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M26 M3 Vol.26No.32003 7AACTAMATHEMATICAEAPPLICATAESINICAJuly,2003 ∗ (czF nWF 100080) c TOK YGdjfQ Ts mVG a T u a ni a C DlogPqG a OK Y(ETEG),W TKmV(DEDS), mVdj1 QM (ETEG) Vh IX VMoX gPetri F aHoXK [1] ETEG( h EventMulti-Graph). R ETEGI sz S M [2,3] (TEG)Ifly L pIAr(Dioid)a 1998 KG[4,5]d [2]I vBg ETEGIAr v C E h d Ogv [6]q g Ix F [7]P x JaI Bg [8]q RJaIAr a 1998 CohenG[9]d ETEGd g v fl O MpIoXj O ETEGIAr v ETEGIoX fl o Ih gR GZl aqB ts e ETEGIflgR X VETEGN wgxh [4] ETEGIfly Ara x=Ax⊕w,(1) dx=:t1,···,tn,ti Nio I !ct0,t1,···;n !wINiof! eqB ci !cwi=:wi,wi,···;n#aQAIE ![∆v]⊗[d]⊗[z−m]⊗[∇u]IFr O$[∇u] haFr [∇u]t0,t1,···=:t0,···t0,t1,···t1,···,tihhu![z−m] #$Fr [z−m]t0,t1,···=:−∞,···,−∞,t0,t1,···,−∞hhm![d] % Fr [d]t0,t1,···=:t0+d,t1+d,···.O$&#[10] [4]dI % dk b % d;[∆v] $&Fr [∆v]t0,t1,···=:tv−1,t2v−1,t3v−1···.⊗ Frh’DF ⊕ AxdI% W DF X(o !)!c ZXf!)!cpE * #Fr+I + F,S (f⊕g)t=:ft⊕gt,f,g∈F,I(F,⊕,⊗) oAr ’Ar IZl(F(1)%d2001,2B9$%B&2002,8B9$%B)jn&∗z-s&F ./(69874040’)0z-1J23qlzx&452*+ ,,(26 I x=A∗w,O$A∗=:+∞k=0Ak.(2)’ A∗)ivZ’ [4–8,10]aq A∗)(2)(I I xIxh -*flgR)a A w ETEGIv r# D./ZG+4[1,4,10],,x# 2 [11] l -QAS og* G(A), eAmvG(A)d+(S5g o g6tuO.(3)*G(A)d. LI5gC NiS7I* Fr[∆v]⊗[di]⊗[z−m]⊗[∇u],i=1,2,···,L.eNiS7I,O i−1,eO i,(0 L- U O).*-jCiIgI* R(i,j),O$gI* ,OCeOp7I*-# yh’DFIFIFr -i.5gk/5CiIgI** g(i),ed(C)=:d=:Li=1di.F*A∗I0i.0jc a∗ij. 10G(A)8 S. LI5gC,Ia∗ijwj=+∞k=0g(i)kR(i,j)wj=−∞,···,−∞,a0,···,a0,a1,···a1,···, da0=wj+r=j+1dr,dr=:dr(modL),ak=a0+kd,k0;y,N oak !cINNk+1o O$Nk=nI+Lk,nr=nr−1ur+mr)/vr,* a ≤aIx S /1Ynj=0;(4) dur=:ur(modL),mr=:mr(modL),vr=:vr(modL),nI+Lkd Aij I=i,Ai≤j I=i+L. !−∞,···,−∞,a,···,a,···I!c 8(oY−∞,a,O$−∞hhN! *OV!c N,a.pqj2Frh’DF*9⊗,(V[∆v][dj+1][z−m][∇u]wj= mj+1/vj+1,wj+dj+1. yy (FR(i,j)wj=[∆v][di][z−m][∇u]···[∆v][dj+2][z−m][∇u]nj+1,wj+dj+1=nI,a0=N0,a0,m Ai=j N0,a0=0,wj. g(i)R(i,j)wj=g(i)N0,a0=[∆v][di][z−m][∇u]···[∇u][∆v][di+1][z−m][∇u]N0,a0=nI+L,a0+d=N1,a1. yy (Fg(i)kR(i,j)wj=g(i)kN0,a0=Nk,ak,3 0cD OK YGdj453F’ lVa∗ij=+∞k=0g(i)kR(i,j), a∗ijwj=+∞k=0g(i)kR(i,j)wj=+∞k=0Nk,ak. 2*P(C)=:P=:L r=1ur/vr,βr=:I+L s=r(us/vs),I Z1dINk p:N0Pk+I+Lr=I+1(mr/vr−1+1/vr)βrk−1s=0Ps≤Nk≤N0Pk+I+Lr=I+1mrβr/vrk−1s=0Ps. *θ aI3 1f I0≤θ1, a=a−θ, Z1dnr= (nr−1ur+mr)/vr=(nr−1ur+mr)/vr−θr,O$θr 2 vr I3 1f θr≤1−1/vr.I ’ Z1 Nk=nI+Lk=···((((Nk−1uI+1+mI+1)/vI+1−θI+1)uI+2+mI+2)/vI+2−θI+2)···uI+L+mI+L)/vI+L−θI+L=Nk−1I+Ls=I+1us/vs+I+Lr=I+1(mr/vr−θr)βr.D yy (FNk=N0Pk+k−1s=0PsI+Lr=I+1mr/vr−θr(s) βr.(5)’0≤θr(s)≤1−1/vr,F Z 3 1 1X !ct0,t1,···,tj,···,03Gk0≥0,4FI j≥k0W tj+N=tj+T,I oj !c λ=:T/N 2iAT=0,N=1 o !c 1Amv(3),P1,I5gC O6 p3O(;G(A)d-5gC gC4IO)iX IxIf!ti v!ct0,t1,···,tq. ’ Z2,(V0≤Nk≤N0Pk+(1−Pk)I+Lr=I+1mrβr/vr(1−P)N0+I+Lr=I+1mrβr/vr(1−P)=:M+∞,’ETEGID./ZV wj8AS IC1D. ,4yNk, GD.k/ 5 NNk+1o56 I Nk+1≥Nk, 53G vZokmvNk+1−Nk≥1,73G oq4Fkq B Nk+1−Nk=0,’ Z1O ak=+∞.’[10]Vak,C6 +∞,I a∗ijwj v!c *5gC OX IxIf!ti=Lj=1a∗ijwj v!c ’ +∞⊕a=+∞,vp3OX IxIf!ti v!c ’ETEGID./ZVNk+1−Nk=0,; D.k+1/ i7565 SZ1 5 qo56 O8Oii7565 O1f T8D. 20G(A)8 S. LI5gC,*xIf!tic v!c I1)P=1 5gC IOiX IxIf!tiB oj !c 2i λ(i), Y v6 FGF454*+ ,,(26 2)P1,mr≥vr,1≤r≤L ti=:t0,t1,···,tq,···IEtq,Gq7f mv |tq−dlogPq|b,O$b qiwI 1)qDwr,’(1)JXPetri Iz Tr8Gwr H 8G tPetri D. T09maxwr|1≤r≤L .E9 T0 -OiC1 Petri D.kj nob Prd56 =Ixh p1≤j≤L,*Pr I TrNs/Nj!m Prd9:56 mr(s,j),TrNs/Nj! IPrd7 56 Mr(s,j),O$Mr(s,j)G 0,hp): Tr−1z Ns/Nj! -O ’Petri D./ZV xIf!tic v!c Tr−1S?! @ 5 56 TrNs/Nj+1! ;IPrd56u mr(s,j)+Mr(s,j).,bi=1, M1(s+1,j)=ML(s,j)+mL(s,j)−mL(s,j+1) uL/vL=ML−1(s,j)+mL−1(s,j)−mL−1(s,j+1) ,uL−1/vL−1+mL(s,j)−mL(s,j+1) uL/vL=···=M1(s,j)P+Lr=1mr(s,j)−mr(s,j+1) αr, dαr=: s=r(us/vs),# (s=1,2,···k,M1(k+1,j)=M1(1,j)Pk+ks=1Pk−sLr=1mr(s,j)−mr(s,j+1) αr,P=1 M1(k+1,j)=M1(1,j)+ks=1Lr=1mr(s,j)−mr(s,j+1) αr.- 1≤j≤L6mr(s,L+1)=mr(s+1,1),I Lj=1M1(k+1,j)=Lj=1M1(1,j)+Lj=1ks=1Lr=1mr(s,j)−mr(s,j+1) αr=Lj=1M1(1,j)+ks=1Lr=1mr(s,1)−mr(s,L+1) αr=Lj=1M1(1,j)+Lr=1mr(1,1)−mr(k+1,1) αr.’ 0≤mr(s,j)vrI 0≤Lj=1M1(k+1,j)Lj=1M1(1,j)+Lr=1vrαr+∞, mr(s,j),M1(k+1,j)5 vZoY M1(k+1,j)k Mi(k+1,j) lV3 0cD OK YGdj455D ’ k ∞o I 3GQ,k04k≥k0 Mr(k+Q,j)=Mr(k,j),mr(k+Q,j)=mr(k,j),1≤r≤n,1≤j≤L,VD Z1VQ/ tqJYdQ,Q/ Ti5 56N(i)o tiB oj !c 2i λ(i)=dQ/N(i),7 v6 FGF 2)P1,mr≥vr,1≤r≤L - Z16(5)GFci k + Ia y=f(x):y=a0+kd,(6)x=N0Pk+k−1s=0PsI+Lr=I+1mr/vr−θr(s) βr.(7)* Z1Ia∗ijwj=:c0,c1,c2,···, 0≤f(q)−cq≤d.(8)’(7)6 Z2Fx=N0Pk+Ck(Pk−1)/(P−1), dI+Lr=I+1(mr/vr−1+1/vr)βr≤Ck≤I+Lr=I+1mrβr/vr, logP(x−N0)=logP(Pk−1)+logPN0+Ck/(P−1) ,k=logPx−logPN0+Ck/(P−1) +o(1), ’(6)Fy=dlogPx+a0−dlogPN0+Ck/(P−1) +o(1),(9)’ mr≥vr,1≤r≤L,I Ck≥I+Lr=I+1βr/vr,;-N0≥0,P1, logPN0+Ck/(P−1) +d≤B, : ’(8),(9),Gq7f F|cq−dlogPq|a0+dB+1,(10)’ ti=Lj=1a∗ijwj, Z1dj,U 8!ca∗ij