignosis, 2006 Amrin Control Confrn, Minnpolis, Jun PI n 1800

37 32 Eri Fr, Alrt Bnvnist [27] G. Roznrg (.) Hnook on Grph Grmmrs n Computing y Grph Trnsformtion 1 (Fountions), Worl Sintifi, [28] R. Su, Distriut Dignosis for Disrt-Evnt Systms, PhD Thsis, Dpt. of El. n Comp. Eng., Univ. of Toronto, Jun [29] R. Su, W.M. Wonhm, J. Kurin, X. Koutsoukos, Distriut Dignosis for Qulittiv Systms, in Pro. 6th Int. Workshop on Disrt Evnt Systms, WODES 02, pp , [30] R. Su, W.M. Wonhm, Hirrhil Fult Dignosis for Disrt-Evnt Systms unr Glol Consistny, J. Disrt Evnt Dynmi Systms, vol. 16(1), pp , Jn [31] S. Tripkis. Unil Prolms of Dntrliz Osrvtion n Control. In IEEE Confrn on Dision n Control, [32] T. Yoo, S. Lfortun, A Gnrl Arhittur for Dntrliz Suprvisory Control of Disrt-Evnt Systms, J. Disrt Evnt Dynmi Systms, vol. 12(3), pp , July, Iris