2 TO APPEAR IN THE APR. 999 ISSUE OF IEEE TRANSACTIONS ON AUTOMATIC CONTROL tiulrly interesting for the suessive evlution of lrge numer of sheules. Fo

Transcription

1 TO APPEAR IN THE APR. 999 ISSUE OF IEEE TRANSACTIONS ON AUTOMATIC CONTROL Moeling n Anlysis of Time Petri Nets using Heps of Piees Stephne Guert, Jen Miresse Astrt We show tht sfe time Petri nets n e represente y speil utomt over the (mx,+) semiring, whih ompute the height of heps of piees. This extens to the time se the lssil representtion l Mzurkiewiz of the ehvior of sfe Petri nets y tre monois n tre lnguges. For sulss inluing ll sfe Free Choie Petri nets, we otin reue hep reliztions using struturl properties of the net (overing y sfe stte mhine omponents). We illustrte the hep-se moeling y the typil se of sfe joshops. For perioi sheule, we otin hep-se throughput formul, whih is simpler to ompute thn its tritionl time event grph version, prtiulrly if one is intereste in the suessive evlution of lrge numer of possile sheules. Keywors Time Petri nets, utomt with multipliities, heps of piees, (mx,+) semiring, sheuling. U I. Introution NTIMED n time Petri nets hve een tively stuie for long time, in prtiulr s moeling n nlysis tool for Disrete Event Systems. Two ifferent kins of lgeri ojets hve een introue for untime n time Petri nets respetively. () The untime ehviors of Petri nets n e represente y lnguges (sets of possile ring sequenes). In the se of net with oune mrkings, the lnguge of the net is reognize y nite utomton, the rehility grph. (2) The time ehvior of net n e represente y ter funtions whih provie the ourrene time of ll the possile events in the system. The most stisftory results re reltive to the sulss of time Event Grphs, whih n e moele y nite imensionl reurrent liner systems over the (mx,+) semiring, see [2], [2]. There exists striking onnetion etween the two ses: oth (mx,+) liner systems n onventionl utomt re speiliztions of (mx,+) utomt, i.e. utomt with multipliities [20] over the (mx,+) semiring. This oservtion les to the nturl question, whih ws left unsolve in [2]: Wht is the moeling power of (mx,+) utomt in terms of time Petri nets? The purpose of this pper is to propose the following nswer: Time sfe Petri nets re speil (mx,+) utomt, whih ompute the height of heps of piees. This work ws prtilly supporte y the Europen Community Frmework IV progrmme through the reserh network ALAPEDES (\The ALgeri Approh to Performne Evlution of Disrete Event Systems") S. Guert is with INRIA, Domine e Volueu, B.P. 05, 7853 Le Chesny Ceex, Frne. E-mil: Stephne.Guert@inri.fr J. Miresse is with CNRS, LIAFA, Universite Pris 7, Cse 704, 2 ple Jussieu, 7525 Pris Ceex 05, Frne. E-mil: miresse@lif.jussieu.fr As y-prout, we will otin new utomt-se performne evlution lgorithms. This representtion theorem is est unerstoo y omprison with the following existing pprohes.. Tre lnguges. In the lnmrk pper [3], Mzurkiewiz oserve tht tre monois (tht is, free prtilly ommuttive monois) n their susets (tre lnguges) re nturl moel of the logil ehvior of sfe Petri nets. Tre monois, in whih ertin letters ommute, n others o not, llow one to ientify the ifferent sequentil representtions of the sme onurrent events. 2. Heps of piees. In [40], Viennot oserve tht tre monois re isomorphi to hep monois, tht is, monois in whih the genertors re piees (in the nerly usul unerstning: these piees re soli retngulr-shpe loks), n where the ontention onsists in piling up one hep ove the other. This yiels very intuitive grphil representtion of tre monois. 3. (Mx,+) liner representtions. A next step ws the oservtion tht the height of heps of piees is reognize y hep utomton, speil type of (mx,+) utomton, the result holing for generl, polyomino-shpe, piees. This ws prove in Guert n Miresse [22], n in ierent form, y Brilmn n Vinent [4], [7]. The hep representtion theorem for sfe time Petri nets tht we give n e seen s synthesis of these three results. The essentil ie is tht onsiering generl piees in hep utomt enles to moel time through the height of piee. More preisely, one of the min results of the pper goes s follows. Let G e sfe time Petri net, with set of trnsitions T n set of ring sequenes L T. Then the mp y G : L! R +, whih ssoites to ring sequene the te of ompletion of the lst event in the net, is reognize y hep utomton (less formlly, piee is ssoite to eh letter in T n y G (w) is the height of the hep otine y piling up the piees orresponing to the letters of w). Hep representtions re prtiulrly well pte to lgeri omputtions. As typil illustrtion, we erive n hep-se performne evlution metho for sfe joshops. The ssignment of the jos on the mhines is xe ut not the orer on whih the jos re proesse y the mhines (the sheule). With eh perioi sheule, the lssil moeling ssoites n event grph (i.e. (mx,+) liner system) whose size grows with the perio of the sheule, see [2], [27], [2]. On the other hn, the representtion y hep moel is inepenent of the (even non perioi) sheule whih is onsiere. This is pr-

2 2 TO APPEAR IN THE APR. 999 ISSUE OF IEEE TRANSACTIONS ON AUTOMATIC CONTROL tiulrly interesting for the suessive evlution of lrge numer of sheules. For perioi sheule, we propose new hep-se lgorithm to ompute the throughput of the joshop, whih is simpler thn the rene vrints of the tritionl (event grph se) metho. It is worth noting tht heps of piees re essentilly n extension of the Gntt hrts tritionlly use in sheuling. Wheres onventionl Gntt hrts only isply the resoure (mhine) ouption times, heps of piees ontin the omplete time informtion for oth resoures n jos, whih llows us to write ynmil equtions. Let us mention the relte inepenent work of Hulgr [28], who stuie the sulss of sfe Free Choie Petri nets. He i not put forwr the utomt or hep moels, ut he i introue (for time nlysis purposes) ter vriles n ynmil equtions similr to the ones use here, see the isussion in Remrk IV.8 elow. A very ierent lgeri pproh is tht of Belli, Foss n Gujl [3]; Cohen, Guert n Qurt [3], [4]; n Lieut n Loiseu, see [29, Chp. 2]. Essentilly, the ounter funtion (vetor of numers of rings, s funtion of time) of generl (non-neessrily sfe) Free Choie or Flui Petri nets stises some omintion of impliit (min,+) n (+,)-liner ynmil equtions, whih eome expliit uner ertin ssumptions on the timing or routing poliy. In ertin sense, this pproh, whih omputes the numer of rings (logil time) s funtion of the physil time, is ul, or inverse, to the utomt pproh propose here, whih omputes the tes (physil times) s funtion of the sheule (logil time). Surprisingly, these two points of view le to ompletely ierent tehnil evelopments. The pper is orgnize s follows. In x II, we introue hep moels n (mx,+) utomt. In x III, we rell si fts out Petri nets n their representtion y tre monois. The min result, \sfe time Petri nets n e represente y heps of piees", is prove in x IV-A. In this representtion, the size of the hep moel is equl to the numer of ples in the Petri net. In x IV-B, we show the existene of muh smller hep representtion for the sulss of nets whih n e overe y sfe Stte Mhine Components. It inlues in prtiulr ll sfe Free Choie Petri nets. In x V, we pply these results to the typil exmple of joshops, proposing new performne evlution lgorithm. In x VI, we isuss t more lgeri level the moels use in the pper n their interply. To onlue, let us mention relte generl referenes. The reer is referre to [2], [5], [42], [30], [25], [23], for n ount of the theory of the (mx,+) semiring. The theory of utomt with multipliities is elt with in [4], [20], [37]. The (mx,+) utomt use in this pper re generliztions of \ost utomt", or utomt with multipliities over the tropil semiring (N [ f+g; min; +). They hve een wiely stuie for their onnetions with lssil eiility prolems in lnguge theory (see [35], [38], n the referenes therein). A Disrete Event Systems oriente presenttion n e foun in [2]. Generl ounts of Petri net theory n e foun in [5], [33], [6] or in the proeeings [6]. II. Hep Moels n (mx,+) Automt The following hep moel generlizes the heps of piees of Viennot [40]. Imgine n horizontl xis with nite numer of slots. A piee is soli (possily non onnete) \lok" oupying some of the slots, with stirse-shpe upper n lower ontours, see Fig.. With n orere sequene of piees, we ssoite hep y piling up the piees, strting from n horizontl groun. This piling ours in the intuitive wy: piee is only sujet to vertil trnsltions n oupies the lowest possile position, provie it is ove the groun n the piees previously pile up. Let us propose more forml enition. Denition II.: A hep moel is 5-tuple H = (T ; R; R; l; u), where: T is nite set whose elements re lle piees. R is nite set whose elements re lle slots. R : T! P(R) gives the suset of slots oupie y piee. We ssume tht eh piee oupies t lest one slot: 8 2 T ; R() 6= ;. l : T R! R [ f g gives the height of the lower ontour of the piee t the ierent slots. u : T R! R[f g (with u l) gives the height of the upper ontour of the piee. By onvention, l(; r) = u(; r) = if r 62 R() n min r2r() l(; r) = 0. A piee oupies region of the R R + plne, of the form f(r; y) 2 R() R + j + l(; r) y + u(; r)g, where 2 R + epens on the piees lrey pile up, n = 0 if there is no other piee. We will interpret length k wor 2 w = ; : : : ; k 2 T s hep, i.e. s sequene of k piees ; : : : ; k pile up in this logil orer. We ene the upper ontour of the hep w s the R- imensionl row vetor x H (w), where x H (w) r is the height of the hep on slot r. The horizontl groun ssumption yiels x H (e) = (0; ; 0) (rell tht e enotes the empty wor). The height of the hep w is y H (w) = mx r2r x H(w) r : A useful interprettion of hep moel onsists in viewing piees s tsks n slots s resoures. Eh tsk requires suset of the resoures (given y R()) uring ertin mount of time (u(; r) l(; r) for resoure r 2 R()). In the simplest se where l(; r) = 0; 8r 2 R(), the exeution of tsk egins s soon s ll the require resoures, use y erlier tsks, eome free. For more etils long these lines, see [22], [7]. It orrespons for exmple to the mehnism of the Tetris gme. 2 We rell the following stnr nottions. Given nite set (lphet) T, we enote y T n the set of wors of length n on T. We enote y T = [ n2nt n the free monoi on T, tht is, the set of nite wors equippe with ontention. The unit (empty wor) will e enote y e. The length of the wor w will e enote y jwj. We shll write jwj for the numer of ourrenes of given letter in w.

3 GAUBERT AND MAIRESSE:MODELING AND ANALYSIS OF TIMED PETRI NETS 3 Borrowing the terminology of [2], the mps y H n x H re lle the ter funtions of the hep moel. The piling mehnism n the ierent nottions re est unerstoo grphilly n on n exmple, see Fig.. Exmple II.2: Let us onsier the following hep moel. T = f; ; ; g, R = f; 2; 3; 4g; R() = f; 2; 3g, R() = f; 2g, R() = f2; 4g, R() = f2; 3; 4g; u(; :) = [; ; 3; ]; l(; :) = [0; 0; 0; ]; u(; :) = [3; 2; ; ]; l(; :) = [0; 0; ; ]; u(; :) = [ ; 2; ; 2]; l(; :) = [ ; 0; ; 0]; u(; :) = [ ; ; 3; ]; l(; :) = [ ; 0; 0; 0] : We hve represente, in Fig., the hep ssoite with the wor w =. Piee is n exmple of non onnete (ut \rigi") piee. Q is nite set (of sttes); I 2 R Q n F 2 R Q re the initil n nl vetors, mx mx respetively; M is morphism T! R QQ mx. The morphism M is uniquely speie y the nite fmily of Q Q mtries, M(); 2 T. Then, for wor w = : : : n, we hve M(w) = M( : : : n ) = M( ) : : : M( n ) ; the mtrix prout eing interprete in the (mx,+) semiring. Let us ene the vetors x A (w) = IM(w) 2 Rmx Q n the slrs y A (w) = IM(w)F 2 R mx ssoite with the (mx,+) utomton A. We sy tht x A n y A re reognize 5 y the utomton A. We hve x A (e) = I ; x A (w) = x A (w)m() ; y A (w) = x A (w)f : Hene (mx,+) utomton my e seen s (mx,+) liner system whose ynmis is riven y letters. With hep moel H = (T ; R; R; l; u), we ssoite the morphism M : T! R RR mx, ene y 8 >< if s = r 62 R(), M() sr = u(; r) l(; s) if r 2 R(); s 2 R(), >: 0 otherwise. Fig.. Hep of piees ssoite with the wor w =. We n re iretly on Fig. the vlues x H (w) = [4; 6; 8; 6] n y H (w) = 8. Denition II.3: The (mx,+) semiring 3 R mx is the set R [ f g, equippe with the opertion mx, written itively (i.e. = mx(; )) n the usul sum, written multiplitively (i.e. = + ). In this semiring, 0 =, = 0. Note tht R mx is ioi, i.e. semiring with n iempotent ition ( = ). We shll use throughout the pper the mtrix n vetor opertions inue y the semiring struture. For mtries A; B of pproprite sizes, (A B) ij = A ij B ij = mx(a ij ; B ij ), (A B) ij = L k A ik B kj = mx k (A ik + B kj ), n for slr, ( A) ij = A ij = + A ij. We will omit the sign, writing for instne AB inste of A B s usul. Given set S, we enote y S the S-imensionl olumn vetor whose entries re ll equl to. Denition II.4: Given nite lphet T, (mx,+) utomton 4 is 4-tuple A = (Q; I; F; M), where 3 A set K equippe with two opertions n is semiring if is ssoitive n ommuttive, is ssoitive n istriutive with respet to, there is zero element 0 ( 0 = ; 0 = 0 = 0) n unit element ( = = ). 4 This is the speiliztion to the (mx,+) semiring of the lssil notion of utomton with multipliity [20], or reognizle series [37], Theorem II.5: Let H = (T ; R; R; l; u) e hep moel. The (mx,+) utomton (R; t R ; R; M) reognizes the upper ontour x H n the height y H : 8w 2 T x ; H (w) = t RM(w) ; y H (w) = t () RM(w) R : A vrint of this result ws prove in [22], see lso [7]. We sy tht (R; t R ; R; M) is the hep utomton ssoite with the hep moel. For the ske of ompleteness, we give n rige version of the rgument. Proof: The following ynmil eqution shoul e ler from the physil esription of the system: 8 >< x H (w) r if r 62 R() x H (w) r = mx s2r() ( x H (w) s + u(; r) l(; s) ) >: if r 2 R(). (2) For exmple, let us onsier the se of piee with n horizontl se (l(; s) = 0; s 2 R()). When ing piee to hep w, one hs rst to ompute the height of the se of the piee whih is equl to mx s2r() x H (w) s. Then [4]. For more etils long these lines, see x VI. 5 Clssilly, in utomt theory, only the mp ya is si to e reognize. It is onvenient here to exten the enition of reognizility.

4 4 TO APPEAR IN THE APR. 999 ISSUE OF IEEE TRANSACTIONS ON AUTOMATIC CONTROL the height of the hep w on slot r 2 R() is otine s mx s2r() x H (w) s + u(; r). Clerly, we hve: 8r 2 R; x H (e) r = ; y H (w) = mx r x H (w) r = x H (w) R : We ientify in (2) n (3) ynmis of the form (). We mention for further use the following elementry ommuttion property, whih hols for ll ; 2 T : (3) R() \ R() = ; =) M()M() = M()M() : (4) An lterntive \ul" utomton representing the hep moel, otine y ssoiting ter vriles to piees, inste of slots, ws given in [22]. Exmple II.6: Let T = f; ; ; g, Q = f; 2; 3; 4g, I = t Q ; F = Q M() = M() = ; M() = 7 5 ; M() = ; ; (the 0 entries re omitte). One esily veries tht the (mx,+) utomton (Q; I; F; M) represents the hep moel given in Ex. II.2. We hve M() = ; x A () = IM() = [4; 6; 8; 6]; y A () = 8 : This provies n lgeri onrmtion of the vlues otine grphilly in Fig.. Remrk II.7: Hep utomt, s introue in Theorem II.5, re (mx,+) utomt of spei form. The morphism M of hep utomton veries: M() = I [ ~ l(; :)] t u(; :) ; where I is the ientity mtrix ene y I ii = ; I ij = 0; i 6= j, where ~ l(; i) = l(; i) if l(; i) 6= 0 n ~ l(; i) = l(; i) = 0 otherwise, n where ~ l(; :); u(; :) re viewe s row vetors. A. Denitions III. Untime n Time Petri Nets We next rell some lssil fts out untime n time Petri nets. Denition III.: A Petri net (PN) is 4-tuple G = (P; T ; F; M), where : P is nite set, whose elements re lle ples. T is nite set, whose elements re lle trnsitions. F (P T ) [ (T P) is reltion etween ples n trnsitions. M is mp P! N. The integer M(p) is lle the initil mrking of ple p. We will use the term Petri net to enote oth the unmrke n the mrke net. We will sometime enote the mrke net y (G; M) inste of G to insist on the vlue of the initil mrking. A Petri net is tritionlly represente s iprtite irete grph. There re two ierent kins of noes, ples p 2 P, (represente y irles) n trnsitions 2 T (represente y retngles). An element of F is n r from ple to trnsition or from trnsition to ple. It is therefore nturl to spek of \input ples", \output trnsitions" n so on. We use the nottions p; p (resp. ; ) for the set of input n output trnsitions of ple p (resp. input n output ples of trnsition ). The mrking M(p) is isplye y rwing M(p) tokens in ple p. We will use the lssil notions of (oriente) pth, iruit, onneteness n strong-onneteness of grph theory. An exmple of strongly onnete Petri net is provie in Fig. 2. p Fig. 2. p 2 p 3 p 5 p 6 p 4 Strongly onnete sfe Petri net A Petri net is ynmi ojet. The unerlying struture (P; T ; F) is never moie, ut the mrking M evolves oring to the following ring rule.. Trnsition is enle t M if there is t lest one token in eh of its input ples. 2. An enle trnsition n re. The ring of trnsforms M into M 0 (written M! M 0 ) y removing one token from eh of the input ples n ing one token in eh of the output ples of. We sy tht wor w = 2 n 2 T is ring sequene strting from mrking M 0 if there is sequene of mrkings M 0 = M 0 ; M ; : : : ; M n = M 00 suh tht trnsition i is enle t M i n M i! M i. We revite this y M 0! w M 00. A mrking M 00 is rehle from mrking M 0 if there is ring sequene w 2 T suh tht M 0! w M 00. We enote y R(M 0 ) the set of mrkings rehle from M 0. We ll lnguge 6 of the Petri net (G; M) the set L T of ring sequenes strting from M. 6 More properly, the P-type free lele lnguge, oring to the terminology of Peterson [34]. Vrious kins of Petri net lnguges hve een ene (oring to vrious kins of eptne onitions i

5 GAUBERT AND MAIRESSE:MODELING AND ANALYSIS OF TIMED PETRI NETS 5 Denition III.2: A Petri net (G; M) with lnguge L is: live if 8w 2 L; 8t 2 T ; 9u 2 T, suh tht wut 2 L, i.e. if whtever the pst rings (= w) re, it is possile to n ring sequene from the urrent stte, ontining trnsition t; oune if the set R(M) is nite. Equivlently, if 9k suh tht 8M 0 2 R(M); 8p 2 P; M 0 (p) k; sfe (or -oune) if ple will not hol more thn one token: 8M 0 2 R(M); 8p 2 P; M 0 (p). Exmple III.3: The Petri net represente in Fig. 2 is live n sfe. Its lnguge is L = ( [ ) (e [ [ ). Let us rell some lssil sulsses of Petri nets.. A iruit is PN suh tht j tj = jt j = j pj = jp j = for ll t 2 T ; p 2 P. 2. A Stte Mhine (SM) is PN suh tht j tj = jt j = for ll t 2 T. 3. An Event Grph (EG) is PN suh tht j pj = jp j = for ll p 2 P. 4. A Free Choie net (FC) is PN suh tht p \ q 6= ; ) p = q for ll p; q 2 P. SM re lso known s S-systems n EG s mrke grphs, eision-free Petri nets or T-systems. Our enition of FC orrespons to wht is often lle extene Free Choie nets in the literture. It follows from the enitions tht EG F C n SM F C. In Fig. 3, we illustrte the si notions of onurreny, hoie (or eision) n synhroniztion. In se (I), trnsitions n re onurrently enle, i.e. they n re inepenently. In se (II), we sy tht there is hoie etween trnsitions n or tht n re in on- it. In se (III), there is synhroniztion t trnsition. Among Petri nets, SM llow hoie ut not synhroniztion wheres EG llow synhroniztion ut not hoie. FC re the nturl generliztions of oth SM n EG. Fig. 3. (I) (II) (III) (I) Conurreny. (II) Choie. (III) Synhroniztion. B. Exeution semntis n tres Let us onsier Petri net where n trnsitions, to n, re onurrently enle, see Fig. 3,(I). We ssume tht ll of them hve to re efore ny new trnsition eomes enle. Then, the sme ehvior, i.e. the ring of the n trnsitions, n e esrie y ny of the following n! ring sequenes: () ; ; (n) where is permuttion of f; : : : ; ng. This simple exmple shows tht ring sequenes, whih provie sequentil esription of the ehvior, re not relly pte in the presene of onurreny. on nl mrkings n the ierent leling funtions tht one my onsier). The prolem of moeling onurreny in more eient wy hs long een onsiere. A lssil pproh, propose y Mzurkiewiz [3], [32], uses the notion of tre monoi. For generl n reent referene on tres, see [8]. Denition III.4: Let T e n lphet equippe with reexive symmetri reltion lle epenene reltion n enote y D. We enote y I the omplement of D, lle inepenene reltion. The tre monoi T = is the quotient of the free monoi T y the lest ongruene ontining the reltions ; 8(; ) 2 I. The elements of T = will e lle tres. Two wors re representtives of the sme tre if they n e otine one from the other y repetely interhnging jent inepenent letters. Inee, one n esily show tht the reexive n trnsitive losure of the reltion uv uv, 8(; ) 2 I; u; v 2 T is omptile with the monoi struture of T. Hene, this reexive n trnsitive losure is preisely the tre reltion. See [8, Chp.,x.3] for etils. Given Petri net (P; T ; F; M), we ene the following inepenene reltion: I = f(; ) 2 T 2 j ( [ ) \ ( [ ) = ;g ; (5) n the ssoite tre monoi T =. Two trnsitions re inepenent i they o not shre input or output ples. For sfe Petri nets 7, the ongruene generte y I ienties ring sequenes whih ier only y the sequentil orering of onurrent events. In prtiulr, if w w 2 then w 2 L, w 2 2 L (where L is the lnguge of the Petri net). A tre whose representtives re ring sequenes is lle ring tre. The set of ring tres is enote y (L= ). It is very onvenient to visulize tre monois using hep moels, s it ws originlly propose y Viennot in [40]. Let us etil this for the tre monoi T = ssoite with sfe Petri net (P; T ; F; M). Consier the hep moel H(T = ) with: set of piees T ; set of slots P; R() = [, 2 T ; l(; r) = 0; u(; r) = ; r 2 R(). Two wors w ; w 2 2 T re equivlent (w w 2 ) if n only if they provie the sme hep. In the hep ssoite with w 2 L, the piees ssoite with one level (i.e. the piees oupying ommon vertil position) orrespon to the events (i.e. the trnsition rings) ourring onurrently. Exmple III.5: Let us onsier the Petri net of Fig. 2. We hve represente in Fig. 4 the hep of piees ssoite with the ring sequene. Note tht there is no onurreny t ll in this Petri net. It is purely sequentil n the inepenene reltion I ene in (5) is empty. 7 In non-sfe Petri net, two trnsitions ; suh tht \ 6= ; n e onurrently enle if their ommon input ples hol more thn one token. It is possile to introue nother epenene reltion to tke re of this, see Diekert [7]. Suh renements re not neee here s we onsier only sfe Petri nets.

6 6 TO APPEAR IN THE APR. 999 ISSUE OF IEEE TRANSACTIONS ON AUTOMATIC CONTROL We ssoite with its trnsitions, the following ring times: = ; = 2; = 2; = : We ssoite with the ples the holing times: Fig. 4. p p 3 p 2 p 4 p 5 p 6 Hep of piees for the wor w =. = ; 2 = 0; 3 = 0; 4 = 2; 5 = 0; 6 = 0 : Let us ssume tht the initil mrking is the one shown in Fig. 2. Trnsitions n re enle. If we hoose to re trnsition, the ring will e initite t time 0, omplete t time 2, n trnsition will eome enle t time 2. C. Timing A time Petri net (TPN) is net with ring times ssoite with trnsitions n/or holing times ssoite with ples. Severl ring semntis hve een onsiere in the literture. We restrit our ttention to sfe Petri nets n onsier semnti whih oinies with the one of Rmhnni [36] for this sulss. Let e trnsition with ring time n whose output ples p 2 hve holing times p. We ssume tht trnsition eomes enle t instnt t. A ring ours in three steps.. At instnt t, the ring of my e initite. If initite, it removes one token from eh input ple. 2. One token is e in eh of the output ple t instnt t The token e in ple p 2 n ontriute to the enling of the trnsitions in p fter instnt t + + p. Between t n t+, the tokens n e onsiere s eing `frozen' in their originl input ple. The tokens n the trnsition n not e involve in ny other ring etween t n t +. A nturl question to sk is wht hppens if trnsition is not initite t instnt t. First, my never re. Seon, in orer to re, trnsition nees to get isle in rst time (euse of the sfeness property, this hppens preisely when the token of one of the ples in prtiiptes in the ring of nother trnsition), then re-enle lter on. Let us investigte some other onsequenes of this semnti. First of ll, if trnsition res, it oes so \s soon s possile". We sy tht the Petri net opertes with n erliest ring rule. Seon, the eisions on whih trnsitions re to re is not se on time onsiertions. All logilly fesile hoies n e onsiere. This ontrsts with severl moels stuie in the literture. For exmple, in the so-lle re poliy, see for instne [], ple with severl output trnsitions llotes its token to the trnsition whih is le to omplete its ring rst. We enote y (G; M; ) time Petri net, where is mp T [P! R +, proviing the ring n holing times of trnsitions n ples. By onvention, the time evolution of the Petri net strts t instnt 0, in mrking M, the holing times of the initil tokens eing omplete. Exmple III.6: Let us onsier the Petri net of Fig. 2. IV. Hep Representtion for Sfe Time Petri Nets A. Hep representtion theorem In this setion, we stte the min representtion theorem of the pper: ring times of sfe time Petri nets re reognize y hep utomt. Let G = (T ; P; F; M; ) enote sfe time Petri net, with set of ring sequenes L T. The time ehvior of the net is ene s follows: for ring sequene or sheule w = : : : k 2 L, we strt t time 0, n re the trnsitions ; : : : ; k in this orer, pplying the erliest ring semnti esrie in x III-C ove. With eh ple p, n for w 2 L, we ssoite the rel nonnegtive numers: z(w) p = instnt t whih the lst token rrive in ple p uner the sheule w eomes ville for the ring of ownstrem trnsitions. z 0 (w) p = lst instnt of presene of token in ple p, uner the sheule w. We set z 0 (w) p = z(w) p = 0, if no token ws ever present in ple p. We set ( z(w) p if M! w M 0, with M 0 (p) = x G (w) p = z 0 (w) p if M! w M 0, with M 0 (6) (p) = 0 In wors, this is the ompletion time of the lst \event" t ple p, uner sheule w, n \event" eing either the vilility, or the eprture of token. We ll x G the ter funtion of the Petri net. The mkespn or exeution time of the ring sequene w is nturlly ene y y G (w) = mx x G (w) p : p This is the ompletion time of the lst \event" in the Petri net uner sheule w. Theorem IV. (Hep Representtion for Sfe TPN) Let G = (T ; P; F; M; ) e sfe time Petri net with lnguge L. Then, the hep moel H = (T ; P; R; l; u), with 8 2 T ; R() = [ ; 8 2 T ; 8p 2 ; u(; p) = + p ; 8 2 T ; 8p 2 n ; u(; p) = 0 ; 8 2 T ; 8p 2 R(); l(; p) = 0 ;

7 GAUBERT AND MAIRESSE:MODELING AND ANALYSIS OF TIMED PETRI NETS 7 is suh tht 8w 2 L; x G (w) = x H (w); y G (w) = y H (w) : (7) Eqution (7) sttes tht the ter vetor of the net oinies with the upper ontour of the ssoite hep. Let us onsier the hep moel otine from H y repling u n l y ^u() = ; 2 R() n ^l() = 0; 2 R(), respetively. This is preisely the hep moel ssoite with the tre monoi T = of the Petri net, i.e. H(T = ), see x III-B. The piees of the hep moel H re otine y eformtion of the piees of the hep moel H(T = ), inorporting the timing informtion. Proof of Theorem IV.: Let us onsier w 2 L; 2 T ; suh tht w 2 L. We hve, 8 >< x G (w) p x G (w) p = mx p 0 2 >: x G (w) p p if p 2 if p 62 [ mx p 0 2 x G (w) p 0 if p 2 n. (8) Inee, the ring of trnsition fter w is initite t instnt T = mx p 0 2 x G(w) p 0 ; from whih (8) follows. The ynmis (8) woul oinie vertim with the one of the hep moel H given ove (see (2)), if the term mx p 0 x G (w) p 2 0 ws reple y mx p 0 2 x G(w) p 0. Hene, it remins to hek tht [ or equivlently mx x G (w) p 0 = mx x G(w) p 0 ; p 0 2 [ p 0 2 8p 00 2 n ; x G (w) p 00 mx p 0 2 x G(w) p 0 : Sine p 00 2 n, the ring of t time T s one token to the mrking of p 00. Sine the net is sfe, there is t most one token in eh ple, for ny logilly missile exeution of the system. We onlue tht the exit time of the lst token in ple p 00 uner the ring sequene w must e stritly less thn T. Using the ening reltion (6), x G (w) p 00 is equl to the lst instnt of presene of token in p 00, uner w. We onlue tht x G (w) p 00 T = mx p 0 2 x G (w) p 0. Due to the ommuttion in (4), the hep ssoite with w 2 T, n fortiori x H (w) n y H (w), epen only on the equivlene lss of w in T =. Although this hep, x H (w), n y H (w) re ene for ll w 2 T, they hve no mening in terms of the Petri net if w 62 L. Exmple IV.2: Let us illustrte the previous onstrution with the Petri net G = (T ; P; F; M) represente in Fig. 2 n the numeril vlues of Ex. III.6. The hep moel ssoite with G is H = (T ; P; R; l; u) with R() = [ = fp ; p 2 ; p 3 ; p 4 g; R() = fp ; p 2 ; p 3 g; R() = fp 3 ; p 5 ; p 6 g; R() = fp 3 ; p 4 ; p 5 ; p 6 g : u(; :) = [0; ; 0; 3; 0; 0]; u(; :) = [3; 0; 2; 0; 0; 0]; u(; :) = [0; 0; 0; 0; 2; 0]; u(; :) = [0; 0; ; 3; 0; ] : We hve represente the hep of piees ssoite with the sheule w = in Fig. 5. For the lrity of the gure, piees with prts of zero with hve een mterilize y repling zero y `smll' ut stritly positive height. This hep is eformtion of the one of Fig. 4. The reer oul hek iretly (y simultion of the Petri Fig. 5. p p 2 p 3 p 4 p 5 p 6 Hep of piees ssoite with the wor w =. net) tht the height of the hep, y G (w) = 6, ttine t slot 4, orrespons to the lst ourrene of n event in the system (the vilility of the token in p 4, fter the ring of the lst ourrene of in the sheule w). B. The miniml reliztion prolem The size of the hep representtion in Theorem IV. is equl to the numer of ples of the Petri net, possily lrge numer. This rises nturlly the following miniml reliztion prolem: wht is the miniml size of hep representtion of given sfe time Petri net? As prtil nswer to this proly iult prolem, we will show tht simple (usully smll) hep representtions n e uilt from struturl invrints, for sulss of nets. To formlize this rigorously, we introue the following enition. Denition IV.3 (Hep Reliztion) We sy tht time Petri net (G; M) with lnguge L, hs Hep Reliztion of size k if there is hep moel H with k slots, suh tht 8w 2 L; y G (w) = y H (w) : Tht is, the exeution time of the ring sequene w oinies with the height of the hep of piees w. It is not require tht x H = x G, whih gives the potentil for smller hep reliztion. The following notions re lssil, see [6, x5.]. Denition IV.4 (Stte Mhine Covering) A stte mhine omponent of Petri net G is sunet G 0 of G, tht is stte mhine, n stises p [ p G 0 for every ple p of G 0.

8 8 TO APPEAR IN THE APR. 999 ISSUE OF IEEE TRANSACTIONS ON AUTOMATIC CONTROL We sy tht G ; : : : ; G k is (rinlity k) stte-mhine overing of Petri net G, if. G ; : : : ; G k re stte-mhine omponents of G; 2. every r of G elongs to t lest one omponent G i, i k. We sy tht the overing is sfe (resp. live) if it is ompose of sfe (resp. live) stte-mhine omponents. It follows from this enition tht stte mhine omponent is uniquely ene y its set of ples. Note tht sfe (resp. live) SM is SM with t most (resp. t lest) one token. Theorem IV.5 (Reue Reliztion Theorem) Let G = (P; T ; F; M; ) e sfe time Petri net with lnguge L n hving sfe stte mhine overing (G ; : : : ; G k ). Then, G mits hep reliztion of size k given y the hep moel H = (T ; f; : : : ; kg; R; l; u), where 8 2 T ; R() = fi j 2 G i g ; 8 2 T ; 8i 2 R(); l(; i) = 0 ; 8 2 T ; 8i 2 R(); u(; i) = + p(;i) ; where p(; i) is the unique ple suh tht (; p(; i)) 2 G i. Proof: We rst prove the result when the sttemhine omponents re not only sfe ut live. Then, there is extly one token t ny time in eh stte-mhine omponent n we my spek unmiguously of the token in G i. With this token, we ssoite ter funtion ~z i. If w is ring sequene, we enote y ~z(w) i the time t whih token i eomes ville in its urrent ple p 2 G i, fter the ring of the lst trnsition 2 w suh tht p 2 (it is the \ompletion time" of sheule w for the token i). It is ler tht mx i ~z(w) i = y G (w). Hene, it is enough to prove tht ~z(w) = x G (w), for w 2 L. We set ~z(e) i = 0 (rell tht e enotes the empty wor). Clerly, ~z(e) = x H (e). Let us onsier w 2 L; 2 T suh tht w 2 L. We hve ( ~z(w) i if i 62 R() ~z(w) i = : mx p 0 2 x G (w) p p if i 2 R() (9) We rell tht p(; i) is the unique ple suh tht (; p(; i)) 2 G i. For ll ples p 0 2, there exists t lest one stte mhine omponent G j suh tht (p 0 ; ) 2 G j. Thus, mx x G(w) p 0 = mx ~z j (w) : (0) p 0 2 j: 9p 00 ;(p 00 ;)2G j Arguing s t the en of the proof of Theorem IV. (using the sfe hrter of the net), we get tht mx ~z(w) j = mx ~z(w) j:9p 00 ;(p 00 ;)2G j j: 9p 00 ;(p 00 ;)2G j or (;p 00 )2G j = mx j2r() ~z(w) j : () Sustituting (0) in (9), n using (), we get tht ~z stises preisely the ynmil equtions (2) of the Hep moel H. This onlues the proof of the theorem, when the stte-mhine overing is live. For generl sfe ut not-live overing, there is either 0 or token in eh stte-mhine omponent. All the trnsitions within unmrke stte-mhine omponents will never re. It is now immeite to pt the ove rgument, setting ~z(w) i =, for ny unmrke sttemhine omponent G i. Exmple IV.6: Let us illustrte Theorem IV.5. We onsier the Petri net G = (T ; P; F; M) of Fig. 2, with the timings ene in Ex. III.6. This net mits eomposition into 4 SM-omponents, G i ; i = ; : : : ; 4, with respetive sets of ples: P = fp ; p 2 g; P 2 = fp 2 ; p 3 ; p 5 g; P 3 = fp 4 g; P 4 = fp 5 ; p 6 g : Let us onsier the ssoite hep moel H s in Theorem IV.5. It is extly the hep moel ene in Ex. II.2 n Ex. II.6. The set of slots is R = f; 2; 3; 4g, slot i orresponing to the SM-omponent G i. The hep ssoite with the sheule ws represente in Fig.. As further illustrtion, the heps ssoite with n re represente in Fig. 6,(,2). Fig. 6. (,2): Heps of piees for the wors n. (3): Miniml Hep reliztion It is interesting to note tht the hep reliztion given ove is not miniml. A smller reliztion is shown on Fig. 6,(3). Note tht this size 3 reliztion is not ssoite with stte mhine overing of the net (here, the rinlity of stte mhine overing is t lest equl to 4). This shows tht Theorem IV.5 provies only prtil nswer to the miniml reliztion prolem. The next lssil result (see for exmple [6, Th. 5.6]), shows tht the reue reliztion of Theorem IV.5 n e pplie to ll live n sfe Free Choie nets. Proposition IV.7: A live n sfe Free Choie net mits live n sfe stte mhine overing. In the se of n event grph, the sme result pplies when repling stte mhine overings y iruit overings. The prolem of ning the overing of miniml rinlity in Prop. IV.7 (or, in generl, for Petri nets mitting suh overings) ppers to e iult one.

9 GAUBERT AND MAIRESSE:MODELING AND ANALYSIS OF TIMED PETRI NETS 9 Remrk IV.8: In his thesis, Hulgr propose similr pproh for sfe FC [28, Chp. 7]. He ene n nlogue of the vetor x G (w) n of the mtries M(), n erive ynmil equtions similr to (9). The min ierene is tht he uses retngulr mtries whih epen not only on the trnsition to re ut lso on the urrent mrking. C. Hep reliztion n P-semiow There is lose onnetion etween the size of the hep reliztion in Theorem IV.5 n lssil invrint, the P-semiow. A P-semiow is olumn vetor x 2 N P suh tht P p2 x(p) = P p2 x(p); 8 2 T. Tht is, the weighte mrking (usul lger) P p x(p)m(p) of the ples is invrint y the ring of trnsition. Let G e sfe time Petri net mitting SM overing fg ; : : : ; G k g. We onsier the stritly positive vetor x = x G + + x Gk, where x Gi 2 R P mx is the hrteristi vetor of G i, ene y: x Gi (p) = ( if p 2 G i 0 otherwise : The vetor x is P-semiow. The invrint P p x(p)m(p) is equl to k times the size of the hep reliztion of G. However, it is not true tht eh P-semiow of Petri net n e represente s the sum x G + +x Gl of the hrteristi vetors of SM overing. A fortiori, the prolem of ning sfe SM overing of miniml rinlity n not e reue to the lssil prolem of ning miniml P-semiow. We ene the expnsion ~G of the Petri net G with respet to x s follows: Eh ple p 2 P suh tht x(p) > is reple y x(p) ples. Eh of these x(p) ples hve the sme input n output trnsitions s the originl ple p. They lso hve the sme numer of tokens n the sme holing time s p. The ring sequenes n the temporl ehviors of ~G n G re extly the sme (given ring sequene w, the ring instnts of the trnsitions re the sme). Furthermore, the grph ~G mits SM prtition 8 ( ~G ; : : : ; G~ k ). This is est unerstoo with n exmple. Exmple IV.9: Let us onsier the Petri net of Fig. 2. A stte mhine overing of this Petri net ws provie in Exmple IV.6. The ssoite hrteristi vetor is x = (; 2; ; ; 2; ). Hene, we hve to uplite p 2 n p 5. We hve represente the expne Petri net with its SM-prtition in Fig. 7. The reue hep utomton H ssoite with ~G is the sme s the one ssoite with G. Now, the numer of tokens of ~G is onstnt n there is simple interprettion for (x H ) i (or (x H ~ ) i): it is the ter funtion of the token of ~G i. 8 Sme enition s SM overing exept tht eh r elongs to extly one SM omponent. G 2 p p 2 p~ 2 p~ 5 p 5 p 6 p 3 p 4 G G 3 Fig. 7. Expnsion of the Petri net of Fig. 2. G 4 V. An Applition to the Moeling n Performne Anlysis of Joshops The results presente ove n nturl omin of pplition in sheuling theory. A goo introution to the sujet is provie y the ooks [8], [0]. We rst show how hep representtions n e use to esign performne evlution methos. We explin informlly the metho on smll mnufturing moel, n we ompre it with the lssil pproh. Then, we onsier the generl sulss of sfe joshops. We esrie the lssil performne evlution lgorithm n new hep utomt se one, n we erive omplexity ouns for oth of them. A. An Elementry exmple Let us onsier the Petri net of Fig. 8, tht the reer ertinly reognizes s eing the one isusse extensively ove. p Fig. 8. J p 2 S p 4 J 2 p 3 p 5 p 6 M Holing n Firing Times p p 2 p 3 p 4 p 5 p A two jos two resoures mnufturing system This Petri net n e interprete s mnufturing system proessing two jo types J, J 2, using two (heterogeneous) resoures: one speilist S n one mhine M. There re four elementry tsks ; ; ;. The proution sequene for jo J is whih mens tht the elementry tsks n hve to e performe in this orer to omplete one jo J. The proution sequene for jo J 2 is.

10 0 TO APPEAR IN THE APR. 999 ISSUE OF IEEE TRANSACTIONS ON AUTOMATIC CONTROL Let w 2 T. For eh jo type J i ; i = ; 2; we set jwj Ji = numer of type J i jos omplete uner the sheule w. (2) Then, given n innite sheule 9 z = 2 3 : : : 2 T!, with ; 2 ; : : : 2 T, we ene the symptoti throughput of the jos of type J i : p p 2 p 32 p 3 p5 p 6 i = lim inf n! j : : : n j Ji exeution time of : : : n : (3) We re intereste in ning the innite sheules mximizing the throughput, uner proution rtio onstrint (e.g. one jo J for one jo J 2, in the verge). We restrin this prolem, y requiring the sheules to e perioi. Tht is, one only onsiers perioi sequenes of the form v! = vvvv : : :, where v is nite proution pttern stisfying the rtio onstrint. In this se, s etile elow, the lim inf in (3) eomes limit (this will follow from the (mx,+) liner representtion, together with the (mx,+) yliity theorem). For instne, let us onsier miniml length ptterns with rtio =. There re two possile forms for suh ptterns: n. Moreover, we note tht the symptoti performne is invrint y yli onjugy of the pttern. Tht is, for ll wors u; v, (uv)! n (vu)! hve the sme symptoti throughput, whih follows from the ientities (uv)! = u(vu)!, (vu)! = v(uv)! (the two ehviors ier only y nite numer of tsks). Hene, there is only one ehvior to onsier: L = ()! : One might of ourse onsier longer ptterns. E.g. perioi sequenes whose pttern onsists in the proution of 2 jos J n 2 jos J 2 re given y: L 2 = ()! [ ()! : We will not onsier s suh the sheule optimiztion prolem (whih is iult omintoril one), ut we will show how the hep-se moeling mkes esier the suprolem of the performne evlution of given perioi sheule. This is est unerstoo y omprison with the time Event Grph moeling, tht we next rell. A. Illustrting the lssil pproh For given perioi sheule, one is le to uil time Event Grph representing the system, n then to ompute the perioi throughput of this time Event Grph. Let us onsier for exmple the sheule ()!. This funtioning is represente y the time Event Grph isplye in Fig. 9, whih is otine from the time Petri net of Fig. 2 y repling the resoure ples p 3 n p 4 y iruits, foring the perioi sequene : : : Let x(n) 2 R T mx e the vetor proviing the tes of the ompletion of the n-th ring of the trnsitions. The vetor 9 We enote y T! the set of innite wors over the lphet T. p 4 p 42 Holing n Firing Times: p p 2 p 3 p 32 p 4 p 42 p 5 p Fig. 9. Time Event Grph for the sheule ()! x(n) evolves oring to (mx,+) liner ynmi: x (n) = ( x (n ) ( 3 4 )x (n )) x (n) = 2 x (n) x (n) = ( 32 x (n) 6 x (n )) x (n) = ( 5 x (n) 42 x (n)) : Setting (n) = [x (n); x (n)] t, eliminting x n x, n tking the numeril vlues of Fig. 9, we otin the susystem (n) = A(n ), with 4 5 A = ; (A) = 8 ; 7 8 where (A) enotes the (unique) (mx,+) eigenvlue of the (irreuile) mtrix A (see [2], [2], prtiulrly the introutive setion x.3 of [2], for etils on the (mx,+) spetrl theory n its pplitions to isrete event systems). It follows from the yliity theorem in [2] tht for i = ; 2; n u 2 T, the symptoti throughputs re n i = lim = n! x(n) u (A) = 8 : (4) For sheule with longer perio, one woul hve to perform similr nlysis on lrger time EG. For instne, the time EG orresponing to the sheule w = ()! is shown on Fig. 0. A.2 Illustrting the utomt pproh We ssoite with the time Petri net of Fig. 8 the reue hep moel n utomton given in Ex. IV.6 ove. As oppose to wht ws one in the lssil pproh, one onsiers single Petri net (the originl one, Fig. 8) n single lgeri representtion (the hep moel). Only the orer in whih the prouts of the mtries M(u); u 2 T ; re performe is moie from one sheule to the other.

11 GAUBERT AND MAIRESSE:MODELING AND ANALYSIS OF TIMED PETRI NETS p 2 p 2 p 42 p 5 2 p p 3 p 4 p 32 p 22 p 34 2 p 33 p 62 2 p 43 p 52 2 p 6 nite set R of resoures (mhines); nite set T of elementry tsks; for eh tsk 2 T, urtion () n single mhine R() 2 R on whih is to e exeute; nite set J T of proution sequenes or jos. Eh jo J = : : : k 2 J is ompose of nite numer of tsks : : : k, to e exeute in this orer. We require 0 tht tsk 2 T elongs to unique jo J() 2 J. We sy tht unit of jo J is proue eh time the proution sequene J is omplete. This moel is equivlent to the one of [27]. We will mke restrition y ssuming tht the work in proess for eh jo is equl to one, tht is, t most one unit of eh jo is proesse simultneously. We ll suh joshops sfe, this ssumption eing equivlent to the sfeness of the nturl Petri net representtion of the system, shown in Fig.. jo jo jj j p 44 unmrke hin Fig. 0. Time Event Grph for the sheule ()! In prtiulr, for perioi sheule (v)!, the symptoti throughput of jo J i is given y i = jvj J i (M(v)) : For instne, the mtrix M() ws given in Ex. II.6. Its eigenvlue is (M()) = 8, proviing throughput i = =8, whih onrms the vlue otine in Eqn (4). Similrly, one my ompute the mtrix M() n otin (M()) = 5, yieling throughput i = 2=5 > =8. This improvement of the throughput n e visulize on the heps of piees of Fig. 6. These omputtions oul e performe equivlently, n more simply, with the three imensionl mtries orresponing to the miniml reliztion, see Fig. 6,(3). More generlly, one n esily hek tht the optiml perioi sheule of perio 4n n stisfying / rtio onstrint is v! n with v n = () n () n. It n e inferre from the heps of Fig. 6 tht the ssoite throughput is = 2 = n 7n + ; so tht i inreses to =7 s n! +. This is of inepenent interest. It shows tht we n lwys improve on the throughput y inresing the length of the pttern. Hene there exists no optiml sheule with nite perio (espite the ft tht ll urtions re integer vlue). An exmple of the sme kin (ut for non-sfe Petri net) ws exhiite in Crlier n Chretienne [9, xvi-]. We next turn our ttention to the generl lss of joshops. B. Performne evlution of sfe joshops Denition V.: A joshop is speie y: mhine Fig.. mhine jrj Generi sfe Petri net ssoite with sfe joshop We lso ssume tht the joshop, or equivlently the Petri net, is onnete. It mens tht there is no proper suset of jos shring no resoures with some other jos. The extension to the non-onnete se is strightforwr. The Petri net mits nturl overing y live n sfe stte mhine omponents, eh jo n mhine orresponing to omponent. Hene, joshops mit reue hep reliztions (see Theorem IV.5): H = (T ; R 0 ; R 0 ; l; u); R 0 = R [ J ; R 0 () = R() [ J(); l(; r) = 0; u(; r) = (); 8r 2 R() : (5) We will e intereste in perioi sheules of the form v! 2 T!, where v elongs to L, the lnguge of the Petri net, n is pttern orresponing to the ext ompletion of severl jos (i.e. without leving some proution sequene unnishe) n meeting xe rtio onstrint. In line with (2), for eh proution sequene J, we will enote y jvj J the numer of units of J omplete uner v. The symptoti throughput of jo J is ene s in (3), repling J i y J. The following result is n immeite onsequene of the hep representtion theorem, together with the yliity theorem for powers of mtries [2, x 3.7.5, Th. 3.2]. Theorem V.2 (Throughput Formul) For sfe joshop with hep reliztion (5) n ssoite mtrix representtion M, the symptoti throughput of jo J is given y J = jvj J (M(v)) ; 0 We my lwys ssume this. If the sme physil tsk ours in two jos, we hve to represent it y two istint letters.

12 2 TO APPEAR IN THE APR. 999 ISSUE OF IEEE TRANSACTIONS ON AUTOMATIC CONTROL where (M(v)) is the (mx,+) eigenvlue of M(v). As the joshop is ssume to e onnete, the mtrix M(v) is irreuile, hene it hs unique eigenvlue, see [2]. As yprout of this theorem, we otin n lgorithm to ompute J. Algorithm V.3 (Automt-se) Input: joshop, pttern v 2 L.. Buil the Hep moel (5), n its ssoite mtries M(), 2 T. 2. Compute the prout of mtries M(v). Complexity : O(jvj(jJ j + jrj)). 3. Compute the eigenvlue of M(v), (M(v)), using Krp lgorithm. Complexity: O(jJ j + jrj) 3. Output: J = jvj J (M(v)). Totl omplexity: O(jvj(jJ j + jrj) + (jj j + jrj) 3 ). The jvj prouts of mtries in M(v) n e ompute in sprse wy. Inee, the mtries M(), 2 T, ier from the ientity mtrix only on two row n two olumn inies, so tht the omplexity of n opertion MM() for ny mtrix M of size jj j+jrj is O(jJ j+jrj). For etils on Krp lgorithm, see e.g. [2, Th. 2.9]. For omprison, we next give the most eient vrint known to us of the \lssil" performne evlution lgorithm. The generl metho is orrowe from [2], ut the renements whih gretly reue the exeution time n not e foun in the literture. Algorithm V.4 (Clssil or Event Grph-se) Sme input n output s Algorithm V.4.. Buil the time EG representtion of the system, following [2, x 2.6] or [27]. 2. Write the (mx,+) liner representtion x(n) = A 0 x(n) A x(n ) ; (6) where x enotes the vetor of ter funtions of the trnsitions of the time Event Grph, [2, x 5.]. 3. Let C enote the set of trnsitions with t lest one token in one ownstrem ple. Compute the C C sumtrix A of A 0 A. Complexity: O(jvj(jJ j + jrj)) + (jj j + jrj) 2 ). 4. Compute the eigenvlue (A), using Krp lgorithm. Complexity: O(jJ j + jrj) 3. Totl omplexity: O(jvj(jJ j + jrj) + (jj j + jrj) 3 ). The time EG uilt in step of Algorithm V.4 hs jvj trnsitions n jj j + jrj tokens. Hene, the mtries A 0 n A in Eqn (6) re of imension jvjjvj. For live time event grphs, the mtrix A 0 hs no iruits. Then, we n erive from (6) the nonil form x(n) = A 0A x(n ) ; where A 0 = A 0 0 A 0 A 2 0 A jvj 0 (see [2, Th. 3.7]). By onstrution of C, if j 62 C, (A 0 A ) ij = 0, for ll i. Hene, A 0A hs only one nontrivil igonl lok A, in position C C, n the growth rte of (A 0A ) k oinies with the growth rte of A k. This is the exeution time, the usul opertions (omprison, ition, et) ounting for one unit. The rinl jcj vries with the mrking ut is of orer jj j + jrj. To ompute (A 0 A ) CC, we hve ) to selet the rows of inies i 2 C of A 0, 2) to multiply eh row of inex i 2 C of A 0 y eh olumn of inex j 2 C of A. Sine the mtrix A 0 hs no iruits, using rnk funtion, we n ompute row of A 0 in time O(E), where E is the numer of rs of the grph of A 0 (see [24, Chp. 2,x2.4]). Here we hve E = O(jvj). We onlue tht the omplexity of omputing jcj rows of A 0 is O(jvj(jJ j + jrj)). On eh olumn of A, there re t most two terms ierent from 0. Thus, omputing the jcj 2 entries of (A 0 A ) CC tkes time O(jvj(jJ j + jrj)) + (jj j + jrj) 2 ). The totl omplexities of oth lgorithms re the sme. However, we i not tke into ount the \moeling" omplexities for oth methos. This spet n e onsiere s strong rgument in fvor of the utomt-se lgorithm. A new time Event grph must e uilt for eh new sheule in the tritionl metho, wheres the hep reliztion M is uilt only one, n remins vli for ll (even non-perioi!) sheules. Moreover, the size of the time EG grows with the length of the pttern, wheres the size of the hep utomton remins onstnt. Remrk V.5: Aitionl fetures n e inorporte to the joshop of Def. V., the ove moeling y hep utomton remining vli. First, tsk my require severl resoures t the sme time to e exeute. Seon, there might e generl preeene reltions etween the tsks of jo (it orrespons to repling the live n sfe iruits in Fig. y live n sfe Event Grphs). Exmple V.6: We illustrte Theorem V.2 with the joshop esrie in [27], x III. There re three mhines: R = fm ; M 2 ; M 3 g, four proution sequenes J = fj ; : : : ; J 4 g, J = 2 3, J 2 = 2, J 3 = 2, J 4 = 2. The resoure llotion n time exeution mps re given in the following tle: Tsk R M M 2 M 3 M 3 M 2 M M 3 M M One requires proution mix of =4; =4; =4; =4 etween J ; J 2 ; J 3 ; J 4. A sheule stisfying this onstrint (n omptile with the orer of preeene of the elementry tsks) is v!, with v = The mtrix of the resoure utomton of is (omitting the 0 entries): M 0 M 2 M 3 J J 2 J 3 J 4 M M 2 M 3 M( ) = J J 2 B J J 4 We leve it to the reer to write the other mtries, n just give M(v) = C A : C A

13 GAUBERT AND MAIRESSE:MODELING AND ANALYSIS OF TIMED PETRI NETS 3 We hve (M(v)) = 9, whih yiels the throughput of =9, s in xv of [27]. We use toy implementtion 2 in Mple V.3. There re ;2;2;2 = 9!=(3!(2!) 3 ) = 7560 sheules ( sheule is wor in the shue prout L of the four wors 2 2, 2, 2, 2 ). But the performne of sheule only epens on its equivlene lss moulo the prtil ommuttions xy xy, for the ouples of tsks (x; y) elonging either to the sme jo or to the sme mhine. There re only 26 suh equivlene lsses. Moreover, sine the perioi throughput is invrint y yli onjugy of the pttern (rell tht two wors of the form uv n vu re yli onjugtes), we only kept one wor y lss of yli onjugy (92 wors remine). Finlly, we h to ompute the mtrix prout M(w) for these 92 wors. We foun n optiml throughput of =7, ttine for instne for the yli sheule ~v!, with ~v = This pproh hs een evelope in [26]. It n e omine with rnh n oun tehniques tht re lssil in sheuling. Remrk V.7: It is essentil to note tht the hep representtion oinies (up to 90 rottion) with version of the Gntt hrts, where oth the ouption of jos n mhines re represente. Tritionl Gntt hrts re extly the restrition to the mhine olumns of heps of piees representtions. As n illustrtion, we hve represente elow the tritionl Gntt hrt for the moel trete t length in this pper (Ex. II.2, Ex. II.6, Ex. III.5, Ex. IV.2, Ex. IV.6, xv- A) n the sheule. The two e piees seem to \ot" in the ir euse one ritil moeling vrile is lking. Inee, the smllest hep representtion is of size three, see Fig. 6,(3). A. (Mx,+) utomton n hep utomton Clssilly, (mx,+) utomton A = (Q; I; F; M) over the lphet T n e represente y nite R mx -vlue n T -lele grph s follows. One rws nite grph with noes q 2 Q. There re three types of rs. For eh q suh tht I q 6= 0, one rws n input r vlue y the slr I q (with no lel). Suh noe q is lle initil. Dully, for eh q suh tht F q 6= 0, one rws n output r vlue y the slr F q (with no lel). Suh noe q is lle nl. For eh triple (q; ; q 0 ) 2 Q T Q suh tht M() qq 0 6= 0, one rws n r from q to q 0, lele with the letter n vlue y the slr M() qq 0.! q 2 : : : q n n! q n+ ) is The weight of pth p = (q the prout w(p) = I q M( ) qq 2 : : : M( n ) qnq n+ F qn+, evlute in the R mx semiring. The lel of this pth is the prout of the lels of the eges: `(p) = : : : n. Then, the multipliity of the wor w is equl to the mx of the weights of the pths of lel w: y A (w) = L p:`(p)=w w(p). Exmple VI.: As n illustrtion, we provie in Fig. 3 the grphil representtion of the (mx,+) utomton of Ex. II.6. ; 2; ; ; 2; ; 3 q q3 ; ; 2 ; 2 3; 3 3 2; q2 q4 ; 3; ; ; 3 ; ; 2; Fig. 2. (Conventionl) Gntt hrts re not (mx,+) liner. Fig. 3. (Mx,+) utomton. VI. Epilogue: Algeri Sttus of Petri Net Hep Representtions In orer to pply the mhinery of utomt to performne evlution prolems, we next isuss t more lgeri level the ierent moels use in this pper. ) At the logil level, the set of missile ehviors of Petri net G is esrie y its lnguge L, whih is reognize y \lssil" utomton (eterministi Boolen utomton), the mrking utomton. 2) At the time level, the exeution time of n missile sequene w 2 L is reognize y hep utomton. It is very nturl to eme oth moels in ommon lgeri frmework, s follows. 2 Aville on the uthor's we pges: n B. Boolen utomton n mrking utomton Strting from (mx,+) utomton, y speiliztion to the Boolen semiring 3, one otins the lssil notion of (noneterministi) utomton: the multipliity of w is if the wor w is epte, i.e. if there is pth with lel w from n initil stte to nl stte, n 0 otherwise. The lnguge of (Boolen) utomton is the set of wors epte y this utomton. A Boolen utomton is eterministi if for ll (q; ) 2 Q T, there is t most one q 0 suh tht M() qq 0 6= 0, n if there is unique q suh tht I q 6= 0. Let G = (P; T ; F; M) e sfe Petri net. The mrking utomton 4 of (G; M) is the eterministi Boolen utomton (R(M); I 0 ; R(M) ; M 0 ), where the initil vetor I 0 3 The Boolen semiring B = (fflse; trueg; or; n) is isomorphi to the susemiring (f0; g; ; ) of R mx. 4 Also known s mrking grph or rehility grph.