Tsung Hsien Yu Department of Management and Information Science National Cheng Chi University Taipei, Taiwan, ROC

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1 Th Closd-Form Solution of th Control Rlatd Stats of Dficint Gn-Lft -th ordr Systm (th ssntial lmnt of non-sharing subnt) of Ptri Nts Tsung Hsin Yu Dpartmnt of Managmnt and Information Scinc National Chng Chi Univrsity Taipi, Taiwan, ROC Abstract Earlir, Chao pionrd th vry first closd-form solution of th numbr of rachabl and othr stats for -th ordr systm which is th first stp that will lt th xponntial computation tim for numrating rachabl stats of a particular larg Ptri Nt b rducd within intra-scond. This papr progrsss on stp furthr on numrating th Control Rlatd Stats for th Dficint Gn-Lft -th ordr systms (th initial maring of idl placs in on procss is lss than ) which is an ssntial lmnt of non-sharing subnt in PNs. Kywords Control systms; discrt vnt systms; flxibl manufacturing systms; Ptri nts I. INTRODUCTION In th light of th rapid innovation of th Intrnt of Things (IoT), robot systms and th cloud computing systm, w nd an fficint mthodology to modl ral-tim rsourc-allocation systms for bottlnc situations, dadloc avoidanc/prvntion policy and othr systm control rlatd problms. As th widly applid instrumnt for modling and analyzing flxibl manufacturing (or rsourc allocation) systms (FMS) (or RAS) [-9], th prsist problm of using Ptri nt (PN) for modling various systms is th larg numbr of stats gnratd (calld th stat xplosion problm). It has bn shown that th complxity of th rachability problm of PN is EXPSPACE-hard [0] and is NP-complt for a liv and saf Fr Choic nt (LSFC) [0]. Evn using mixd intgr programming(mip) [] to solv th rachability problm, it still is an NP-hard problm [2]. Hnc th bottlnc is how can gnrat th rachability rlatd information fficintly, which will consum th xponntial computation tim, so that w can apply th information for th rsourc allocation/dallocation dcisionmaing for a larg ral-tim, dynamic rsourc allocation systm. To ovrcom this barrir, applying graph thory Chao [] constructd th concpt of complt rachability graph; split th rachability graph of th control nt into rachabl, liv, forbiddn, dadloc, non-rachabl and mpty-siphon+nonrachabl stats (w call all of diffrnt typs of control stats as Control Rlatd Stats, dnotd as CRSs) and pionrd th vry first closd-form solution of th numbr of CRSs for -th ordr systm (dfind in Dfinition ) (.g. Fig.). Th most important contribution is that th xponntial computation tim of driving a particular and vry larg PN s CRSs information can b rducd within intra-scond. W hav also xtndd and applid Chao s [] y mthodology to construct th closd-form formulas of th CRSs of various -th ordr systms with a non-sharing rsourc r* in diffrnt spcific locations [4-24], rspctivly. Howvr, th rsarch domain is limitd on th issu of th ffct of non-sharing rsourc in -th ordr systm.(in this papr, th closd-form formulas /solution man th closd-form formulas/solution of CRSs for PNs) Th dadloc avoidanc/prvntion policy for PNs prsntly is focus on how to find critic first-mt bad marings (FBMs) for maximally prmissiv control pur which is th policy basd on th nt structur [25]. Chao [26] showd that for th dadlocs prvntion policy of a -th ordr systm with =5, it nds additional 0 controllrs. Hnc it will b havy cost in modling larg ral-tim rsourc allocation systms and adopting full dadlocs prvntion policy for both th spac and tim implmntation problms of th controllrs allocation. Furthrmor, modling th collision problms of robots collaborativ manufacturing systms, w cannot add additional controllrs to prvnt collision but nd find a mchanism to avoid collision problms. Basd on th contributions of closd-form solution, w pro a nw concpt - th momnt to launch rsourc allocation (MLR) for a partial dadloc avoidanc\prvntion policy to sav th cost of controllrs implmntation for larg ral-tim systms [2, 27]. In [28]applying th closd-form solution of Gn-Lft and Gn-Right, th MLR concpt can b compltd for th non-sharing rsourc allocation loading balanc in whol variant -th ordr systm; rgarding th dadloc thrad-holdr (dnotd as DTH) as a non-sharing waiting dummy rsourc that can provid procss holding this DTH and wait for th othr procss s wor flow, a simpl dadloc avoiding algorithm is prod. Th dcisionmaing of a DTH allocation is basd on th maximum valu of th rachabl stats of Gn-Lft or Gn-Right (dfind in Dfinition ) subnt and th loading balanc of procsss. Howvr xtnding th application of MLR to th variant -th ordr systm containing a non-sharing subnt, th first

2 problm to b solvd is to construct th closd-form solution of th Dficint -th ordr systm with a non-sharing rsourc. (.g. Fig.2). As shown in Fig., unlss thr is a mchanism of dynamic assigning th tons of init maring, othrwis th procss of non-sharing subnt (NNS) will shar th tons that flow into th -th ordr li systm. Hnc th Dficint -th ordr systm is th ssntial lmnt of non-sharing subnt in variant -th ordr systm. This papr will rport th construction of th closd-form solution of a Dficint Gn- Lft systm. Du to th closd-form solution w can nhanc th capability of dynamic modling a PN containing a nonsharing subnt and rsourc xtndd from variant -th ordr systms. Th approach is xplaind as follows. Chao [] has provd that in a -th ordr systm th numbr of nonrachabl stats is th numbr of total stats Ŗ (Ŗ is th numbr of rachabl stats); F=Ŗ Ļ whr F (rsp. Ļ) is th numbr of forbiddn (rsp. liv) stats. Lt Ŗ(,h) (rsp. Ļ(,h)) b th numbr of rachabl stats of a Dficint -th ordr systm with h (h ) tons in th lft idl plac. W hav shown th stats pattrns of th Additional Rstrictd Rachabl (rsp. Liv) Stats drivd from non-rachabl (rsp. non-liv) stats of th quivalnt (dfind in Dfinition ) causd by a non-sharing rsourc, ARRS ΔŖ( rsp. ARLS ΔĻ) [2], so that th numbr of rachabl (rsp. liv) stats of Dficint Gn-Lft will b Ŗ(,h) + Ŗ(,h ) +ΔŖ(rsp. Ļ(,h) II. LITERATURE REVIEW Dfinition : A variant Dficint -th ordr systm is a subclass of S PR with rsourc placs r, r 2,, r shard btwn two procsss N L and N R and on non-sharing rsourc plac r (=r*) usd by an opration plac p * in P L [2].. r P R, M 0 (r)= []. 2. N L (rsp. N R ) uss r, r 2,, r (rsp. r, r, r 2, r ) in that ordr [].. M 0 (p 0 )=h, h<, M 0 (p 0)=, whr p 0 and p 0 ar th idl placs in procsss N L and N R, rspctivly. Th systm is dnotd as a h systm []. 4. Whn M 0 (p 0 )=M 0 (p 0)= and non-sharing rsourc dos not xist, th systm is calld a -th ordr systm [2]. 5. Holdr placs of r j in N L and N R ar dnotd as p j and p j rspctivly []. 6. Th compound circuit containing r i, r i+,, r j, r j is calld (r i -r j )-rgion []. 7.Thr ar sibilitis for th ton initially at r i to sit at: p i (N L ), r i and p i (N R ). Th corrsponding ton or r i stat is dnotd by, 0 and rspctivly []. 8.d n mans r is at d stat (d=,0, ) and r is at n stat (n=, 0), whr is th location of non-sharing rsourc bing usd by an opration plac p *, [2]. Th systm is dnotd as a (Dficint) Gn-Lft -th ordr N L N R N L N R t r p t' p' t p t* 2 r t' p p 0 t 2 t r 2 t' 2 p 2 p 2 p 0 t' p 0 p* t 2 p 2 r * =r r 2 t' 2 p 2 p 0 p r p t p r t' p t 4 t' 4 t 4 t' 4 Fig..rd ordr systm N. Fig. 2 Dficint) Gn-Lft systm whr =. + Ļ(,h )+ΔĻ), whr th valu of h of th part of Ŗ(,h ) (rsp. Ļ(,h )) is causd by th situation whn th holdr of non-sharing rsourc contains th ton will consum on ton of h. systm whn p* in P L []. Exampls ar shown in Figs., 2. Dfinition []: s = (d d 2 d ), x i =, 0 or, i = to, is a stat for a -th ordr systm N, d i is th ton at r i to sit at: p i (N L ), r i or p i (N R ) rspctivly. (d i d i+ d q d q+ ), i, q i (mbddd in s) is a substat of s.

3 Lt N r b th rvrs nt of a PN N. Chao [] constructd th concpt of a complt rachability graph and split th rachability graph of th PN N into rachabl (from th initial stat), forbiddn (rachabl to dadloc stats only), dadloc (rachabl to non stat), non-rachabl (from th initial stat), and mpty-siphon+non-rachabl stats (both nonrachabl stats in N and N r ) p5 p NSS t p2 p t t t 2 p 4 t 4 p 5 t 5 p 6 p Fig. [29]. a-nt :comd by Dficint -th ordr systm and non-sharing subnt(nss) p 2 p According to graph thory, Chao found Lmma and Lmma 2. Lmma []: Any forbiddn stat in N is non-rachabl in N r. Lmma 2 []: Any non-rachabl stat s in N is a forbiddn on or a non-rachabl on in N r. Lmma []: ) s is a liv stat if and only if (iff) s = {(y... y ) y i = or 0}, or s = {(x... x ) x i = or 0}. 2) Th st of liv stats L = {(x... x ) x i = or 0} {(y... y ) y i = or 0} = L a L b. ) Th total numbr of liv stats is 2. Lmma 4 []: Th sibl rachabl stats ar s = {( x x 2... x j y j+... y ) 0 j } ={(x... x j y j+2... y ) j } {(y... y )}, whr x i = or 0 (i = to j) and y p = 0 or (p = j+2 to ) = L c L d. Lt Ŗ, Ļ, F, Ŭ, Ɓ and Ď b th functions of th total numbr of rachabl(ŗ), liv(ļ), forbiddn(f), nonrachabl(ŭ), mpty-siphon+non-rachabl(ɓ) and dadloc(ď) stats of a variant -th ordr systm, rspctivly. In a -th ordr systm [], th total numbr of stats is ; Ļ()=2 + ; Ŗ()=(+2)2 ( ) ; F()=( 2)2 ( ) +; Ŭ()= (+2)2 ( ) ; Ɓ()= 2 ; Ď()=. p 4 p 7 p 8 p 9 t 6 t 0 -th ordr systm comition at t t 7 2 t 8 t 9 p 0 Dfinition : Th quivalnt N = ( P P R, T, F ) of a nt N = ( P P, T, F) is dfind as R ( P = P \ P ; 2. P = P \ H ( r) ; r P. T = T \ r ; R. R F = ( F r P ( r, r r P ( H( r), H( r)) ( r, r) ( r, r ) ( r, r P is th st of non-sharing placs) ) ( ( r ), r) \ r P [( H( r), H( r) ) ) ( ( r ), r )]; W can say that th nt in Fig. is th quivalnt of th nt in Fig. 2. Lt N G b th Gn-Lft systm; N b th -th ordr systm; N r b th rvrs nt of N ; Ŗ(,h) (rsp. Ļ(,h)) b th numbr of rachabl stats of a Dficint -th ordr systm with h (h ) tons in th lft idl plac. In [2], w showd th stats pattrn and th numbr of Additional Rstrictd Rachabl (rsp. Liv) Stats drivd from non-rachabl (rsp. non-liv) stats of th quivalnt (dfind in Dfinition ) causd by a non-sharing rsourc, ARRS ΔŖ( rsp. ARLS ΔĻ) of Gn-Lft systm in Lmma 5 (rsp. Lmma 7 and 8); th numbr of rachabl (rsp. liv) stats is 2Ŗ()+ΔŖ (rsp. 2Ļ()+ΔĻ), whr 2 is th ton distribution of th xisting on non-sharing rsourc undr th condition that thr ar sufficint tons in th lft idl plac. In Dficint Gn-Lft th numbr of rachabl (rsp. liv) will b Ŗ(,h) + Ŗ(,h ) +ΔŖ D (rsp. Ļ(,h) + Ļ(,h )+ΔĻ D ) whr ΔŖ D (rsp. ΔĻ D ) is th numbr of ARRS(rsp. ARLS) of Dficint Gn-Lft. As shown in Fig.2, whn p* dos not contains th ton, th quivalnt of Fig.2 will b systm, whr th numbr of rachabl stats is Ŗ(,); p* contains th ton, th quivalnt will b 2 systm, whr th numbr of rachabl stats is Ŗ(,2). Lmma 5 [2]: Lt s = (d m 0 0 n d n+ d n+2 d ) whr m ; + n, such that only th r m -r n siphon in N r is unmard. M is non-rachabl in N, M*=M+r* is rachabl in N G, th total numbr of such M* is(2 ( n) )Ŗ(m ). Lmma 6 [2]: Th total numbr of rachabl stats in N G m= n= + is ( n) Ŗ(, ) = 2Ŗ( ) + Ŗ( m ) 2. In Lmma 5, d i =0 or, m+ i which is th tons distribution of right hand sid procss; (d d m ) is a rachabl stats pattrn of an (m )-th ordr systm. Hnc Th total numbr of such M* in Dficint Gn-Lft will b (2 ( n) )Ŗ(m,h ) du to th fact that n consum on ton of h. Hnc w hav th total numbr of rachabl stats in Dficint Gn-Lft will b

4 (,, ) = (, ) + (, ) Ŗ h Ŗ h Ŗ h + Ŗ( m, h ) 2 m= n= + ( n) Lmma 7 [2]: Lt s=(d d 2 m n d ), m, + n corrspond to Maring M such that thr ar unmard siphons in only th(r m -r n )-rgion in N. Th total numbr of sibl liv marings undr M is 2 ( n). Lmma 8 [2]: Lt s=(d d 2 m 0 m n n d ) m, + n corrspond to non-rachabl maring M such that thr ar unmard siphons in only th r m -r n siphon in N r. Th total numbr of sibl liv marings undr M is 2 (m ). Lmma 9 : Th total numbr of liv stats of Gn-Lft is Ļ(,)=2(2 (+) )+()(2 ( ) )+( )(2 () ), whr ()(2 ( ) )+( )(2 () ) is th total numbr of ARLS of Gn-Lft. In Lmma 7 and 8 th total numbr of sibl liv marings of Gn-Lft is th valu of th tons distribution of right hand sid procss, hnc it is indpndnt to th tons distribution of lft hand sid procss, th numbr of ARLS of Dficint Gn-Lft will b qual to Gn-Lft. Hnc th total numbr of liv stats in Dficint Gn-Lft is III. (,, ) = (, ) + (, ) Ļ h Ļ h Ļ h. ( ) ( 2 ) ( 2 ) + + COMPUTATION OF THE REACHABLE AND LIVE STATES OF DEFICIENT GEN-LEFT K-TH ORDER SYSTEM Lt h systm b a Dficint -th ordr systm with h (h ) tons in th lft idl plac; i.., M(p 0 )=h and M(p 0 )=. Thorm []: Lt Ŗ() b th rachabl stats of -th ordr systm, Ŗ(,h) b th numbr of rachabl stats of h, a Dficint -th ordr systm. l h l Ŗ( h, ) = Ŗ( ) Cl (, l i) ( 2 ), l= h+ i= 0 whr Cl (, l i) = ( l )! [( l i)! i! ] is a binomial cofficint. Thorm 2 []: Lt Ļ() b th liv stats of -th ordr systm, Ļ(,h) b th numbr of liv stats of -th ordr systm with ton numbr of lft idl plac bing h. l h Ļ( h, ) = Ļ( ) C( l, l i) l= h+ i= 0 Cl (, l i) = l! ( l i)! i! is a binomial cofficint., whr [ ] Thorm []: F (, h) = Ŗ (, h) Ļ (, h). = ( Ŗ( ) Ļ( ) ) l h l C( l, l i) ( 2 ). l= h+ i= 0. Tabl shows th total rachabl, liv and forbiddn stats of h systm, whr =7 and 8. Plas rfr to th abov tabl for Ŗ(,h), Ļ(,h) and (,h), whr =7 and 8; h = to 8; thy ar consistnt with th rachability analysis using th INA [0] (Intgratd Nt Analyzr) tool. Lmma 0: Lt s = (x m n x n+ x n+2 x ) whr m ; + n, such that only th r m -r n siphon in N r is unmard. M is non-rachabl in N, M*=M+r* is rachabl in Dficint Gn-Lft, th total numbr of such M* is (2 ( n) )Ŗ(m,h ). TABLE I. STATES FOR K H - SYSTEM, WHER K=7 AND 8. h Ŗ(7,h) Ļ(7,h) (7,h) Ŗ(8,h) Ļ(8,h) F(8,h) Thorm 4: Th total numbr of rachabl stats in Dficint Gn-Lft is Rhgn (,, ) = Rh (, ) + Rh (, ). n + Rm (, h ) ( 2 ) m= n= + Thorm 5:Th total numbr of liv stats in Dficint Gn-Lft is Ļ( h,, ) = Ļ( h, ) + Ļ( h, ) ( ) ( ) +( )(2 )+( )(2 ). Thorm 6: F (, h, ) = R(, h, ) L(, h, ) = + n + R( m, h ) ( 2 ) 2 2. ( R(, h) R(, h ) L(, h) L(, h ) ) m= n= + ( ) ( ) ( ) ( ) According to Thorm 4, Ŗ(8,7,)=Ŗ(8,7)+Ŗ(8,6)+ i= to ( j=4 to 8 (Ŗ(i,h )(2 (-j) )))= (Ŗ(0,6)+Ŗ(,6)+ Ŗ(2,6))( )=292; according to Thorm 5, Ļ(8,7,)=Ļ(8,7)+Ļ(8,6)+()(2 ( ) )+( )(2 () )=

5 40. Th valus of Ŗ(8,7,) and Ļ(8,7,) ar consistnt with th rachability analysis using th INA. IV. CONCLUSION Th charactristic of ral-tim rsourc allocation PNs modl is that th nt structur or th tons of rsourcs ar changd dynamically. By MLR concpt w rport th vry first mthod to comput in closd-form th numbr of rachabl stats of Dficint Gn-Lft -th ordr systm (a simpl vrsion of S PR) without constructing a rachability graph. This hlps to raliz th MLR partial dadloc avoidanc concpt to a variant -th ordr systm with nonsharing subnt as shown in Fig. du to that w can dynamically and ral-tim allocat th dadloc thrad-holdr basd on th maximum rachabl stats numbr by closdform formula and avoid th dir situation of mid-run abortion of rachability analysis du to xhaustd mmory. Futur wor should xtnd th rsarch on MLR concpt to Dficint -th ordr systm with multi-tons. REFERENCES [] D. Y. Chao, Rachability of nonsynchronizd choic Ptri nts and its applications, IEEE Transactions on Systms, Man and Cybrntics, Part B (Cybrntics), vol. 5, no. 6, pp. 20 2, Dc [2] J. Ezplta, J. M. Colom, and J. Martinz, A Ptri nt basd dadloc prvntion policy for flxibl manufacturing systms, IEEE Transactions Robot. Automat., vol., pp. 7 84, Apr [] A. Zimmrmann, TimNT, Tchnisch Univrsitat Ilmnau, Systm and Softwar Enginring, D Ilmnau, Grmany. Availabl: [4] D. Y. Chao, Computation of lmntary siphons in Ptri nts for dadloc control, Th Computr Journal, vol. 49, no. 4, pp , May [5] Y.-Y. Shih and D. Y. Chao, Squnc of control in SPMR, Th Computr Journal, vol. 5, no. 0, pp , Sp [6] M. Uzam and M. C. Zhou, An improvd itrativ synthsis mthod for livnss nforcing suprvisors of flxibl manufacturing systms, Intrnational Journal of Production Rsarch, vol. 44, no. 0, pp , May [7] D. Y. Chao, "Improvmnt of suboptimal siphon- and FBM-basd control modl of a wll-nown S PR," IEEE Trans. Automat. Sci. Eng., vol. 8, no. 2, pp , Apr. 20. [8] D. Y. Chao, Enumration of lost stats of a suboptimal control modl of a wll-nown SPR, IET Control Thory Appl., 20, vol. 5, no., pp [9] J. S. L, M. C. Zhou, and P. L. Hsu, An application of Ptri nts to suprvisory control for human-computr intractiv systms, IEEE Transactions on Industrial Elctronics, vol. 52, no. 5, pp , Oct [0] D. I. L, S. Kumagai, and S. Kodama, Rachability of LSFC nts, IEEE Intrnational Symium on Circuits and Systms, vol. 4, pp , May 990. [] Z.W. Li and M.C. Zhou, Two-Stag Mthod For Synthsizing Livnss-Enforcing Suprvisors For Flxibl Manufacturing Systms Using Ptri Nts, IEEE Transactions on Industrial Informatics vol.2, no. 4, pp. 25, [2] A. Schrijvr, Thory of linar and intgr programming. John Wily & Sons, 998. [] D. Y. Chao, Enumration of rachabl (forbiddn, liv, and dadloc) stats of -th ordr systm of Ptri nts, IMA Journal of Mathmatical Control and Information, vol. 4, no. 4, pp , 205. [4] D. Y. Chao and T. H. Yu, Enumration of rachabl (forbiddn, liv, and dadloc) stats of top -th ordr systm (with a non-sharing rsourc plac) of Ptri nts, 9th Annual Confrnc of th IEEE Industrial Elctronics Socity (IECON 20), Nov. 20, pp [5] D. Y. Chao, T. H. Yu, and S. C. Wu, Closd form formula construction to numrat control rlatd stats of -th ordr SPR systm (with a Top Lft sid non-sharing rsourc plac) of Ptri nts, Procdings of th th IEEE Intrnational Confrnc on Ntworing, Snsing and Control, Apr [6] D. Y. Chao and T. H. Yu, Computation of control rlatd stats of bottom th-ordr systm (with a non-sharing rsourc plac) of Ptri nts, Transactions of th Institut of Masurmnt and Control, vol. 7, no., pp , Jul [7] D. Y. Chao, T. H. Yu, and C. C. Liou, Enumration of rachabl, forbiddn, liv, and dadloc stats of bottom -th ordr systm (with a lft sid non-sharing rsourc plac) of Ptri nts, Procdings of th th IEEE Intrnational Confrnc on Ntworing, Snsing and Control, Apr [8] D. Y. Chao, T. H. Yu, and T. F. Lin, Closd form formula construction to numrat control rlatd stats of middl-lft -th ordr SPR systm of Ptri nts, 204 Intrnational Confrnc on Computational Scinc and Computational Intllignc, Mar. 204, vol., pp [9] D. Y. Chao, T. H. Yu, and T. Y. Chn, Computation of control rlatd stats of middl -th ordr systm (with a non-sharing rsourc plac) of Ptri nts, 204 Intrnational Symium on Computr, Consumr and Control, Jun. 204, pp [20] D. Y. Chao, T. H. Yu, and D. C. Huang Computation of control rlatd stats of top lft -nt systm (with a nonsharing rsourc plac) of Ptri nts, 204 World Congrss in Computr Scinc, Computr Enginring, and Applid Computing (WORLDCOMP 4). [2] D.Y. Chao and T.H. Yu, Paramtrizd of control rlatd stats of Middl Lft K-nt systm (with a nonsharing rsourc plac) of Ptri nts, 204 CACS Intrnational Automatic Control Confrnc (CACS 204), Nov [22] D. Y. Chao and T. H. Yu, "Closd form formula construction to numrat control rlatd stats of Bottom Lft -nt S PR systm (with a Bottom Lft sid non-sharing rsourc plac) of Ptri nts." 205 IEEE 2th Intrnational Confrnc on Ntworing, Snsing and Control, Apr. 205 [2] D. Y. Chao and T. H. Yu, "Th fundamntal closd-form solution of control rlatd stats of -th ordr SPR systm with lft-sid nonsharing rsourc placs of Ptri nts, Intrnational Journal of Control, vol 89, no., pp 69-78, 206. [24] D. Y. Chao, Y. P. Chi, T. H. Yu, L. C. Yu, and M. Y. J. L, Proof by Modl: A nw nowldg-basd rachability analysis mthodology for Ptri nt, IMA Journal of Mathmatical Control and Information, vol. 2, pp. 22, 206. [25] D. Y. Chao, and G. Y. Liu. "Furthr rduction of minimal first-mt bad marings for th computationally fficint synthsis of a maximally prmissiv controllr." Intrnational Journal of Control, pp. 4, 204. [26] D.Y. Chao, "Closd-form solution of controllr synthsis for infinitly larg systms of rsourc sharing systms of a subclass of Ptri nts." Transactions of th Institut of Masurmnt and Control, , 205. [27] D. Y. Chao and T. H. Yu," MLR: a nw concpt to launch a partial dadloc avoidanc policy for -th ordr systm of Ptri nts," 4th Annual Confrnc on IEEE Industrial Elctronics Socity (IECON 205). [28] T. H. Yu, Paramtrizd of Control Rlatd Stats of Gn-Right -th ordr Systm of Ptri Nts Basd on Proof by Modl of Gn-Lft, IECON th Annual Confrnc of th IEEE Industrial Elctronics Socity, in prss. [29] Liu M, Wang S, and Li ZW (20) Suprvisor Rconfiguration for Dadloc Prvntion by Rsourcs Rallocation Journal of Applid Mathmatics, Articl ID 5894, [0] P. H. Star, INA: Intgratd Nt Analyzr, 992. Handbuch, Grmany. [] D. Y. Chao and T. H. Yu, Construct th closd-form solution of A-nt of Ptri nts by cas study, unpublishd.

6 APPENDIX Indx of trms Tabl A. Th indx of th notation of PN. H(r) Th st of holdr placs that us r. H(r) = r P M Maring M 0 Initial maring M 0 (p) All tons initially in p N G Ptri nt (P, T, F, W), usd in Sction II for a Gn-Lft -th ordr systm. N i An S 2 PR in N, i =, 2 (N, M 0 ) Calld a Ptri nt, a mard nt, or a nt systm P Th st of placs p Th st of output transitions of a plac p p Th st of input transitions of a plac p P i St of opration placs of N i,, i = L, R p i An opration plac, i =, 2,, n R(N,M 0 ) Th st of rachabl marings in th nt (N, M 0 ) S 2 PR Simpl squntial procss with rsourcs S PR Systms of simpl squntial procss with rsourcs siphon For a Ptri nt (N, M 0 ), a non-mpty subst S and trap (rsp. τ) of placs is calld a siphon (rsp. trap) if S S (rsp. τ τ)