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Everett Warren
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1 OPERTIONS RESERH o.287/opr.8.559c pp. c c8 -copao ONLY VILLE IN ELETRONI FORM fors 28 INFORMS Elctroc opao Optzato Mols of scrt-evt Syst yacs by Wa K (Vctor ha a L Schrub, Opratos Rsarch, o.287/opr

2 OPTIMIZTION MOELS OF ISRETE-EVENT SYSTEM YNMIS Wa K (Vctor ha * L Schrub ** * partt of cso Sccs a Egrg Systs, Rsslar Polytchc Isttut, II 55, Eght Strt, Troy, NY 28 ** partt of Iustral Egrg a Opratos Rsarch, Uvrsty of alfora, rkly, 45 Etchvrry Hall, rkly, 9472 PPENIX tals for gratg qualty costrats ERG2MP ar gv th followg. ot by X a grc tgr stat varabl th sulato talz at. Lt X(t b th valu of X at t t, a b a gv tgr ubr. X ca b crt or crt by ffrt vts; Fgur.2 lsts cass for scrt-vt systs cosr ths papr (otato ths fgur wll b f latr wh thy ar us. ass (a ( ar th four basc stuatos whr actly two vts ca chag th valu of X: (a th schulg a schul vts (vts a ; (b th schul vt (vt a o aothr vt (vt ; (c th schulg vt (vt a o aothr vt (vt ; a ( ay two vts (vts a clug th schulg a schul vts (vts a. Sytrc cass (.., th rols of crasg a crasg th valu of X ar trchag ar ot clu sc thr costrats ar rv a slar way. ass whch four vts ca chag X s valu ar pct by cass ( (h. Thy ca also b gralz to cass wth or tha four vts chagg X s valu. W oly cosr cass whr a vrt s schul by ( o cotoal arc or (2 ultpl cotoal arcs wth sa cotos (ths covrs all th systs cosr ths papr. For cass wth a vrt schul by ultpl cotoal arcs wth ffrt cotos, o ca splt th vrt to a group of vrtcs, ach of whch s schul by o cotoal arc (Yucsa a Schrub 992. tals for Stp : I Stp, sc th t lay t( s rao, th th ( vt to b schul ay ot b th th ( vt to occur; o apl s ob "ovrtakg" quug systs: for stac, a ult-srvr quug syst, th frst start ob ay ot b th frst ob to fsh. O as to tr whch vt schuls whch vt s to us bary varabls. Lt bary varabl = f th th O( vt schuls th th ( vt, a othrws. Ths assgt O (:, (: s ( O ( t (-M(- O (:, (: O ( ( - t ( ( - å O (:, (: O (:, (: å O (: I, (:.

3 HN N SHRUEN Optzato Mols of scrt-evt Syst yacs whr a M ar sutabl ral ubrs,.g., th lowr bou a uppr bou of O( ( t(, rspctvly, or sply th ubr of vts th sulato sapl path. If thr s o ovrtakg for apl, a sgl-srvr ta quu or a G//R quu, th th start ob ust b th th ob to fsh th sply rop th bary varabls a st = Stp. Th frst two costrats ar to ak sur that at ost o par of a ca achv both qualts, a hc a qualty. I so cass, havg o qualty forc woul b suffc, whch s th cas Eapl 2.2. tals for Stp 2: Spl ass For cas (a Fgur.2, aftr th occurrc of vt, f coto (X s tru, th a vt s schul to occur. t th stat ust bfor th cuto of th th vt,.., -, w ust hav Û ( X ( = ( - ( Û =, =,..., whr s th ubr of occurrcs of vt th sapl path, capturg th fact that th th occurrc of vt ust occur aftr th th occurrc of vt. Th frst qualty follows bcaus s th ubr of vts that ust occur bfor t -. Th last qualty s bcaus vt ts ar cotuous (ot to b cofus wth th tgr valu stat varabls. as (b Fgur.2 s hal slarly. ostrats wth ary Varabls O as to rvg costrats wth bary varabls s as follows. W scuss cas (c frst. If th coto o arc (, s tru rght aftr th th occurrc of vt (at t, whch s qual to, th usg (2., w hav X ( = ( - ( Û ( - Û =. - Usg a bary varabl z, :, w ca prss ths rlato as th logc: - «z =, whch s quvalt to, : - M c (- z -, : c c -, : (-, : z z 2

4 HN N SHRUEN Optzato Mols of scrt-evt Syst yacs whr M c a c ar, rspctvly, th uppr bou a lowr bou of - - (or sply th ubr of vts th sapl path of th sulato ru, a c s a postv sall ral ubr such that - c ca b cosr as - > (s Hookr 22 or Wllas 995 for th us of logc tgr prograg. So vt wll b schul f a oly f z, : =. Ths s costra by å s :, : = z, : whr s :, : s ar bary varabls forcg th logc s :, : = =, whch s quvalt to c 2 :, : c 2 :, : -M (- s. (- s. whr M 2 c a 2 c ar, rspctvly, th uppr bou a lowr bou of -. Iqualts å s :, : a å s :, : ca b a to sur that th o-to-o schulg rlatoshp btw vt a vt, v though å s :, : s ruat u to co- strat :, : = z, :. To sur that a vt wth a lowr ust b schul ar- å s å= å ca b us. = lr tha vts wth hghr s, costrats s :, : s :, : lso, f thr s o othr vt that wll schul vt, th costrats - å s pq :, : {, }( s :, : p= å ca b appl to forc th orrg q= of all vts. Th costrat rvato procur for cas ( s or coplcat a rqurs th us of four sts of bary varabls. Rght aftr th th occurrc of vt (at t, f th coto o arc (, s th tru, w hav X ( = ( - ( Thr two vt coutg fuctos valuat at t. For vt, w hav ( ( ( - - Û =. If ( = k, th k, for so k. Ths allows us to valuat th vt couts for vt as follows.

5 HN N SHRUEN Optzato Mols of scrt-evt Syst yacs ( ( - ( k - k Û k - Û > Û =. whr 2 s a postv sall ral ubr such that - 2 ca b cosr ( as ( - >. ostrats k a k - 2 sply tst whthr th coto o arc (, s tru urg th t trval btw k a k -. If th th vt occurs urg ths t pro, th so vt, say th th vt, wll b schul to occur. To tst what k s satsfy such coto, w us two bary varabls, z k :, : a h k :, : -, a plt th logc: k «zk :, : = a k - «h - = usg th followg qualts, k :, : k - (- k :, : k k :, : (- k :, : k k :, : - M z z z M ( h k - 2 k :, : - h. whr M a ar, rspctvly, th uppr bou a lowr bou of k -, a M 2 a ar, rspctvly, th uppr bou a lowr bou of k - -, a s a 2 postv sall ral ubr such that wthout whch th corrspog qualty k ca b rgar as k >. If both z k :, :, a h k :, : - ar qual to, th th coto o arc (, s tru a th th vt wll b schul to occur. y usg bary varabls g k :, : a s :, :, ths ca b suarz as zk :, : = & hk :, : - = «gk :, : = a å g «å s =, gvg qualts k k :, : :, : gk :, : -( zk :, : h k :, : - 2 gk -( zk h k - :, : :, : :, : å å å s å g, gk :, : - s :, : k :, : - k :, : k a th followg bouary costrats for k =,,, 4

6 HN N SHRUEN Optzato Mols of scrt-evt Syst yacs g g k :, : - zk :, : k :, : zk :, : -. To schul th th vt, w us th logc s :, : = =, whch gv costrats -M (- s. :, : (- s. :, : whr M a ar, rspctvly, th uppr bou a lowr bou of -, Slar to cas (c, th followg qualts ca b a to sur th thr rqurts: th o-too rlatoshp, 2 lowr- vts shoul b schul frst, a th orrg of vts. å s :, : å s :, : å å s :, : s = = :, : - å s pq :, : {, }( s :, : p= å q= ovoluto ostrats ass ( (h volv or tha two vts that coul chag th valu of X. W rly o covolutos to grat costrats for all th volvg vts. W scuss cas (. ass (f (h ca b hal by cobg th covoluto costrats rv cas ( a th costrats rv cass (b (. Rght aftr th th occurrc of vt (at t, f th coto o arc (, s th tru, w hav X ( = ( ( -( - ( Û ( - ( whr W = È s a coutr vt, for vts a, that occurs wh thr vt or vt occurs,.., W (t = (t (t, " t. Thrfor, by fg such a coutr vt, cass whch th valu of a varabl ca b chag by or tha two vts ca b hal usg costrats vlop cas (. Nt, w scuss th costrats grato procur for vts,, a W. Th rlato ( = ( ( s a covoluto rlatoshp. For apl, th W st W vt coul occur wh th st vt occurs or wh th st vt occurs, but ot both; usg th laguag of ath prograg a troucg cotuous varabls P,, a P,, (th frst, sco, a thr subscrpts ar th cs for th th W vt, th vt, a th 2 vt, rspctvly, such that = 2, ths as P,, =, P,, =, a o of th co- W 5

7 HN N SHRUEN Optzato Mols of scrt-evt Syst yacs strats, W P,, a W P,,, s bg (.., W = {P,,, P,, }. For th 2 W vt (wth w cotuous varabls P 2,,, 2 2 =, ths covoluto rlatoshps ar P2,2, = 2, P2,,2 = 2, o of th costrats, P2,, a P2,,, s bg (.., P 2,, = a{, }, a o of th costrats, W2 P2,2,, W2 P2,,, a W2 P2,,2, s bg (.., W 2 = {P 2,2,, P 2,,, P 2,,2 }. Rpatg ths costructo utl Z wll grat å ( costrats. Th bg costrats ca b fou by usg bary varabls th = sa way as bfor. For apl, to sur P 2,, = a{, }, lt bary varabl a 2,, = «a apply th followg costrats: -M (- a2,,, a2,, (- a2,,, P2,, -M2(- a2,,, P2,, 2( - a2,,, P2,, - Ma2,,, P2,, a2,,, whr M, M 2, M,, 2, a, ar th uppr bou a lowr bou of th corrspog costrats rspctvly, a s a postv sall ral ubr such that wthout whch th corrspog qualty ca b rgar as >. ll th w cotuous varabls ca b put to th obctv fucto to b z. For a cas wth thr vts that coul crt X,.., lt {,, E} b ths thr vts (ot that ths cas s ot pct Fgur.2, ot by Z = È È E th coutr vt for vts,, a E. To rprst th rlatoshps btw Z a th thr vts that vt Z couts (,, a E, w ot that thr ar ((2/2 possbl prutatos that coul caus Z to happ (th ubr of tgr pots o a hyprpla crat by th cs of ths thr vts, pct Fgur.. For apl, cosr Z Fgur., P,,, rprsts th au of,, a E ; all th tgr pots o th tragl-pla ar th possbl prutatos of Z ; a th u of th s th t for Z. Lt L Z ( b th st of all possbl prutatos of Z a P,, 2, b o of ths prutatos. For ach =,...,, w hav: P,, 2, P -, -, 2, P,, 2, P -,, 2-, P P Z P, (,, L (.,, 2, -,, 2, -,,, " 2 Î Z 2 (. Th bouary qualts ar: P,,, = P,, = P,,, = E, =,...,. ga, o of th costrats govrg Z a P,, 2, ust b bg, whch ca b forc by usg bary varabls. Rug th sulato to solv th rla athatcal progra wthout ths bary varabl wll sply tr th bg costrat, but w wll forc t hr wth th bary varabl for copltss. 6

8 HN N SHRUEN Optzato Mols of scrt-evt Syst yacs Now, for a gral cas wth vts crtg X(t (.., {,,..., } å å å - 2 = = = bco: =, thr ar prutatos of ach. Th abov two sts of costrats wll th P,,..., P -, -,..., P P Z P, (,..., L (,,..., -,,..., -,,..., " Î Z P,,,..., P,,...,,, =,...,. W ot that th ubr of such costrats grows potally th ubr of vts. Howvr, w ar ot trst solvg th athatcal progras ffctly sc rug th sulato ca prov solutos to th rla ath progras. Th bfts of stablshg th act athatcal rprstatos for ERGs ar suarz Scto. a applcatos ar llustrat Scto. P,,, E Fgur. : Th Gotrc Itrprtato of ostrat (. 7

9 HN N SHRUEN Optzato Mols of scrt-evt Syst yacs (a ( (b (f E (c (g E ( (h E F Fgur.2: Gratg ostrats ur ffrt Stuatos 8

Correlation in tree The (ferromagnetic) Ising model

Correlation in tree The (ferromagnetic) Ising model 5/3/00 :\ jh\slf\nots.oc\7 Chaptr 7 Blf propagato corrlato tr Corrlato tr Th (frromagtc) Isg mol Th Isg mol s a graphcal mol or par ws raom Markov fl cosstg of a urct graph wth varabls assocat wth th vrtcs.