Distriut Pross Disovry n Conormn Chkin Wil M.P. vn r Alst 1,2 1 Einhovn Univrsity o Thnoloy, Einhovn, Th Nthrlns 2 Qunsln Univrsity o Thnoloy, Brisn, Austrli www.vlst.om Astrt. Pross minin thniqus hv mtur ovr th lst n mor n mor orniztion strt to us this nw thnoloy. Th two most importnt typs o pross minin r pross isovry (i.., lrnin pross mol rom xmpl hvior ror in n vnt lo) n onormn hkin (i.., omprin mol hvior with osrv hvior). Pross minin is motivt y th vilility o vnt t. Howvr, s vnt los om lrr (sy tryts), prormn oms onrn. Th only wy to hnl lrr pplitions whil nsurin ptl rspons tims, is to istriut nlysis ovr ntwork o omputrs (.., multior systms, ris, n lous). This ppr provis n ovrviw o th irnt wys in whih pross minin prolms n istriut. W intiy thr typs o istriution: rplition, horizontl prtitionin o th vnt lo, n vrtil prtitionin o th vnt lo. Ths typs r isuss in th ontxt o oth prourl (.., Ptri nts) n lrtiv pross mols. Most hllnin is th horizontl prtitionin o vnt los in th ontxt o prourl mols. Thror, nw pproh to ompos Ptri nts n ssoit vnt los is prsnt. This pproh illustrts tht pross minin prolms n istriut in vrious wys. Kywors: pross minin, istriut omputin, ri omputin, pross isovry, onormn hkin, usinss pross mnmnt 1 Introution Diitl t is vrywhr in vry stor, in vry onomy, in vry orniztion, n in vry hom n will ontinu to row xponntilly [22]. Som lim tht ll o th worl s musi n stor on $600 isk riv. Howvr, spit Moor s Lw, stor sp n omputin powr nnot kp up with th rowth o vnt t. Thror, nlysis thniqus lin with i t [22] n to rsort to istriut omputin. This ppr ouss on pross minin, i.., th nlysis o prosss s on vnt t [3]. Pross minin thniqus im to isovr, monitor, n improv prosss usin vnt los. Pross minin is rltivly youn rsrh isiplin tht sits twn mhin lrnin n t minin on th on hn, n pross
nlysis n orml mthos on th othr hn. Th i o pross minin is to isovr, monitor n improv rl prosss (i.., not ssum prosss) y xtrtin knowl rom vnt los rily vill in toy s (inormtion) systms. Pross minin inlus (utomt) pross isovry (i.., xtrtin pross mols rom n vnt lo), onormn hkin (i.., monitorin vitions y omprin mol n lo), soil ntwork/orniztionl minin, utomt onstrution o simultion mols, mol xtnsion, mol rpir, s prition, n history-s rommntions. ijkl hkjil jkil khijl ijkl jikl... pross isovry skip xtr insurn onormn hkin hn ookin 4 5 xtr insurn skip xtr insurn h i 8 9 slt r in ook r 1 xtr insurn 2 onirm 3 initit hk-in 6 j hk rivr s lins 10 l supply r out 7 k hr rit r 11 Fi. 1. Exmpl illustrtin two typs o pross minin: pross isovry n onormn hkin. Fiur 1 illustrts th two most importnt typs o pross minin: pross isovry n onormn hkin. Strtin point or pross minin is n vnt lo. Eh vnt in suh lo rrs to n tivity (i.., wll-in stp in som pross) n is rlt to prtiulr s (i.., pross instn). Th vnts lonin to s r orr n n sn s on run o th pross. For xmpl, th irst s in th vnt lo shown in Fi. 1 n sri y th tr,,,,, i, j, k, l. This is th snrio whr r is ook (tivity ), xtr insurn is (tivity ), th ookin is onirm (tivity ), th hk-in pross is initit (tivity ), mor insurn is (tivity ), r is slt (tivity i), th lins is hk (tivity j), th rit r is hr (tivity k), n th r is suppli (tivity l). Th son s is sri y th tr,,,,,, h, k, j, i, l. In this snrio, th ookin ws hn two tims (tivity ) n no xtr insurn ws tkn t hk-in (tivity h). It is importnt to not tht n vnt lo ontins only xmpl
hvior, i.., w nnot ssum tht ll possil runs hv n osrv. In t, n vnt lo otn ontins only rtion o th possil hvior [3]. Pross isovry thniqus utomtilly rt mol s on th xmpl hvior sn in th vnt lo. For xmpl, s on th vnt lo shown in Fi. 1 th orrsponin Ptri nt is rt. Not tht th Ptri nt shown in Fi. 1 is in l to nrt th hvior in th vnt lo. Th mol llows or mor hvior, ut this is otn sirl s th mol shoul nrliz th osrv hvior. Whrs pross isovry onstruts mol without ny priori inormtion (othr thn th vnt lo), onormn hkin uss mol n n vnt lo s input. Th mol my hv n m y hn or isovr throuh pross isovry. For onormn hkin, th mol hvior n th osrv hvior (i.., vnt lo) r ompr. Thr r vrious pprohs to inos n quntiy onormn. For xmpl, on n msur th rtion o ss in th lo tht n nrt y th mol. In Fi. 1, ll ss it th mol prtly. Howvr, i thr woul hv n s ollowin tr,,, h, k, j, i, l, thn onormn hkin thniqus woul intiy tht in this tr tivity (th onirmtion) is missin. Givn smll vnt lo, lik th on shown in Fi. 1, nlysis is simpl. Howvr, in rlity, pross mols my hv hunrs o irnt tivitis n thr my millions o vnts n thousns o uniqu ss. In suh ss, pross minin thniqus my hv prolms to prou mninul rsults in rsonl tim. This is why w r intrst in istriut pross minin, i.., omposin hllnin pross isovry n onormn hkin prolms into smllr prolms tht n istriut ovr ntwork o omputrs. Toy, thr r mny irnt typs o istriut systms, i.., systms ompos o multipl utonomous omputtionl ntitis ommunitin throuh ntwork. Multior omputin, lustr omputin, ri omputin, lou omputin, t. ll rr to systms whr irnt rsours r us onurrntly to improv prormn n slility. Most t minin thniqus n istriut [16],.., thr r vrious thniqus or istriut lssiition, istriut lustrin, n istriut ssoition rul minin [13]. Howvr, in th ontxt o pross minin only istriut nti lorithms hv n xmin in til [15]. Yt, thr is n ovious n or istriut pross minin. This ppr xplors th irnt wys in whih pross isovry n onormn hkin prolms n istriut. W will not ous on th thnil spts (.., th typ o istriut systm to us) nor on spii minin lorithms. Inst, w systmtilly xplor th irnt wys in whih vnt los n mols n prtition. Th rminr o this ppr is orniz s ollows. First, in Stion 2, w isuss th irnt wys in whih pross minin thniqus n istriut. Bsis rplition, w in two typs o istriution: vrtil istriution n horizontl istriution. In Stion 3 w lort on th rprsnttion o vnt los n pross mols. Hr, w lso isuss th irns twn prourl mols n lrtiv mols. W us Ptri nts s typil rprsnttivs o
onvntionl prourl mols. To illustrt th us o lrtiv mols in th ontxt o istriut pross minin, w lort on th Dlr lnu [8]. Stion 4 isusss irnt wys o msurin onormn whil zoomin in on th notion o itnss. Th horizontl istriution o pross minin tsks is promisin, ut lso prtiulrly hllnin or prourl mols. Thror, w lort on prtiulr thniqu to ompos vnt los n prosss (Stion 5). Hr w us th notion o psss or Ptri nts whih nls us to split vnt los n pross mols horizontlly. Stion 6 onlus th ppr. 2 Distriut Pross Minin: An Ovrviw This stion introus som si pross minin onpts (Stion 2.1) n s on ths onpts it is shown tht vnt los n pross mols n istriut in vrious wys (Stion 2.2). 2.1 Pross Disovry n Conormn Chkin As xplin in th introution thr r two si typs o pross minin: pross isovry n onormn hkin. 3 Fiur 2 shows oth typs. pross isovry vnt lo onormn hkin pross mol inostis Fi. 2. Positionin pross minin thniqus. Pross isovry thniqus tk n vnt lo n prou pross mol in som nottion. Fiur 1 lry illustrt th si i o isovry: lrn pross mol rom xmpl trs. Conormn hkin thniqus tk n vnt lo n pross mol n ompr th osrv hvior with th mol hvior. As Fi. 2 shows 3 As sri in [3], pross minin is not limit to pross isovry n onormn hkin n lso inlus nhnmnt (.., xtnin or rpirin mols s on vnt t) n oprtionl support (on-th-ly onormn hkin, prition, n rommntion). Ths r out-o-sop or this ppr.
th pross mol my th rsult o pross isovry or m y hn. Bsilly, thr typs o onormn-rlt inostis n nrt. First o ll, thr my ovrll mtris sriin th r o onormn,.., 80% o ll ss n rply y th mol rom in to n. Son, th non-onormin hvior my hihliht in th vnt lo. Thir, th nononormin hvior my rvl y nnottin th pross mol. Not tht onormn n viw rom two nls: () th mol os not ptur th rl hvior ( th mol is wron ) n () rlity vits rom th sir mol ( th vnt lo is wron ). Th irst viwpoint is tkn whn th mol is suppos to sriptiv, i.., ptur or prit rlity. Th son viwpoint is tkn whn th mol is normtiv, i.., us to inlun or ontrol rlity. in 1 2 3 4 5 6 out Fi. 3. Exmpl illustrtin pross isovry. To urthr illustrt th notion o pross isovry onsir th xmpl shown in Fi. 3. Bs on th vnt lo shown, Ptri nt is lrn. Not tht ll trs in th vnt lo strt with tivity n n with tivity. This is lso th s in th Ptri nt (onsir ll ull irin squns strtin with tokn in pl in n nin with tokn in out). Atr, tivity n xut. Trnsition in th Ptri nt is so-ll AND-split, i.., tr xutin, oth n n xut onurrntly. Trnsition is so-ll AND-join. Atr xutin hoi is m: ithr ours n th s omplts or is xut n th stt with tokn in pl 1 is rvisit. Mny pross isovry lorithms hv n propos in litrtur [9, 10, 12, 17 19, 23, 28 30]. Most o ths lorithms hv no prolms lin with this smll xmpl. Fiur 4 illustrts onormn hkin usin th sm xmpl. Now th vnt lo ontins som trs tht r not possil orin to th pross mol shown in Fi. 4. As isuss in th ontxt o Fi. 2, thr r thr typs o inostis possil. First o ll, w n us mtris to sri th r o onormn. For xmpl, 10 o th 16 ss (i.., 62.5 prnt) in
in 1 2 3 4 5 6 out ours too otn 2 4 in 1 6 out is otn skipp 3 5 Fi. 4. Exmpl illustrtin onormn hkin. Fi. 4 r prtly ittin. Son, w n split th lo into two smllr vnt los: on onsistin o onormin ss n on onsistin o non-onormin ss. Ths los n us or urthr nlysis,.., isovr ommonlitis mon non-onormin ss usin pross isovry. Thir, w n hihliht prolms in th mol. As Fi. 4 shows, thr is prolm with tivity : orin to th mol shoul xut or n ut in th vnt lo this is not lwys th s. Thr is lso prolm with tivity : it shoul only xut tr, ut in th lo it lso pprs t othr pls. Fiurs 3 n 4 show th si i o pross minin. Not tht th xmpl is ovrsimplii. For xmpl, most vnt los ontin muh mor inormtion. In th xmpl lo n vnt is ully sri y n tivity nm. Howvr, otn thr is itionl inormtion out n vnt suh s th rsour (i.., prson or vi) xutin or inititin th tivity, th timstmp o th vnt, or t lmnts ror with th vnt (.., th siz o n orr). Th pross mols shown thus r r ll Ptri nts (WF-nts [1, 6] to pris). Dirnt pross minin lorithms my us irnt rprsnttions.
Morovr, th nottion us to visuliz th rsult my vry irnt rom th rprsnttion us urin th tul isovry pross. All minstrm BPM nottions (Ptri nts, EPCs, BPMN, YAWL, UML tivity irms, t.) n us to show isovr prosss [3, 31]. In t, ltr w will lso lort on so-ll lrtiv pross mols. Howvr, to xplin th onpt o istriut pross minin, suh irns r lss rlvnt. Thror, w r isussion on th irn twn prourl mols n lrtiv mols to Stion 3.4. 2.2 Distriutin Evnt Los n Pross Mols Nw omputin prims suh s lou omputin, ri omputin, lustr omputin, t. hv mr to prorm rsour-intnsiv IT tsks. Morn omputrs (vn lowr-n lptops n hih-n phons) hv multipl prossor ors. Thror, th istriution o omputin-intnsiv tsks, lik pross minin on i t, is omin mor importnt. At th sm tim, thr is n xponntilly rowin torrnt o vnt t. MGI stimts tht ntrpriss lolly stor mor thn 7 xyts o nw t on isk rivs in 2010, whil onsumrs stor mor thn 6 xyts o nw t on vis suh s PCs n notooks [22]. A rnt stuy in Sin susts tht th totl lol stor pity inrs rom 2.6 xyts in 1986 to 295 xyts in 2007 [20]. Ths stuis illustrt th rowin potntil o pross minin. Givn ths osrvtions, it is intrstin to vlop thniqus or istriut pross minin. In rnt yrs, istriut t minin thniqus hv n vlop n orrsponin inrstruturs hv n rliz [16]. Ths thniqus typilly prtition th input t ovr multipl omputin nos. Eh o th nos omputs lol mol n ths lol mols r rt into n ovrll mol. In [15], w show tht it is irly sy to istriut nti pross minin lorithms. In this ppr (i.., [15]), w rplit th ntir lo suh tht h no hs opy o ll input t. Eh no runs th sm nti lorithm, uss th whol vnt lo, ut, u to rnomiztion, works with irnt iniviuls (i.., pross mols). Prioilly, th st iniviuls r xhn twn nos. It is lso possil to prtition th input t (i.., th vnt lo) ovr ll nos. Exprimntl rsults show tht istriut nti pross minin siniintly sps-up th isovry pross. This mks sns us th itnss tst is most tim-onsumin. Howvr, iniviul itnss tsts r ompltly inpnnt. Althouh nti pross minin lorithms n istriut sily, thy r not usl or lr n omplx t sts. Othr pross minin lorithms tn to outprorm nti lorithms [3]. Thror, w lso n to onsir th istriution o othr pross minin thniqus. To isuss th irnt wys o istriutin pross minin thniqus w pproh th prolm rom th viwpoint o th vnt lo. W onsir thr si typs o istriution:
Rplition. I th pross minin lorithm is non-trministi, thn th sm tsk n xut on ll nos n in th n th st rsult n tkn. In this s, th vnt lo n simply rplit, i.., ll nos hv opy o th whol vnt lo. Vrtil prtitionin. Evnt los r ompos o ss. Thr my thousns or vn millions o ss. Ths n istriut ovr th nos in th ntwork, i.., h s is ssin to on omputin no. All nos work on sust o th whol lo n in th n th rsults n to mr. Horizontl prtitionin. Css r ompos o multipl vnts. Thror, w n lso prtition ss, i.., prt o s is nlyz on on no whrs nothr prt o th sm s is nlyz on nothr no. In prinipl, h no ns to onsir ll ss. Howvr, th ttntion o on omputin no is limit to prtiulr sust o vnts pr s. O ours it is possil to omin th thr typs o istriution. Fi. 5. Prtitionin th vnt lo vrtilly: ss r istriut ritrrily. Fiur 5 illustrts th vrtil prtitionin o n vnt lo usin our runnin xmpl. Th oriinl vnt lo ontin 16 ss. Assumin tht thr r two omputin nos, w n prtition th ss ovr ths two nos. Eh s rsis in xtly on lotion, i.., th nos oprt on isjoint sulos. Eh no omputs pross minin rsult or sulo n in th n th rsults r mr. Dpnin on th typ o pross minin rsult, mrin my simpl or omplx. For xmpl, it w r intrst in th prnt o ittin ss it is sy to omput th ovrll prnt. Suppos thr r n nos n h no i {1... n} rports on th numr o ittin ss (x i ) n non-ittin ss (y i ) in th sulo. Th rtion o ittin ss n omput sily: ( i x i )/( i x i + y i ). Whn h no prous pross mol, it is mor iiult to prou n ovrll rsult. Howvr, y usin lowr-lvl output suh
s th pnny mtris us y minin lorithms lik th huristi minr n uzzy minr [3], on n mr th rsults. Fi. 6. Prtitionin th vnt lo vrtilly: ss r istriut s on prtiulr tur (in this s th lnth o th s). In Fi. 5 th ss r prtition ovr th los without onsirin prtiulr turs, i.., th irst iht ss r ssin to th irst no n th rminin iht ss r ssin to th son no. As Fi. 6 shows, on n lso istriut ss s on prtiulr tur. In this s ll ss o lnth 6 r mov to th irst no, ss o lnth 11 r mov to th son no, n ss o lnth 16 r mov to th thir no. Vrious turs n us,.., th typ o ustomr (on no nlyzs th pross or ol ustomrs, on or silvr ustomrs, t.), th low tim o th s, th strt tim o th s, th montry vlu o th s, t. Suh vrtil prtitionin my provi itionl insihts. An xmpl is th us o th strt tim o ss whn istriutin th vnt lo. Now it is intrstin to s whthr thr r siniint irns twn th rsults. Th trm onpt rit rrs to th sitution in whih th pross is hnin whil in nlyz [14]. For instn, in th innin o th vnt lo two tivitis my onurrnt whrs ltr in th lo ths tivitis om squntil. Prosss my hn u to prioi/ssonl hns (.., in Dmr thr is mor mn or on Friy trnoon thr r wr mploys vill ) or u to hnin onitions (.., th mrkt is ttin mor omptitiv ). A vrtil prtitionin s on th strt tim o ss my rvl onpt rit or th intiition o prios with svr onormn prolms. Fiur 7 illustrts th horizontl prtitionin o vnt los. Th irst sulo ontins ll vnts tht orrspon to tivitis,,,, n. Th son sulo ontins ll vnts tht orrspon to tivitis,,, n. Not tht h s pprs in h o th sulos. Howvr, h sulo ontins only sltion o vnts pr s. In othr wors, vnts r prtition horizontlly
Fi. 7. Prtitionin th vnt lo horizontlly. inst o vrtilly. Eh no omputs rsults or prtiulr sulo. In th n, ll rsults r mr. Fiur 8 shows n xmpl o two pross rmnts isovr y two irnt nos. Th pross rmnts r lu tothr usin th ommon vnts. In Stion 5 w will urthr lort on this. in 1 2 4 6 out 3 5 Fi. 8. Horizontlly prtition vnt los r us to isovr pross rmnts tht n mr into omplt mol. 3 Rprsnttion o Evnt los n Pross Mols Thus r, w hv only isuss thins inormlly. In this stion, w ormliz som o th notions introu or. For xmpl, w ormliz th notion o n vnt lo n provi som Ptri nt sis. Morovr, w show n xmpl o lrtiv lnu (Dlr [8]) roun in LTL.
3.1 Multists Multists r us to rprsnt th stt o Ptri nt n to sri vnt los whr th sm tr my ppr multipl tims. B(A) is th st o ll multists ovr som st A. For som multist B(A), () nots th numr o tims lmnt A pprs in. Som xmpls: 1 = [ ], 2 = [x, x, y], 3 = [x, y, z], 4 = [x, x, y, x, y, z], 5 = [x 3, y 2, z] r multists ovr A = {x, y, z}. 1 is th mpty multist, 2 n 3 oth onsist o thr lmnts, n 4 = 5, i.., th orrin o lmnts is irrlvnt n mor ompt nottion my us or rptin lmnts. Th stnr st oprtors n xtn to multists,.., x 2, 2 3 = 4, 5 \ 2 = 3, 5 = 6, t. { } nots th st with ll lmnts or whih () 1. [() ] nots th multist whr lmnt () pprs x (x)=() (x) tims. 3.2 Evnt Los As init rlir, vnt los srv s th strtin point or pross minin. An vnt lo is multist o trs. Eh tr sris th li-yl o prtiulr s (i.., pross instn) in trms o th tivitis xut. Dinition 1 (Tr, Evnt Lo). Lt A st o tivitis. A tr σ A is squn o tivitis. L B(A ) is n vnt lo, i.., multist o trs. An vnt lo is multist o trs us thr n multipl ss hvin th sm tr. In this simpl inition o n vnt lo, n vnt rrs to just n tivity. Otn vnt los my stor itionl inormtion out vnts. For xmpl, mny pross minin thniqus us xtr inormtion suh s th rsour (i.., prson or vi) xutin or inititin th tivity, th timstmp o th vnt, or t lmnts ror with th vnt. In this ppr, w strt rom suh inormtion. Howvr, th rsults prsnt in this ppr n sily xtn to vnt los with mor inormtion. An xmpl lo is L 1 = [,,,,, 30,,,,,, 20,,,,,,,,,,, 5,,,,,,,,,,, 3,,,,,,,,,,, 2 ]. L 1 ontins inormtion out 60 ss,.., 30 ss ollow tr,,,,,. Dinition 2 (Projtion). Lt A st n X A sust. X A X is projtion untion n is in rursivly: () X = n () or σ A n A: (σ; ) X = σ X i X n (σ; ) X = σ X ; i X. Th projtion untion is nrliz to vnt los, i.., or som vnt lo L B(A ) n st X A: L X = [σ X σ L]. For vnt lo L 1 in rlir: L 1 {,,} = [, 50,,, 10 ].
3.3 Prourl Mols A wi vrity o pross molin lnus r us in th ontxt o pross minin,.., Ptri nts, EPCs, C-nts, BPMN, YAWL, n UML tivity irms [3, 31]. Most o ths lnus r prourl lnus (lso rrr to s imprtiv lnus). In this ppr, w us Ptri nts s typil rprsnttiv o suh lnus. Howvr, th is n sily pt to it othr lnus. Ltr w will ormliz slt istriution onpts in trms o Ptri nts. Thror, w introu som stnr nottions. Dinition 3 (Ptri Nt). A Ptri nt is tupl PN = (P, T, F ) with P th st o pls, T th st o trnsitions, n F (P T ) (T P ) th low rltion. Fiur 9 shows n xmpl Ptri nt. Th stt o Ptri nt, ll mrkin, is multist o pls initin how mny tokns h pl ontins. [in] is th initil mrkin shown in Fi. 9. Anothr potntil mrkin is [2 10, 3 5, 5 5 ]. This is th stt with tn tokns in 2, iv tokns in 3, n iv tokns in 5. 2 4 in 1 6 out 3 5 Fi. 9. A Ptri nt PN = (P, T, F ) with P = {in, 1, 2, 3, 4, 5, 6, out}, T = {,,,,,, }, n F = {(in, ), (, 1), (1, ),..., (, out)}. Dinition 4 (Mrkin). Lt PN = (P, T, F ) Ptri nt. A mrkin M is multist o pls, i.., M B(P ). As usul w in th prst n postst o no (pl or trnsition) in th Ptri nt rph. For ny x P T, x = {y (y, x) F } (input nos) n x = {y (x, y) F } (output nos). A trnsition t T is nl in mrkin M, not s M[t, i h o its input pls t ontins t lst on tokn. Consir th Ptri nt in Fi. 9 with M = [3, 4]: M[ us oth input pls r mrk. An nl trnsition t my ir, i.., on tokn is rmov rom h o th input pls t n on tokn is prou or h o th output pls t. Formlly: M = (M \ t) t is th mrkin rsultin rom irin nl trnsition t in mrkin M. M[t M nots tht t is nl in M n irin t rsults in mrkin M. For xmpl, [in][ [1] n [1][ [2, 3] or th nt in Fi. 9.
Lt σ = t 1, t 2,..., t n T squn o trnsitions. M[σ M nots tht thr is st o mrkins M 0, M 1,..., M n suh tht M 0 = M, M n = M, n M i [t i+1 M i+1 or 0 i < n. A mrkin M is rhl rom M i thr xists σ suh tht M[σ M. For xmpl, [in][σ [out] or σ =,,,,,. Dinition 5 (Ll Ptri Nt). A ll Ptri nt PN = (P, T, F, T v ) is Ptri nt (P, T, F ) with visil lls T v T. Lt σ v = t 1, t 2,..., t n T v squn o visil trnsitions. M[σ v M i n only i thr is squn σ T suh tht M[σ M n th projtion o σ on T v yils σ v (i.., σ v = σ Tv ). I w ssum T v = {,,, } or th Ptri nt in Fi. 9, thn [in][σ v [out] or σ v =,,,,,, (i..,,, n r invisil). In th ontxt o pross minin, w lwys onsir prosss tht strt in n initil stt n n in wll-in n stt. For xmpl, ivn th nt in Fi. 9 w r intrst in irin squns strtin in M i = [in] n nin in M o = [out]. Thror, w in th notion o systm nt. Dinition 6 (Systm Nt). A systm nt is triplt SN = (PN, M i, M o ) whr PN = (P, T, F, T v ) is Ptri nt with visil lls T v, M i B(P ) is th initil mrkin, n M o B(P ) is th inl mrkin. Givn systm nt, τ(sn ) is th st o ll possil visil ull trs, i.., irin squns strtin in M i n nin in M o projt onto th st o visil trnsitions. Dinition 7 (Trs). Lt SN = (PN, M i, M o ) systm nt. τ(sn ) = {σ v M i [σ v M o } is th st o visil trs strtin in M i n nin in M o. I w ssum T v = {,, } or th Ptri nt in Fi. 9, thn τ(sn ) = {,,,,,,,,,,,,...}. Th Ptri nt in Fi. 9 hs sint sour pl (in), sint sour pl (out), n ll nos r on pth rom in to out. Suh nts r ll WF-nts [1, 6]. Dinition 8 (WF-nt). WF = (PN, in, T i, out, T o ) is worklow nt (WFnt) i PN = (P, T, F, T v ) is ll Ptri nt, in P is sour pl suh tht in = n in = T i, out P is sink pl suh tht out = n out = T o, T i T v is th st o initil trnsitions n T i = {in}, T o T v is th st o inl trnsitions n T o = {out}, n ll nos r on som pth rom sour pl in to sink pl out. WF-nts r otn us in th ontxt o usinss pross molin n pross minin. Compr to th stnr inition o WF-nts [1, 6] w th rquirmnt tht th initil n inl trnsitions n to visil. A WF-nt WF = (PN, in, T i, out, T o ) ins th systm SN = (PN, M i, M o ) with M i = [in] n M o = [out]. Illy WF-nts r lso soun, i.., r o
loks, livloks, n othr nomlis [1, 6]. Formlly, this mns tht it is possil to rh M o rom ny stt rhl rom M i. Pross mols isovr usin xistin pross minin thniqus my unsoun. Thror, w nnot ssum/rquir ll WF-nts to soun. 3.4 Dlrtiv Mols Prourl pross mols (lik Ptri nts) tk n insi-to-outsi pproh, i.., ll xution ltrntivs n to spii xpliitly n nw ltrntivs must xpliitly to th mol. Dlrtiv mols us n outsi-toinsi pproh: nythin is possil unlss xpliitly orin. Dlrtiv mols r prtiulrly usul or onormn hkin. Thror, w lort on Dlr. Dlr is oth lnu (in t mily o lnus) n ully untionl WFM systm [8, 24]. non o-xistn: tivitis n nnot hppn oth skip xtr insurn xtr insurn rspons ook r prn: vry ourrn o ns to pr y xtr insurn onirm rspons: vry ourrn o shoul vntully ollow y h h skip xtr insurn prn non o-xistn Fi. 10. Exmpl o Dlr mol onsistin o six tivitis n iht onstrints. Dlr uss rphil nottion n its smntis r s on LTL (Linr Tmporl Loi) [8]. Fiur 10 shows Dlr spiition onsistin o iht onstrints. Th onstrut onntin tivitis n is so-ll nonoxistn onstrint. In trms o LTL this onstrint mns (( ) ( )) ; n nnot oth tru, i.., it nnot th s tht oth n hppn or th sm s. Thr is lso non-oxistn onstrint prvntin th xution o oth n h or th sm s. Thr r thr prn onstrints. Th smntis o th prn onstrint onntin to n lso xprss in trms o LTL: ( ) W, i.., shoul not hppn or hs hppn. Sin th wk until (W ) is us in ( ) W, trs without ny n vnts lso stisy th onstrint. Similrly, shoul not hppn or hs hppn: ( ) W. Thr r thr rspons onstrints. Th LTL ormliztion o th prn onstrint onntin to is ( ( )),
i.., vry ourrn o shoul vntully ollow y. Not tht th hvior nrt y th WF-nt in Fi. 1 stisis ll onstrints spii in th Dlr mol, i.., non o th iht onstrints is violt y ny o th trs. Howvr, th Dlr mol shown in Fiur 10 llows or ll kins o hviors not possil in Fi. 1. For xmpl, tr,,,,,, is llow. Whrs in prourl mol, vrythin is orin unlss xpliitly nl, lrtiv mol llows or nythin unlss xpliitly orin. For prosss with lot o lxiility, lrtiv mols r mor pproprit [8, 24]. In [5] it is sri how Dlr/LTL onstrints n hk or ivn lo. This n lso xtn to th on-th-ly onormn hkin. Consir som runnin s hvin prtil tr σ p A listin th vnts tht hv hppn thus r. Eh onstrint is in on o th ollowin stts or σ p : Stisi: th LTL ormul orrsponin to vluts to tru or th prtil tr σ p. Tmporrily violt: th LTL ormul orrsponin to vluts to ls or σ p, howvr, thr is lonr tr σ p tht hs σ p s prix n or whih th LTL ormul orrsponin to vluts to tru. Prmnntly violt: th LTL ormul orrsponin to vluts to ls or σ p n ll its xtnsions, i.., thr is no σ p tht hs σ p s prix n or whih th LTL ormul vluts to tru. Ths thr notions n lit rom th lvl o sinl onstrint to th lvl o omplt Dlr spiition,.., Dlr spiition is stisi or s i ll o its onstrints r stisi. This wy it is possil to hk onormn on-th-ly n nrt wrnins th momnt onstrints r prmnntly/tmporrily violt [3]. pppph pppphp hpp ppph ppp p... non o-xistn onstrint is violt y th irst two ss (pppph n pppphp) 0..1 urs h om holy p pry prn onstrint is violt y th thir s (hpp) Fi. 11. Conormn hkin usin lrtiv mol. W us th smllr xmpl shown in Fi. 11 to illustrt onormn hkin in th ontxt o Dlr. Th pross mol shows our onstrints: th sm prson nnot urs n om holy (non-oxistn onstrint), tr on urss on shoul vntully pry (rspons onstrint), on n
only om holy tr hvin pry t lst on (prn onstrint), n tivity h ( om holy ) n xut t most on (rinlity onstrint). Two o th our onstrints r violt y th vnt lo shown in Fi. 11. Th irst two trs/prsons urs n m holy t th sm tim. Th thir tr/prson m holy without hvin pry or. Conormn hkin n istriut sily or lrtiv mols. On n prtition th lo vrtilly n simply hk pr omputin no ll onstrints on th orrsponin sulo. On n lso prtition th st o onstrints. Eh no o th omputr ntwork is rsponsil or sust o th onstrints n uss lo projt onto th rlvnt tivitis, i.., th vnt lo is istriut horizontlly. In oth ss, it is sy to rt th rsults into ovrll inostis. ppppp pppph pp ppph ppp p... 0..1 urs h om holy p pry Fi. 12. Disovrin lrtiv mol. Fiur 12 illustrts th isovry o Dlr onstrints rom vnt los [21]. A primitiv isovry pproh is to simply invstit lr olltion o nit onstrints usin onormn hkin. This n istriut vrtilly or horizontlly s just sri. It is lso possil to us smrtr pprohs usin th intrstinnss o potntil onstrints. Hr is rom istriut ssoition rul minin [13] n mploy. 4 Msurin Conormn Conormn hkin thniqus n us to invstit how wll n vnt lo L B(A ) n th hvior llow y mol it tothr. Fiur 4 shows n xmpl whr vitions twn n vnt lo n Ptri nt r inos. Fiur 11 shows similr xmpl ut now usin Dlr mol. Both xmpls ous on prtiulr onormn notion: itnss. A mol with oo itnss llows or most o th hvior sn in th vnt lo. A mol hs prt itnss i ll trs in th lo n rply y th mol rom innin to n. This notion n ormliz s ollows.
Dinition 9 (Prtly Fittin Lo). Lt L B(A ) n vnt lo n lt SN = (PN, M i, M o ) systm nt. L is prtly ittin SN i n only i {σ L} τ(sn ). Th ov inition ssums Ptri nt s pross mol. Howvr, th sm i n oprtionliz or Dlr mols [5], i.., or h onstrint n vry s th orrsponin LTL ormul shoul hol. Consir two vnt los L 1 = [,,, 30,,,, 20,,,,,,, 5,,,,,,, 3,,,,,,, 2 ] n L 2 = [,,, 8,,, 6,,,,, 5 ] n th systm nt SN o th WF-nt pit in Fi. 9 with T v = {,,,, }. Clrly, L 1 is prtly ittin SN whrs L 2 is not. Thr r vrious wys to quntiy itnss [3, 4, 11, 19, 23, 25 27], typilly on sl rom 0 to 1 whr 1 mns prt itnss. To msur itnss, on ns to lin trs in th vnt lo to trs o th pross mol. Som xmpl linmnts or L 2 n SN : γ 1 = γ 2 = γ 3 = γ 4 = Th top row o h linmnt orrspons to movs in th lo n th ottom row orrspons to movs in th mol. I mov in th lo nnot mimik y mov in th mol, thn ( no mov ) pprs in th ottom row. For xmpl, in γ 3 th mol is unl to o in-twn n. I mov in th mol nnot mimik y mov in th lo, thn ( no mov ) pprs in th top row. For xmpl, in γ 2 th lo i not o mov whrs th mol hs to mk this mov to nl n rh th n. Givn tr in th vnt lo, thr my mny possil linmnts. Th ol is to in th linmnt with th lst numr o lmnts,.., γ 3 sms ttr thn γ 4. Finin optiml linmnt n viw s n optimiztion prolm [4, 11]. Atr sltin n optiml linmnt, th numr o lmnts n us to quntiy itnss. Fitnss is just on o th our si onormn imnsions in in [3]. Othr qulity imnsions or omprin mol n lo r simpliity, prision, n nrliztion. Th simplst mol tht n xplin th hvior sn in th lo is th st mol. This prinipl is known s Om s Rzor. Thr r vrious mtris to quntiy th omplxity o mol (.., siz, nsity, t.). Th prision imnsion is rlt to th sir to voi unrittin. It is vry sy to onstrut n xtrmly simpl Ptri nt ( lowr mol ) tht is l to rply ll trs in n vnt lo (ut lso ny othr vnt lo rrrin to th sm st o tivitis). S [4, 25 27] or mtris quntiyin this imnsion. Th nrliztion imnsion is rlt to th sir to voi ovrittin. In nrl it is unsirl to hv mol tht only llows or th xt hvior sn in th vnt lo. Rmmr tht th lo ontins only xmpl hvior n tht mny trs tht r possil my not hv n sn yt. Conormn hkin n on or vrious rsons. First o ll, it my us to uit prosss to s whthr rlity onorms to som normtiv o sriptiv mol [7]. Dvitions my point to ru, iniinis, n poorly
sin or outt prours. Son, onormn hkin n us to vlut th prormn o pross isovry thniqu. In t, nti pross minin lorithms us onormn hkin to slt th nit mols us to rt th nxt nrtion o mols [23]. 5 Exmpl: Horizontl Distriution Usin Psss Th vrtil istriution o pross minin tsks is otn irly strihtorwr; just prtition th vnt lo n run th usul lorithms on h sulo rsiin t prtiulr no in th omputr ntwork. Th horizontl prtitionin o vnt los is mor hllnin, ut potntilly vry ttrtiv s th ous o nlysis n limit to w tivitis pr no. Thror, w sri nri istriution pproh s on th notion o psss. 5.1 Psss in Grphs A rph is pir G = (N, E) omprisin st N o nos n st E N N o s. A Ptri nt (P, T, F ) n sn s prtiulr rph with nos N = P T n s E = F. Lik or Ptri nts, w in prst n = {n N (n, n) E} (irt prssors) n postst n = {n N (n, n ) E} (irt sussors). This n nrliz to sts, i.., or X N: X = n X n n X = n X n. To ompos pross minin prolms into smllr prolms, w prtition pross mols usin th notion psss. A pss is pir o non-mpty sts o nos (X, Y ) suh tht th st o irt sussors o X is Y n th st o irt prssors o Y is X. Dinition 10 (Pss). Lt G = (N, E) rph. P = (X, Y ) is pss i n only i X N, Y N, X = Y, n X = Y. ps(g) is th st o ll psss o G. Consir th sts X = {,,,,, } n Y = {,,, h, i} in th rph rmnt shown in Fi. 13. (X, Y ) is pss. As init, thr my no s lvin rom X to nos outsi Y n thr my no s into Y rom nos outsi X. Dinition 11 (Oprtions on Psss). Lt P 1 = (X 1, Y 1 ) n P 2 = (X 2, Y 2 ) two psss. P 1 P 2 i n only i X 1 X 2 n Y 1 Y 2, P 1 < P 2 i n only i P 1 P 2 n P 1 P 2, P 1 P 2 = (X 1 X 2, Y 1 Y 2 ), P 1 \ P 2 = (X 1 \ X 2, Y 1 \ Y 2 ). Th union o two psss P 1 P 2 is in pss. Th irn o two psss P 1 \ P 2 is pss i P 2 < P 1. Sin th union o two psss is in pss, it is intrstin to onsir miniml psss. A pss is miniml i it os not ontin smllr pss.
X Y h i Fi. 13. (X, Y ) is pss us X = {,,,,, } = {,,, h, i} = Y n X = {,,,,, } = {,,, h, i} = Y. Dinition 12 (Miniml Pss). Lt G = (N, E) rph with psss ps(g). P ps(g) is miniml i thr is no P ps(g) suh tht P < P. ps min (G) is th st o miniml psss. Th pss in Fiur 13 is not miniml. It n split into th psss ({,, }, {, }) n ({,, }, {, h, i}). An uniquly trmins on miniml pss. Lmm 1. Lt G = (N, E) rph n (x, y) E. Thr is prisly on miniml pss P (x,y) = (X, Y ) ps min (G) suh tht x X n y Y. Psss in n quivln rltion on th s in rph: (x 1, y 1 ) (x 2, y 2 ) i n only i P (x1,y 1) = P (x2,y 2). For ny {(x, y), (x, y), (x, y )} E: P (x,y) = P (x,y) = P (x,y ), i.., P (x,y) is uniquly trmin y x n P (x,y) is lso uniquly trmin y y. Morovr, ps min (G) = {P (x,y) (x, y) E}. 5.2 Distriut Conormn Chkin Usin Psss Now w show tht it is possil to ompos n istriut onormn hkin prolms usin th notion o psss. In orr to o this w ous on th visil trnsitions n rt th so-ll sklton o th pross mol. To in skltons, w introu th nottion x σ:e#q y whih stts tht thr is non-mpty pth σ rom no x to no y whr th st o intrmit nos visit y pth σ os not inlu ny nos in Q. Dinition 13 (Pth). Lt G = (N, E) rph with x, y N n Q N. y i n only i thr is squn σ = n 1, n 2,... n k with k > 1 suh tht x = n 1, y = n k, or ll 1 i < k: (n i, n i+1 ) E, n or ll 1 < i < k: n i Q. Driv nottions: x σ:e#q x E#Q y i n only i thr xists pth σ suh tht x σ:e#q y, nos(x E#Q y) = {n σ σ N x σ:e#q y}, n
or X, Y N: nos(x E#Q Y ) = (x,y) X Y nos(x E#Q y). Dinition 14 (Sklton). Lt PN = (P, T, F, T v ) ll Ptri nt. Th sklton o PN is th rph skl(pn ) = (N, E) with N = T v n E = {(x, y) T v T v x F #Tv y}. Fiur 14 shows th sklton o th WF-nt in Fi. 1 ssumin tht T v = {,,,,,, l}. Th rsultin rph hs our miniml psss. skip xtr insurn hn ookin l ook r xtr insurn onirm initit hk-in supply r Fi. 14. Th sklton o th ll Ptri nt in Fi. 1 (ssumin tht T v = {,,,,,, l}). Thr r our miniml psss: ({}, {, }), ({,, }, {, }), ({}, {}), n ({}, {l}). Not tht only th visil trnsitions T v ppr in th sklton. For xmpl, i w ssum tht T v = {,, l} in Fi. 1, thn th sklton is ({,, l}, {(, ), (, l)}) with only two psss ({}, {}) n ({}, {l}). I thr r multipl miniml psss in th sklton, w n ompos onormn hkin prolms into smllr prolms y prtitionin th Ptri nt into nt rmnts n th vnt lo into sulos. Eh pss (X, Y ) ins on nt rmnt PN (X,Y ) n on sulo L X Y. W will show tht onormn n hk pr pss. Consir vnt lo L = [,,,, l 20,,,,, l 15,,,,,, l 5,,,,,, l 3,,,,,,, l 2 ], th WF-nt PN shown in Fi. 1 with T v = {,,,,,, l}, n th sklton shown in Fi. 14. Bs on th our psss, w in our nt rmnts PN 1, PN 2, PN 3 n PN 4 s shown in Fi. 15. Morovr, w in our sulos: L 1 = [, 27,, 18 ], L 2 = [, 20,, 15,,, 5,,, 3,,,, 2 ], L 3 = [, 45 ], n L 4 = [, l 45 ]. To hk th onormn o th ovrll vnt lo on th ovrll mol, w hk th onormn o L i on PN i or i {1, 2, 3, 4}. Sin L i is prtly ittin PN i or ll i, w n onlu tht L is prtly ittin PN. This illustrts tht onormn hkin n in ompos. To ormliz this rsult, w in th notion o nt rmnt orrsponin to pss. Dinition 15 (Nt Frmnt). Lt PN = (P, T, F, T v ) ll Ptri nt. For ny two sts o trnsitions X, Y T v, w in th nt rmnt PN (X,Y ) = (P, T, F, T v) with:
() skip xtr insurn () 4 xtr insurn 8 ook r 1 xtr insurn skip xtr insurn h () skip xtr insurn xtr insurn 2 hn ookin initit hk-in 5 6 i slt r j hk rivr s lins 9 10 l supply r () onirm 3 onirm initit hk-in 7 k hr rit r 11 Fi. 15. Four nt rmnts orrsponin to th our psss o th sklton in Fi. 14: () PN 1 = PN ({},{,}), () PN 2 = PN ({,,},{,}), () PN 3 = PN ({},{}), n () PN 4 = PN ({},{l}). Th invisil trnsitions, i.., th trnsitions in T \ T v, r sh. Z = nos(x F #Tv Y ) \ (X Y ) r th intrnl nos o th rmnt, P = P Z, T = (T Z) X Y, F = F ((P T ) (T P )), n T v = X Y. A pross mol n ompos into nt rmnts orrsponin to miniml psss n n vnt lo n ompos y projtin th trs on th tivitis in ths miniml psss. Th ollowin thorm shows tht onormn hkin n on pr pss. Thorm 1 (Distriut Conormn Chkin). Lt L B(A ) n vnt lo n lt WF = (PN, in, T i, out, T o ) WF-nt with PN = (P, T, F, T v ). L is prtly ittin systm nt SN = (PN, [in], [out]) i n only i or ny 1, 2,... k L: 1 T i n k T o, n or ny (X, Y ) ps min (skl(pn )): L X Y is prtly ittin SN (X,Y ) = (PN (X,Y ), [ ], [ ]). For orml proo, w rr to [2]. Althouh th thorm only rsss th notion o prt itnss, othr onormn notions n ompos in
similr mnnr. Mtris n omput pr pss n thn rt into n ovrll mtri. Assumin pross mol with mny psss, th tim n or onormn hkin n ru siniintly. Thr r two rsons or this. First o ll, s Thorm 1 shows, lrr prolms n ompos into st o inpnnt smllr prolms. Thror, onormn hkin n istriut ovr multipl omputrs. Son, u to th xponntil ntur o most onormn hkin thniqus, th tim n to solv mny smllr prolms is lss thn th tim n to solv on i prolm. Existin pprohs us stt-sp nlysis (.., in [27] th shortst pth nlin trnsition is omput) or optimiztion ovr ll possil linmnts (.., in [11] th A lorithm is us to in th st linmnt). Ths thniqus o not sl linrly in th numr o tivitis. Thror, omposition is usul vn i th hks pr pss r on on sinl omputr............. in out 2 4 in 1 6 out 3 5 Fi. 16. Distriut isovry s on our miniml psss: ({, }, {}), ({}, {, }), ({, }, {}), n ({}, {, }). A pross rmnt is isovr or h pss. Susquntly, th rmnts r mr into on ovrll pross.
5.3 Distriut Pross Disovry Usin Psss As xplin or, onormn hkin n pross isovry r losly rlt. Thror, w n xploit th pproh us in Thorm 1 or pross isovry provi tht som ors usl strutur (omprl to th sklton in Stion 5.2) is known. Thr r vrious thniqus to xtrt suh usl strutur, s or xmpl th pnny rltions us y th huristi minr [29]. Th usl strutur ins olltion o psss n th til isovry n on pr pss. Hn, th isovry pross n istriut. Th i is illustrt in Fi. 16. Th pproh is inpnnt o th isovry lorithm us. Th only ssumption is tht th sul strutur n trmin upront. S [2] or mor tils. By omposin th ovrll isovry prolm into olltion o smllr isovry prolms, it is possil to o mor rin nlysis n hiv siniint sp-ups. Th isovry lorithm is ppli to n vnt lo onsistin o just th tivitis involv in th pss unr invstition. Hn, pross isovry tsks n istriut ovr ntwork o omputrs (ssumin thr r multipl psss). Morovr, most isovry lorithms r xponntil in th numr o tivitis. Thror, th squntil isovry o ll iniviul psss is still str thn solvin on i isovry prolm. 6 Conlusion This ppr provis n ovrviw o th irnt mhnisms to istriut pross minin tsks ovr st o omputin nos. Evnt los n ompos vrtilly n horizontlly. In vrtilly istriut vnt lo, h s is nlyz y sint omputin no in th ntwork n h no onsirs th whol pross mol (ll tivitis). In horizontlly istriut vnt lo, th ss thmslvs r prtition n h no onsirs only prt o th ovrll pross mol. Ths istriution pprohs r irly inpnnt o th minin lorithm n pply to oth prourl n lrtiv lnus. Most hllnin is th horizontl istriution o vnt los whil usin prourl lnu. Howvr, s shown in this ppr, it is still possil to horizontlly istriut pross isovry n onormn hkin tsks usin th notion o psss. Aknowlmnts. Th uthor woul lik to thnk ll tht ontriut to th ProM toolst. Mny o thir ontriutions r rrr to in this ppr. Spil thnks o to Bouwijn vn Donn n Eri Vrk (or thir work on th ProM inrstrutur), Crmn Brtosin (or hr work on istriut nti minin), Ary Arinsyh n Ann Rozint (or thir work on onormn hkin), n Mj Psi, Frizio Mi, n Mihl Wstrr (or thir work on Dlr).
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