Fluid Limit of Generalized Jackson Queueing Networks with Stationary and Ergodic Arrivals and Service Times
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Fluid Limit of Generalized Jackson Queueing Networks with Stationary and Ergodic Arrivals and Service Times Marc Lelarge. N° 5069 January 2004. ISSN 0249-6399. ISRN INRIA/RR--5069--FR+ENG. THÈME 1. apport de recherche.
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(14) . 1. 2. +. 1≤i≤n Ui. i. ∗ 3. 3. i−1. +. t 0. {Ui ≤t}. 1≤i≤n 3. i. ∗. ∗. !#"$"%. ∗ 1. ∗ 2. ∗ 3. º db ~z\bd\Lqo\¢g \sgqZlSf~kos\f~\+§eg Z| L\|~qo\\cÉZbdx2~\N\N}B|·kd ¢g\cm m QW=X~\ τSkd\,qoσx~qoªr\k§ekdX~ \csgqd2\ b%τegqd\ =kdX~\0{τ¤³sz }s§Z& }ge Zf Zbn| kosgσ[E\=q {σ } eyqoqd ¢g\Nb½ +kdX~\Sf~\cf~\Aekkd [EA\ τ egZ|µkbub{\c¯ qd¢©}\£kd [E\4 b σ ¯ i º bdbdsi}©eko\c|jkds'eEzfZ\f~\ Q ªi§ \4|i\ÉZ~\9kdXZ\4sgfikox~fikux~qosi}\cbob {τ } ∈ A m 1äM 3 τ =τ +σ , τ = max[τ , τ ] + σ , 2 ≤ i ≤ n. ©bkdX~\|~\xZegq{kof~qd\£kd [E\4sy¤¼}fZb{kdsz[E\q i ¯ v¼hixZeyZ|~~EkoX~©bqd\N}f~qbdszjm\c |Zb τ ¤³sgq 1 ≤ i ≤ n, 1®Á 3 τ = max (τ + σ(j, i)), A. i 1≤i≤n. 2. 1. D 1 D i. A i. A. i D 1≤i≤n i. A 1. 1 A i. D i−1. A i 1≤i≤n. 2. i. D i. D i. j=1...i. A j. ʳË5ÌÀʹÍ.
(15) .
(16) !#"%$ & '( *)+-, .& '. §kdX+koX~\4~sykekosz koXZek . σ(j, i) = σj + · · · + σi. ¯ l9 \Z}\4§ \|~\ÉZ~\N|eE[Eegx~x~ ~. Φ : A × A 7→ A. bdfZ}X 1® 3. τ D = {τiD }1≤i≤n = Φ(Q).. \4§ Àf2b{\CkdXZ\9¤³sg s§ ~~sykekd sg¤³sgqkoX~\|i¸L\cqd\cSkl}sgfZzkoZ!¤³fZZ}BkoszZb A(t) = P 11 L L Σ(t) = P 11 D(t) = P 11 ¯ ?ZsgqegSm ~sg~°|i\N}qo\cezb{ ~¤³f~Z}Bkosz ª §0\|i\Zsykd\'m kdXZ\xZbd\fZ|isg F (t) ≥ x} ¢g\cqobd\sy¤ F 1®§X~©}X b \¤¹k{ }szSkd SfZFsgfZ!b 3 ¯ \CXZe±¢z\ F F(x)(x)≤ u=⇔inf{t, 8¡szqd\cs¢g\qNª§0\lfZbd\ koX~\4~sgkoekosz ∧ ¤³sgq min eyZ| ∨ ¤³sgq max ¯ WX~\9¤³sg s§~ \[E['eEg x¢g\c≤beF~(u)\c§ ¯ |~\cbo}qoxikoszsg¤%koX~\ szfikdxZfikux~qos}\cbob0+kd\cqd[¶sy¤ }sgfZzkoZ!¤³fZZ}BkoszZb ¯ à · G ( ( Q ∈ A × A ( D = Φ(Q) ,
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(22) .&j] v zf2ekd sg2b 1®Á 3ÀegZ| 1ä% 3Vg ¢g\rkn§ su\NSf~¢eg\czk5|i\ÉZZµkoszZbÀsg¤ikdX~\ [Eegx~x~ ~ Φ : A ×A → ª«sz~m¡v zf2ekd sg 1®Á 3uz¢z\cbkoX~\Eqd gXSk9|i\ÉZZµkoszPsy¤ ¯4» PxZegq{ko }f~ egql~sgkd©}\ C0fikC¤³sgq A¯ koXZeku§0\4ey §0e±τmib ∈XZe±A¢g\ τ ≥ σ(1, i) ∨ τ ªi¤³qosg[§X~ }X§0\4|i\qo¢z\ D(t) ≤ΦΣ(t) ∧ A(t) ¯
(23) ' $ )( +* -,/.0(21)35476 9! 8 ;:=<>6 )354 \qo\c}egX~\cqd\0koX~\~sykekd sg zkoqdsi|if2}\c|! +¾Ü±¿°ªkos9|~\cbo}qoL\e£z\~\cqoeg ¥\N|#Seg}¦ibdsg#Z\kn§0sgqo¦C§kdX ~si|i\Nb K WXZ\ ~\kn§ ¯ szqd¦ib§ \ }szZbd |i\cq+egqd\ }XZegqoez}Bkd\cqd ¥\N| m kdX~\¬¤®eg}kkoXZekbd\qo¢ }\kd [E\cb¡egZ| qosgfikoZ |~\c}©bdszZbeyqo\egbobds} eykd\N| §kdX·b{koeykd sgZb!eyZ| Zsyk#§kdX·}fZbnkosg[E\qb ¯ WX~©b![\NeyZb4kdXZeykkdX~\ j äkoX bd\qo¢ }\sg¬bnkekosz k koeg¦g\Nb σ f~~kobCsg¤¼kd [\zª«§X~\qo\ {σ } b9ejx~qd\N|i\É2~\c|bd\cSf~\cZ}\ ¯9» koX~\'bdeg[E\!§e±mgª«§X~\cPkoX~©b4b{\cqd¢©}\#©b4}sz[Ex~\kd\N|«ª«kdXZ\ \ce±¢ ~}fZb{kdsz[\cq9 b9bd\Sk9kdsbnkekd sg ν 1®sgq£\Ne±¢g\cbkdX~\!~\kn§0sgqo¦¤ 3uegZ| b£x~fik9ek£kdX~\!\Z|¡sg¤¼kdX~\Sf~\f~\!sgkoX~ bCb{koekoszÀª ν = K+1 §XZ\qo\ {ν } bCey©b{s+ex~qo\c|~\ÉZ~\N|Pbd\cSf~\cZ}\zªV}cey \N|kdXZ\!qosgfikoZbd\cSf~\2}\ ¯ WX~\Ebd\cSf~\2}\cb egZ| ª§X~\cqd\ qey~z\cbs¢g\cqEkdX~\Pbd\ksg¤9b{koeykd sgZbcªeyqo\¡}ey \c|kdX~\ |iqd ¢~ {σ } bd\cSf~\cZ}\Nbsy¤%kdX~\4{ν~\k ¯r} º z\~\cqoeg ¥\Nk|jSeg}¦ib{szj~\kn§ szqd¦'§ À2\|~\ÉZ~\N|+Sm ∞ i=1. {τiA ≤t}. ∞ n=1. {σ(1,n)≤t}. ∞ i=1. {τiD ≤t}. ←. ←. ∗. ←. 0≤s≤t. ∗. A. D i. A i. (k) j. (k) j≥1 j. (k) j. (k) j. (k) j≥1 j. (k) j≥1 j. (k) j≥1 j. n o (k) (k) JN = {σj }j≥1 , {νj }j≥1 , n(k) , 0 ≤ k ≤ K .. Ì%Ì@å NZÓBÞ>O>P.
(24) . . F( . § XZ\qo\ (n , n , . . . , n ) |i\cbo}qoL\cbukdX~\!Zµko eg5}sgZ|ikd sg ¯ WX~\# Skd\qox~qo\kekd sg¡©blezb¤³sz s§ub ¤³szq k 6= 0 ªgeyk kd [\ t = 0 ªy~si|i\ k ªykoX~\qo\eyqo\ n }fZb{kdsz[E\qb%§kdX'bd\qo¢ }\½ko[E\cb σ , . . . , σ 1®µ¤¼si|ieyxZ\ x~0qdsz[x~siqo |iey\ckd \gb0ª koσX~\4\h['koe±\qomZLey\4Àeg zqdqoko\¢qoegx~Lqosg\kd¤5\N}|fZbnezkobsgeE[Eqd\\Nqb{b©|i fZegkoVX~\4bd\~qo¢\kn }§ \Cszkdqd ¦ [¯ \G9u3 ¯\2}\gª µ¤ n = 0 ªkoX~\qo\4 bZs\hkd\qoZegÀeyqoqd ¢ey ¯ µ¤ ∞ > n ≥ 1 ªkdX~\c+¤³sgqey 1 ≤ j ≤ n ªkoX~\eyqoqd ¢ey2kd [E\4sy¤ koX~\ j äkoX+}fZb{kdsg[E\cq0 +kdX~\ ~\kn§0sgqo¦4key¦g\Nb¼x~©eg}\uek σ + · · · + σ eyZ|EkÀtnszZb¼kdX~\u\cZ|sy¤VkdXZ\lSf~\f~\sg¤Vb{koeykd sg ν ¯ 9u\cZ}\ bukdX~\ TkdX zko\qd°egqdqo ¢±egLkd [\ sykd\kdXZeykukoX~ bl}egbd\gª~kdX~\cqd\4['e±mL\#e!ÉZ~kd\ f~[#L\qCσ sy¤½}fZbnkosg[Ej\qbux2egbob{ ~'koX~qdszf~gX e'¯ g ¢g\cPbnkekoszPb{skdXZeyklkdXZ\!~\kn§ szqd¦+©b£ez}Bkdf2ey m § \c%|i\ÉZZ\c|+sgZ}\4e#É2~µko\#bd\cSf~\2}\9sy¤ qdszfikd ~'|i\N}©b{ sgZbegZ|b{\cqd¢©}\Ckd [E\cbuegqd\Cg ¢g\csz kdX~©bub{koeykd sg ¯ µx~¤ qosn}\cbob= b∞eªÀqo\ko~X~\c\§0 eg§«X~x~\cqos¬}ko\ceybob0¦ \~ko+} ¤³sgq Zb{koegZ}\kdXZ\b{\NzfZ\Z}\ {σ } ¯ ¯ | ¯ ªÀkoX~\jegqdqo¢eg ¯ W%s \cez}X´~si|i\sy¤9e¬g\Z\qey ¥c\c|´Seg}¦ibdsg·Z\kn§0sgqo¦Lª½§ \¡}egezbdbdsi}©ekd\kdXZ\¤³sg s§ ~ }sgfZzkoZ ¤³fZZ}BkoszZb A ] ¯ K + 1 ¤³f~Z}kd sgZblegbob{si}©eko\c|kdsEkdXZ\b{\cqd¢©}\£kd [\Nb σ 1äegb +kdX~\bdZg \9bd\qo¢g\cqzfZ\f~G\ 3>L ¯ K(K+1) ¤³f~Z}kd sgZb kdX2ek%}sgf~SkobÀkdX~\ f~[#L\q%sy¤~}fZb{kdsz[E\qbÀqosgfiko\c|C¤³qosg[ e~si|i\ {0, . . . , K} kdse~si|i\ {1, . . . , K} L Á ¯ K + 1 ¤³f~Z}kd sgZblegbob{si}©eko\c|kds n ¯ 9l\Z}\4eEg\Z\qey ¥c\c|jSeg}¦ibdsgZ\kn§0sgqo¦'§µko X ~si|i\Nb beg+szitn\c}k =A ¯ \4§ Àf2b{\CkdXZ\9¤³sg s§ ~~sykekd sg¤³sgq\Neg}X+sy¤5kdX~\Nb{\4}sgf~Skd ~¤³f~Z}kd AsgZ b N = (n , . . . , n ) ª~§kdX n ≥ 0 L σ = {σ } egZ| σ (1, n) = P σ ª¤³sgq 0 ≤ k ≤ K L ªi¤³sgq 0 ≤ i ≤ K L Σ (t) = P 11 ªi¤³sgq 0 ≤ i ≤ K, 1 ≤ j ≤ K + 1 ¯ P (n) = P 11 \l|i\c~syko\kdXZ\uZx~fik0egZ|Esgf~kdx~fik x~qdsi}\cbob{\Nb sy¤«\cez}XESf~\cf~\ sg¤LkoX~\u~\kn§ szqd¦ib¼m egZ| qo\cbdxL\c}Bko¢z\ mgªÀ§kdX¬koX~\E¤³sg s§ ~+~sykekosz A = (A , . . . ,kA ) egZ| D = (D A , . . . , D D ) ¯ xZqdsi}\N|if~qo\lkoXZekl}sgZb{kdqofZ}kob0kdXZ\4x~qdsi}\cbob{\Nb egZ| bz¢z\+ x~x2\cZ|ih+ º \Ckoeg¦g\CkdX~\C¤³sz s§Z!~sgkoeykd sg g ¢g\c+e|i\cxZAeyqdkdf~qo\9Dx~qos}\cbob ¤³sgqSf~º \f~\ 0 Σ ¯ ª~eg¯ Z|+|i\xZegq{kof~qo\ xZqdsi}\Nbdbd\cb¤³sgq£kdX~\zfZ\f~\Nb i ∈ [1, K] X = {X } ªLegZ|egZµko eg%f~[!2\cq£sg¤½}fZb{kdsz[\cqob \cez}XSf~\fZ\ n ªZ§ \4}szZb{kdqofZ}BkkdXZ\9¤³sg s§ ~~xZfiklx~qosi}\cbobd\cb Y = {Y } (0). (1). (K). (k) 1. (k). (k) n(k). (k) 1. (0). (0). (0). (0) 1. (0) j. (0) j. (0) j. (0) j≥1 j. (0). (k). (k). (K+1)(K+2). (0). (k) (i). (K). (k) j≥1 j n. i,j. JN. (i). (k). n j=1. (k) j. {σ (i) (1,n)≤t}. l≤n. (i). {νl =j}. (k). (1). (K). (1). (k). (K). (0). (i). (i). 1≤i≤K. Y (i) (t) = n(i) + P0,i (Σ(0) (t) ∧ n(0) ) +. K X. (i). Pj,i (X (j) (t)).. 1≤i≤K. 1ä%3. j=1. ʳË5ÌÀʹÍ.
(25)
(26) !#"%$ & '( *)+-, .& '. @. \ |i\c~syko\£koX~©bm Y = Γ(X, JN) ¯ 5C ¢g\c¡eyeyqoqo¢ey«xZqdsi}\Nbdb¤³sgqu\Neg}XSf~\fZ\ ªL§ \|~\ÉZ~\kdX~\}sgqoqo\cbdx2szZ|i ~Esgfikox~fiklx~qosi}\Nbdb egZ|j|~\~sgkd\Cµkm X = Φ(Y, JN) ¯ 9u\cZ}\zªi§ Y\£XZe±¢g\ X = Φ(Y , Σ ) ªi§X~\qo\ Φ §egb|i\ÉZ~\cX| ¤³szqkdX~\bd ~g \4b{\cqd¢z\qzfZ\f~\4 1®%Á 3 ¯ Ç ½Ã ± Ä D
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(36) .&&0WX~©brxZqdszx2\cq{knmg ¢g\Nb kdX~\C}sz~~\c}kd sgEL\kn§0\\kn§ s4xLszbob{ ~ \l|i\Nbd}qd xikd sg2brsy¤Àe4z\~\cq{ eg ¥\N|Seg}¦ib{sz~\kn§ szqd¦ ¯ H Z\£sg¤ÀkdX~\Nb{\9|i\Nbd}qd xikd sg2b½XZezb02\c\jz¢z\§ szqo|~b0ek0kdXZ\£L\z~Z~ sg¤ koX~©bb{\N}Bkosz¬egZ|¬ b|~\x~©}Bko\c|¬§µkoX [szqd\Eqozsgq9 º x~xL\Z|ih² ¯ ¯ WX~\'sgkdX~\cq|i\cbo}qox~kd sg¬ b ko\qo[*sy¤ É~hi\c|xLsg Skv½SfZekosz 1-@M3 §X~ }XXZegbegqo\cez|imE2\c\j Skdqosi|ifZ}\N|jm 8eytn\§ubd¦2 ¾ ]N±¿ ¯ WXZ\cbd\Ekn§ s¡|i\Nbd}qd xikd sg2b4eyqo\\NSf~¢eg\czk ¬kdX~\b{xL\c} eg½}ezb{\Esy¤|i©bd}qd\kd\' ~x~fikbey2|¬eg \[Exiknm Z\kn§0sgqo¦ekkd [E\ t = 0− ¯ ?~sgque[Esgqo\9g\c~\qeyVqo\cbdf~µkszkoX~©bb{f~~tn\c}BkNª~§0\Cqo\¤³\cqkds¡¾ ]g]B¿ ¯
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(47) Aˆ $! $! E ' '( $ !#'( $!+
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(54) 7. F( . ÇÀÇ ªVX~\cZ}\![Eeg¦~kdX~\ D (t) ≤ Σ (t) ∧ A (t) ¨2f~ |bo}eg ~'eyZ|key¦Z!koX~\4 [Ekl n ª~§0\9XZe±¢g\ D(t) r p d q z s 2 x c \ q{knm+!¤³sg s§ub0 kdX~\ }cegbd\ µ = 0 m \[E['e9 ¯ \}szZbd |i\cq%Zs§²kdX~\}cegbdˆ\ µ >≤0µteyZ∧|A(t) ÉZˆqob{kr¯ egbob{fZ[\½kdXZeyk Q ∈ A ×A ¤³szquey n egZ| A(0) ˆ 4 Z}\ ˆ ª =©b0}0sg¯ Skd f~sgf2brsg R ey2| \c[E[Eez¢z\cb Aˆ → Aˆ f ¯ s ¯ } ¯ 8¡sgqo\s¢z\q¼koXZey~¦ib A(0) = 0 Aˆ kos \c[['eEÁ~ª~kdX~\bd\cSf~\cZ}\Nb Σˆ egZ| Σˆ }sg¢z\qog\lf ¯ s ¯ } ¯ kdsEkoX~\4qd\Nb{xL\c}kd ¢g\C¤³f~Z}kd sgZb t 7→ µt egZ| t 7→ ¯ ?ZsgqÉ~hi\N| ªi§ \9XZe±¢z\£koXZeyZ¦b0kosEf~~¤³sgqo[knmjsz+}sg[ExZez}Bkub{\kobcª t≥0 ?5 qob{klszZb{\cqd¢z\9kdXZeyklkoXZey~¦ibukdsj_\c['eyqo¦ ]zª2§0\XZe±¢z\. n. n. n. ∗. n. n. + n←. n. t µ. lim. n→∞. Dn (nt) n. 1 n An (nt) Σ [n(t − u) + (Σn )← (An (nu))] ∧ n→∞ 0≤u≤t n n 1 n An (nt) inf lim Σ [n(t − u) + (Σn )← (An (nu))] ∧ lim n→∞ 0≤u≤t n→∞ n n ˆ ˆ inf [µ(t − u) + A(u)] ∧ A(t). =. lim. = =. inf. 0≤u≤t. ˆ = µt ∧ A(t),. §XZ\qo\lkoX~\£©egb{k\cSfZey knmE¤³sg s§ub½¤³qdsz[*}szZ}e±¢knmsg¤ Aˆ ¯ s§ fZbdZ \[E['e~ªzkoX~\9qd\Nb{fZµk0¤³sg s§ub +kdX~©b}ezb{\ ¯ W%s'\hko\Z|kdXZ\qd\Nb{fZµkkosEkdX~\#}cegbd\ ªZ§0\}szZb{©|i\cqkdX~\#bd\cSf~\cZ}\ 1/i §XZ }X+2\csz~zb0kos A ¯ ?Zsgqueym >Q0 ªi§0∈\9AXZe±×¢g\£A¤³sgq n ≥ 1/ ª A (n(t − )) ≤ τB (nt)= τ≤ A +(nt) ¯ 9l\Z}\ ˆ ˆ ˆ egZ|¡b{ Z}\ Aˆ ©bC}sgSkof~szfZbª2§ \#XZe±¢g\ B ˆ = Aˆ ¯ 8¡sgqo\s¢z\qNª~b{ Z}\ A(t − ) ≤ B(t) ≤ A(t) ªi§0\CXZe±¢g\ D = Φ(B , Σ ) ≤ Φ(A , Σ ) ª~eyZ|§ \9}egegx~x~ m'kdX~\CÉZqb{k0xZegq{ksy¤%kdX~\ τ xZqdssy¤%≥kds τ Bˆ ªZX~\Z}\ D (t) ˆ ∧ µt eyZ|jkdX~\4qo\cbdf~k¤³sg s§ub0 +kdX~©bu}ezb{\ ¯ → A(t) WXZ\+}cegbd\ A(0) c } y e · L \ i | \cegµkE§µkoX·koX~\¡bdeg[\[Esg~sgkdsgZ }µknm²eyqogfZ[\cSk ¯ ?~sgqEeym > 0 ª ˆ 6= 0 }sgZbd©|i\q9koX~\jbd\cSf~\2}\ τ = τ ∨ i ¯ \XZe±¢z\ C(t) egZ| τ ≥ τ ¯ \ ˆ ˆ = ∧ A(t) }ceyPegx~x~ m+kdX~\!ÉZqbnkCxZeyqdk£sg¤¼kdX~\x~qosSsg¤¼kds Cˆ ªVX~\Z}\ D (t) → C(t) ˆ ∧ µt ¯ ?~sgq ≤ µ ªV§ \!g\k n. B,n i n. A,n i. t . C,n i. A,n i −1. n. ∗. B,n i. A,n i. n B. n B. n. C,n i. n. n. n. B,n i. n C. ˆ ˆ D(t) ≥ µt ∧ A(t). . >6 :7: $%-1 3 ( 4 +4 \9ÉZqbnku|i\ÉZ~\9koX~\sg !eyxikof~qd\4}sgZ|ikd sgegb0¤³sz s§ub !Ç È
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(57) . n. . i,j 1≤i,j≤K. . p1,1. .... R= pK,1 0. pi,j ... .... p1,K pK,K 0. 1− 1−. ¯¯. P P 1. i. i. p1,i pK,i. bdeykd©bnÉZ\Nb ±ªi¤ P b. . ʳË5ÌÀʹÍ.
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(66) !# ( - F ◦ G (x) = x
(67) ' ( *'( =G( (α, y) 7→ x(α, y) ' $! D D ' % $(.!G'> x(α, y) α. !$- &
(68) . & Á. y. WX~\Nb{\qd\c eykd sgZbey qd\Neg|im eyxZx2\Neyqo\c|² 8Pegbob{\cm¾ ]cÁ±¿legZ|w X~\cegZ| 8egZ|i\cZegf~[ ܾ N¿b{\c\jb{\N}Bkosz Á ¯ ] ¯» ·¤®eg}Bkegb4xLsg Skd\c| szfik# >¾ ¿Tª5§0\j}eg¬f2b{\jW5eyqbd¦S b9É~hi\c|¬xLsg zk!kdX~\csgqo\[ 1äW%egqobd¦r¾ ]c±¿ 30kdsjg\klkdX~\\hi b{kd\cZ}\4sg¤5koX~©bÉ~hi\c|xLsg Sk+1®}cey \N|~¨Zs§ ²¾ÜN¿ 3 C0fikl§0\g ¢g\9X~\cqd\ eEbd\¤¹°}sgSkoegZ\c|xZqdssy¤%kdXZeyklb{XZs§ub}sgSkof~knmjey2|j[EsgZsykdsz~©}knmx~qosgxL\qdkd \cb0¯ sy¤5kdX~\bdsg fikosz ¯ ÇÀÇ vrh©b{kd\2}\sg¤¼ejb{szfikosz+kds'koX~\4É~hi\c|x2szSkl\Nzf2ekd sg©bleg¡\cegbdm}sgZbd\cSf~\cZ}\4sg¤¼[sz~sykosg~©}knm ¯ 4 Z}\ eyZ| G egqd\~szi |i\c}qd\Negbd~P¤³f~Z}kd sgZbegZ| F ◦ G (0) ≥ 0 ª0§0\bd\\+koXZek (F ◦ F \4XZe±¢g\ egZ| F ◦ G (b) = b ¯ G ) (0) % b ¯ ?Zsgqueg ¢g\c+bdf~Zbd\k sy¤ b ≤ F ey(y) 2 | §0\|i\ÉZ~\ F : R → R m ∆ [1, K] y∈R y. n. α. y. α. α K +. (F∆ α,y )i (x1 , . . . , xK ) = αi +. x1 . Ì%Ì@å NZÓBÞ>O>P. xK. = α1 +. ¯¯. X. ¯. ∅ α = Fα,y ∆ α,y (x) = x. c. i. K +. P. = αK +. j∈∆. P. y. ∆ α,y. K +. pj,i yj +. pj,1 yj +. j∈∆. α. pj,i xj .. ¯. P. pj,K yj +. X. K +. j∈∆c. j∈∆. |i\xL\Z|+sz~mjsg eyZ| 9\ É~h y ∈ R egZ|ÉZqbnku{xb{kd,fZ|ii mj∈ kd∆X~\4}}cegbd\ FF WXZ b\Nzf2ekd sg+©b. F•∆ (•). α. y. j∈∆c. P. pj,1 xj ,. j∈∆c. pj,K xj ..
(69) ]N. . F( . egZ|9koX~\Àªg§ \ sgikey #¤®eg}kcª§0\½sgZm9XZe±¢z\rkdsC}eg }f~ eykd\ _\cSfZ[#L\qo~ ok» X~\4 Z|i\hi\Nbsy¤ x ªZeyZ|jkoeg¦~E Skds'{xeg}c}, sgifZ∈zk∆sz~}mkdX~sSb{\9 ∆ ª~§0\9XZe±{x¢g\ , i ∈ ∆} ¯ c. i. x1 . = λ1 (α, y) +. ¯¯. xn. i. c. = λn (α, y) +. Pn. j=1. Pn. j=1. p∆ j,1 xj ,. 17%3. p∆ j,n xj .. b'e bdf~Zb{kdsi}XZegb{kd©}[Eeykdqoµh eyZ| P bE ¢g\cq{ko~ \ 1³\c¢g\´¤³sgq db \\ \[E['e ¯ u\cZ}\zª ¤ λ(α, y) = (λ (α, y), . . . , λ I(α,−y)) ª v½SfZekosz 1 7%3£XZezb9sg~ m sz~\4b{szf~kd sg+g ¢g\+m . P ∆ = (p∆ i,j ; i, j = 1,3 . . .9 , n) ∆=∅. ∆. 1. ˜ ∆ = λ(α, y) + x ˜∆P ∆ x. n. ˜ ∆ = λ(α, y)(I − P ∆ )−1 . x. ⇔. \~s§·qo\kof~qd#kdsCsgf~q5É~h!x2szSk x~qdsz~ \[ W%sCbdX~s§·f~Z Sf~\c~\cbob5sy¤ZkoX~\b{szfikoszÀª Z}\ ¯ F ◦ G (z) ≥ F ◦ G (0) egZ|EkoX~\ key¦z\£eymEb{szf~kd sg z = F ◦ G (z) ¯ \CXZxe±¢g=\ Fz ≥◦G0 XZ\(x) \k eyZ| B = {i, b > y } ¯ H ¤r}szf~qobd\gªZ§ \X2e±¢g\ B ⊂ A egZ| b = x˜ z ≥ b¯ y} bd Z}\ F (b)A == F{i, ◦z G> (b) 8sgqo\s¢g\cqcªS§0\CX2e±¢g\ = b¯ α. α. i. B α,y. α. i. α. y. y. i. α. y. y. B. i. y. zi. = αi +. (FB α,y )i (z). = αi +. X. X. pj,i yj +. pj,i yj +. X. j∈B. j∈A\B. j ∈A /. X. X. X. rj,i yj +. j∈B. pj,i zj +. j∈A\B. pj,i zj , pj,i zj ,. j ∈A /. XZ\Z}\gªg§0\XZe±¢g\ F (z) ≥ z ¯ C0fik½bdZ}\ (F ) (z) % x˜ = b ªg§0\XZe±¢z\ b ≥ z ¯ ?52ey m z = b ¯ ?Zsgq0eym ª ©b½e}sgSkd f~sgfZbr~sgi |i\N}qo\cegbd ~C¤³f~Z}Bkosz ¯ y) = λ(α, y)(I − P ) \+X2e±¢g\jsg¤ ?5h´eym ∆ (α, ªy)egZ7→|´x˜|~\ÉZ(α, ~ \ ª (α, y) A = {i, x (α, y) ≥ y } B = {i, x (α, y) > y } ¯ }sgf~qbd\ x(α, y e Z P | ³ ¤ g s q ² + e ~ \ g X L g s o q ~ X s s P | y s ¤ ª5§0\X2e±¢g\ ˜ (α, y) = x ˜ (α, y) y) = x (β, z) (α, y) À ª g e Z | d k Z X ! \ } z s S d k S Z f µ n k m y s ¤ ³ ¤ z s s u § l b ³ ¤ d q z s [ koXZek9sy¤ ˜ (β, z)} x(β, z) ∈ {˜ x (β, z), x (α, y) 7→ x(α, y) (α, y) 7→ x s§ kosEbd\˜\Cko(α,XZeky)koX~¯ b¤³f~Z}Bkosz bZsg|i\c}qd\Negbd~2ªSkoey¦z\ (β, z) ≥ (α, y) ª2§ \9XZe±¢z\ B α,y. B n α,y ∆ −1. ∆. A. A. i. B. B. i. i. i. B. ∆. Fβ ◦ Gz (x(α, y)) ≥ Fα ◦ Gy (x(α, y)) = x(α, y). egZ|jkdX~\bd\cSf~\cZ}\ {(F ◦ G ) (x(α, y))} Z}qd\Negbd\cb kds x(β, z) ¯
(70) . "
(71) , :6 ' % )( * -, .0(21 3"47698!;:=<>6 )354 \}szZbd |i\cq0kdX~\9¤³sz s§ZEbd\cSf~\cZ}\9sg¤5Seg}¦ib{szj~\kn§ szqd¦ib §kdX , JN = {σ , ν , N }, β. n. n. lim. n→∞. N n. z. n. n. n≥0. n. n. = (n(0) , n(1) , . . . , n(K) ),. n(0) ≤ +∞,. n(i) < ∞, i 6= 0.. ʳË5ÌÀʹÍ.
(72) ]z].
(73) !#"%$ & '( *)+-, .& '. WX2ey~¦ibkos z¢z\ eyx~xL\Z|~µh«ªZ§0\}eg}szZbnkoqdf2}BkkdXZ\!}szqdqo\cbdxLsgZ|i ~Zx~fikCegZ| szfikdxZfikux~qos}Procedure \cbob{\Nb A eyZ1| D ¯ \ezbdbdf~[E\lkoXZekkoX~\|iqo¢ ~'b{\NSf~\Z}\cbboeko b{¤³m §X~\cqd\ t 7→ Σ (t) ∧ n beE}sgZ}ce±¢g\£¤³f~Z}BkoszÀª ˆ Σ (t) → Σ (t), n. ∀k ≥ 1,. n. (0),n. (0). ˆ (k),n. (k). Σ. (0). (t) → µ t, ∀t ≥ 0 (µ n ˆ Pi,j (t) → pi,j t ∀t ≥ 0.. \ bdf~x~xLszbd\£koXZekkoX~\4qosgfiko~['ekoqdh ǽà ± Ä
(74) 0. $ '!'(>' A % ! n. (k). (0). ≥ 0),. bdeykd©bnÉZ\Nb ¯ $! G( >
(75) :$!
(76) %$ '((' <. P = (pi,j )1≤i,j≤K Dn. Aˆ(i) (t). = n(i) + p0,i (Σ(0) (t) ∧ n(0) ) +. ˆ (i) (t) D. = Aˆ(i) (t) ∧ µ(i) t.. K X. +
(77) 1ä%3. ˆ (j) (t), pj,i D. 1{]c%3. j=1. &.
(78) .&. ] v¼hi b{kd\cZ}\ eyZ| f~~©Sf~\Z\cbob'sy¤4b{szf~kd sgZbkos²v½SfZekoszZb:1® 3EeyZ| 1{]c%3!¤³sz s§ i| qd\N}Bkd m¤³qosg¯ [prqosgxL\qdknm¡Ájegb9bdX~s§ koX~\Ex~qdssg¤ ¯ 8sgqo\s¢g\cqcª2µk4\Negbd m+¤³sg s§ubl¤³qdsz[ kdX~\ x~qosSsg¤«kdXZeyk0\Neg}Xj}sg[ExLsg~\czk0sy¤ A eyZ| D ©b}szZ}e±¢z\ueyZ|jµ¤ Σ ©b x~\N}\T§©b{\£ ~\NeyqrkoX~\ b{s'egqd\CkdXZ\4x~qdsi}\cbob{\Nb A egZ| D ¯ ¯ WX~\szqd\c[ @ ¯ ]#sy¤¾ ¿%z¢z\cbkoX~\4¨ZfZ |egx~x~qos±hi['ekoszsy¤re'g\c~\qey ¥\c|Seg}¦ibdsg~\kn§0sgqo¦Vª~¤ § \key¦g\'e+ ~\NeyqC¤³f~Z}kd sg¬¤³sgq Σ ª«koX~\¬¤³qosg[ (A,ˆ D) ˆ ª §0\E}cey }eg }f~ eykd\\hix~ }µkomPkdX~\ b{szf~kd sg+sy¤5kdX~\4\Nzf2ekd sg2b sg¤%koX~©buWX~\szqd\c[ ¯ ÇÀÇ ?ZsgqÀegSmlÉ~hi\c| ªN§0\r|i\ÉZZ\ kdX~\½bd\cSf~\cZ}\Nb«sy¤x~qosi}\cbobd\cb {A (k), D (k)} egZ| {A (k), D (k)} §kdX+koX~\bdeg[En\Cqo≥\c}1f~qoqd\cZ}\C\cSfZeykd sg (0). (0). n t. . An (k + 1) = Γ(Dn (k), JNn ), Dn (k + 1) = Φ(An (k + 1), JNn ),. Zfiku§µkoX¡|iµ¸V\qo\Sku ~kd©eyÀ}szZ|ikd sgZb 4\ qo\c}egLkoX~\4~sykekosz Γi (X, JNn )(t). Dnt (0) = (Σ(1),n , . . . , Σ(K),n ). n = n(i),n + P0,i (Σ(0),n (t) ∧ n(0),n ) +. n t. egZ| K X. n b. n b. k≥0. Dnb (0) = (0, . . . , 0). ¯. n Pj,i (Xj (t)),. j=1. n. Φi (X, JN )(t). = Φ(Xi , σ. (i),n. )(t),. \ ge Z|§0\E§ ¼fZbd\kdX~\'bo}ey \c| b{\NSf~\Z}\cb ge Z| Skdqosi|ifZ}\£kdXZ\C['egx~x~ ~zb Γ : C → C Aˆey2(k)(t) | Φ : C= → C kdXZeyk egx~DˆxL\cey(k)(t) q v =zf2ekd sg2b 1ä%30¯ ye 2| 1{]c 3 1®§X~\qo\ ©b0kdXZ\b{\kusy¤5}sgSkd f~sgf2b0¤³f~Z}BkoszZbsz 3 C R n. K. K. An (k)(nt) n K K. Dn (k)(nt) n. n. +. Γsi (x1 , . . . , xK )(t). = n(i) + p0,i (Σ(0) (t) ∧ n(0) ) +. K X j=1. Φsi (x1 , . . . , xK )(t). Ì%Ì@å NZÓBÞ>O>P. = xi (t) ∧ µ(i) t.. pj,i xj (t),. k≥0.
(79) ]±. . F( . W XZ\£¤³sz s§ ~E\c[['eX~sz |~b0¤³szquLsykdXkdszx¡eyZ|2sgk{kosg[¶bd\cSf~\cZ}\NbªiXZ\Z}\4§ \9sg[EkkdX~\ . szq . ¯ à ·; '!'!
(80) <
(81) # ! k Dˆ (k) → D(k) & $
(82) %$
(83) $
(84) D ˆ '6#$! $ =G !$-
(85) (2,
(86) ˆ D(k) t. b. n. & $ . n→∞ ˆ n (k + 1) − ˆ ˆ + 1) A −−−→ Γs (D(k)) = A(k. . n→∞ ˆ n (k + 1) − ˆ + 1)) = D(k ˆ + 1) D −−−→ Φs (A(k. $
(87) D'6. ÇÀÇÇ Ã + à ´« ª~§ \9XZe±¢z\ t. ˆ + 1) A(k. . ˆ + 1) D(k. & $ . . $! $ =G ! $(- ' . ?ZsgquegSmEÉZh\N|. K n n P0,i (Σ(0),n (nt) ∧ n(0),n ) X Pi,j (D(j),n (k)(nt)) A(i),n (k + 1)(nt) n(i),n = + + . n n n n j=1. 9l\Z}\9kdXZeg~¦ib0kds \[E['eiªi§0\4XZe±¢g\ ˆ f ¯ s ¯ } ¯ eyZ|\cez}X+}sg[Ex2sz~\Sk ˆ A (k + 1) −−−−→ Γ (D(k)) gs ¤ A(k © u b } c \ g e d q j m E e } z s Z } ± e z ¢ l \ ³ ¤ Z f Z B } o k z s s § d k 2 X y e ~ ¦ib0kds'p½qdszx2\cq{knm#koX~\4qd\Nb{fZµk ˆ ¯ ¤³sz ˆs§ub ¯ +1) = Γ (D(k)) \4~s§qd\kdf~qojkoskdX~\4x~qossy¤5sy¤ prqdszx2\cq{knm' ¯ \XZe±¢z\ A(k WX~©b¼\cSfZeykd sgg ¢g\cb%koX~\qd\c eykd sg2\kn§ \c\l¤³f~Z}kd sgZb¼sg¤Le ˆ qo\cegxZeyqey[Eˆ\kd\+q t1)¯ C0=fik Γ§0◦\rΦ}eyC(A(k)) É~hCkdXZ ¯ b xZeyqey[E\ko\qÀegZ|£kdX~\c4§ \¼szikoegC¤³sgq%eymuÉ~hi\N| t eg4\cSfZekosz L\kn§0\\c4qd\NeyzSfZ[#L\qb«kdX2ek §0\r§qoµko\ ˆ 1®\¢g\c4µ¤ b%b{f~xZx2sSb{\N| ˆ koseg}krsg¤³f~Z}kd sgZ!b 3 ¯ 8¡szqd\cs¢g\q¼ezbre4A(k+1)(t) }szZb{\NSf~\Z}=\syΓ¤«pr◦ΦqosgxL(\A(k)(t)) qdknm#Á~ªz§ \¦~s§kdXZΓeykr◦ΦkdXZ\É~hi\c|ExLsg zk \NSfZekosz Γ ◦ Φ (ζ(t)) = ζ(t) XZezb£egf~~©Sf~\Eb{szfikoszÀªVZey[E\cm ζ(t) = x(α, µ t, . . . , µ t) ª §kdX α = (n + p (Σ (t) ∧ n ), . . . , n + p (Σ (t) ∧ n ), . . . , n + p (Σ (t) ∧ ?~szq½eym ªzkdX~\lbd\cSf~\cZ}\ ˆ 1®qd\Nb{x ˆ 3 ©b¼~szE|~\c}qo\cezb{ ~ 1³qo\cbdx ¯ ¯ n )) ¯ t {A (k)(t)} {A (k)(t)} Zsg+Z}qd\NegbdZ 3 ¯ \4XZe±¢g\ Aˆ (k)(t) −−−−→ ζ(t) eyZ| Aˆ (k)(t) −−−−→ ζ(t) eyZ| Dˆ (k)(t) −−−−→ ey2| ˆ Φ (ζ(t)) x~xZ~S¯b . 7→ Γ(., JN ) egZ| . 7→ Φ(., JN ) eyqo\l~sz|i\N}qo\cegbd ~ 8sgqo\s¢g\cqcªyÉ~hjDey(k)(t) m n ≥ −1−ªS−kd−X~→\C['Φey(ζ(t)) egZ| n→∞. n. s. s. s. s. s. s. s. s. s. (1). (1). 0,1. (0). (0). (0). (i). b. (0). 0,i. k≥1. (0). t. k→∞. b. s. 9l\Z}\gªi¤³szquey . s. k→∞. t. k≥0. (K). 0,K. k≥1 k→∞. (0). k→∞. t. b. s. n. ª~§0\9XZe±¢g\ . (K). . n. An = Γ(Dn , JNn ), Dn = Φ(An , JNn ).. Anb (k) ≤ An Dnb (k) ≤ Dn. ≤ Ant (k), ≤ Dnt (k).. ʳË5ÌÀʹÍ.
(88) ]NÁ.
(89) !#"%$ & '( *)+-, .& '. \4XZe±¢z\ An b (k)(nt) n. ˆ b (k)(t) A. XZ\Z}\gªi§0\4XZe±¢g\. An (nt) n n An (nt) ≤ lim supn A n(nt) n. ≤ ≤ lim inf n. ∀t,. lim. WXZ\Cqo\cbdf~k¤³sg s§ub ¤³qosg[ \[E['e ¯. n. ≤ ≤. An t (k)(nt) , n. ˆ t (k)(t), A. An (nt) = ζ(t). n. . #
(90) 4%¶? &À:
(91) %9%"
(92) 62: ( : +6 \É2qob{k#qo\c}egrkdX~\+|i\ÉZZµkosz·sy¤ubd[Ex~ \jvrfZ\cq#~\kn§ szqd¦P¤³qdsz[ 4 \N}Bkosz· ¯ ]'sy¤¾ÜN¿ ¯ w sgZbd©|i\q#e qosgf~kd\ p = (p , . . . , p ) §kdX 1 ≤ p ≤ K ¤³szq i = 2, . . . , L − 1 ¯ 4 fZ}X ejqosgf~kd\ b4bdfZ}}\cbobn¤³fZ5¤ \9}egjegbob{si} eykd\kos!bdfZ}Xjeqosgf~kd\le#qdszfikd ~!bd\cSf~\cZ}\ ν eyZ|je¢g\N}Bkosgq egZ| p eg= 0sz s§ub p1 =[EK + 1 }¯ sgZ}cekd\cZekoszD 0 b ³ ¤ N \ y e Z b 3 φ ⊕ 1. 1. L. i. L. Procedure 2(p) : −1−. for k = 0 . . . K. do. (k). ν := ∅; φ(k) := 0; −2−. od for i = 1 . . . L − 1 do ν (pi ) := ν (pi ) ⊕ pi+1 ; φ(pi ) := φ(pi ) + 1;. ys db ko \£[koxZXZ\rek v½φf~\cqÀ~©b\knkd§ XZ\4szqdS¦ufZ©[#b e0L\zq\~sy\c¤5qo¢eg bd ¥µk\Nb|£kdSsEeg~}¦isibd|isg\ Cj~\¡kn§0bdsgfZqo}¦ X+eEqosgf~kd\ ¯ ªN§µkoX º E = {σ, ν, N } od. (j). W XZ\!qosgfikoZjb{\NSf~\Z}\ ν = {ν } ©bCg\c~\qeko\c|mejbdfZ}}\cbobn¤³fZ%qosgfiko\!egZ| Nσ==(1,{σ0, . }. . , 0) ¯ ©beEb{\NSf~\Z}\9sy¤ qd\NeyT¢ey f~\N|~sziTZ\zeykd ¢g\Cf~[#L\qbªiqo\xZqd\Nb{\czkoZbd\qo¢ }\lko[E\cb ¯ w szZbd |i\cq«~s§ eb{\NzfZ\Z}\¼sy¤ib{ [Ex~\¼v½f~\cq ~\kn§ szqd¦ibcªboe±m §X~\qo\ = {σ(l), ν(l), 1} ¯ \|i\ÉZ~\ σ ey2| ν kos'2\4koX~\ iÉZ~kd\}sgZ}ceko\Zekoszsg¤ {E(l)} kdX~\ {σ(l)} egZE(l) | {ν(l)} ¯ ) \Zsykd\ m σ koX~\b{\NSf~\Z}\9sgikey ~\c|j¤³qdsz[ σ +kdXZ\9¤³sg s§ ~E['ey~~\cq (k) φ(k) i=1 i. +∞ l=1. (k) φ(k) i=1 i. +∞ l=1. +∞ l=1. c. σc = (cσ (0) , σ (1) , . . . , σ (K) ).. \²}szZb{©|i\cqjkoX~\²}sgqoqo\cbdx2szZ|i ~´bd\cSf~\2}\¬sy¤!Seg}¦ibdsg>~\kn§ szqd¦ib JN = {σ , ν, N } ªl§kdX WXZ\Seg}¦ib{sz ~\kn§0sgqo¦ JN }sgqoqo\cbdx2szZ|~b9kos ey´\[Exiknm ~\kn§0sgqo¦¬§µkoX n N = (n, 0, . . . , 0) ¯ n. Ì%Ì@å NZÓBÞ>O>P. n c. n c. c. n.
(93) ]c. . F( . } fZb{kdsg[E\cqobC ¬Zs|~\ 0 eyk9kd [\ t = 0 ¯ \'§ ½|i\Zsykd\Em X koX~\Ekd [E\Ekds¡\[ExiknmkdX~\bdmibnko\[ ªL}ey \c|[Eeyhi['ey%|~eko\qusg¤%koX~\~\kn§ szqd¦ WXZeg~¦ib0kds'koX~\4vrf~ \q£x~qdszx2\cq{knmjsy¤ {E(i)} ª~§ \ JN ¦~s§ kdXZeyk¤³sgquey n ª X < +∞ 1äb{\c\E¾ ¿ 3 ¯ ¯ \bdf~x~xLszbd\CkdXZeyk n c. n c. i≥1. n c. (0). σc (1, n) n→∞ n (k) σ (1, n) lim n→∞ n Pi,j (n) lim n→∞ n. =. lim. \egbobdf~[E\koXZek ¤³sz s§ ~bdmibnko\[. =. c , λ 1 , µ(k). = pi,j ,. P = (pi,j )1≤i,j≤K. ∀i ∈ [1, K],. 1{]g] 3. 0 ≤ i ≤ K,. K X. 1{]cÁ 3. 1 ≤ j ≤ K + 1.. bdeykd©bnÉZ\Nb ¯ \|i\c~syko\¡m πi = p0,i +. 1n]±M3. (µ(k) > 0). 1 ≤ k ≤ K,. πi. koX~\bdsg fikd sg sy¤CkdX~\. 1{] 3. pj,i πj .. WXZ\ }szZb{koeySk ©b kdX~\0\hixL\c}Bko\c|4f~[!2\cq%sg¤~¢ bdµkbÀkos£bdµko\ ¤³sgq kdX~\8egqd¦zs¢£}XZeg4§kdX#kdqeyZbdµkosz '[ eykdqoµh P eyZ|>π §kdX ~kd©ey9|i©bnkoqd ~fikosz p 1äb{\c\Px~qossyi¤4sy¤ \[E['e² 3 ¯ \ § Cx~qds¢z\kdX~\ ¤³sz s§ ~#koX~\szqd\c[ ÃÇ Ã %=
(94) " . '+$! - ' ,#
(95) =G + c ≥ 0 j=1. i. 0,i. Xcn c πi = max (i) ∨ . n→∞ n 1≤i≤K µ λ lim.
(96) )6"6 6 6 5C ¢g\ce¬qosgfiko~¬['ekoqdh koXZekboekd©b{ÉZ\cb ey2|e ¢g\N}Bkdszq P = (p ; i, j = 0, . . . , K + 1) 2 ª § \ ~ | \ ~ g s d k \ m d k ~ X \ { b z s ~ f d k g s y s ¤ kdXZ\4¤³sg s§ ~bdmibnko\[ 1äb{\c\ \c[E[Ee α π 3 = (α , . . . , α ) ∈ R 1. K. i,j. K +. ∀i ∈ [1, K],. α i. πiα = αi +. K X. pj,i πjα .. j=1. ǽà ± Ä '!%6<' ( $!6 ="%$ & '( :-, .& '= ' . (> '> $
(97) *
(98) µ ,
(99) ,
(100)
(101) $! D $ 1/0 = +∞ k Σ (t) = λt/c c ≥ 0 + , M( λ > 0 ,
(102) X < +∞ n α = (n + n p , . . . , n + n p ). : . (k). (0). n. (1). (0). 0,1. (K). (0). . >0. 0,K. Xcn cn(0) πiα ∨ = max (i) . n→∞ n 1≤i≤K µ λ lim. ʳË5ÌÀʹÍ.
(103) ]±.
(104) !#"%$ & '( *)+-, .& '. ÇÀÇ ÇÆ#Ã Ç È w szZbd |i\cq ko~X \ueyf~h egqdm4Seg}¦ib{sz4~\kn§0sgqo¦ ˜ = {0, ν , N } ªyegZ|koX~\egbob{si} eykd\c|¢g\c}kdszq Y(n) ª §XZ\qo\ Y (n) © b0kdXZ\£kosykeyVf~[#L\qsy¤ }fZb{JN kdsz[E\qbrkoXZekgs!kdXZqdszf~gX~si|i\ i kdX~©b~\kn§ szqd¦ ¯ \ X2e±¢g\ n. n. n. (i). n Y (i) (n) = n(i),n + P0,i (n(0),n ) +. K X. n Pj,i (Y (j) (n)).. j=1. 9l\Z}\. bob{f~[Exikosz usz P ¯ ©bsbn§ ko } sglimZbd©|i\qkoX~9u\\sg2=qo} \Cgπ § Z\Ceyko«XZXZ~eye±\~¢gkn\l¦i§ b0kdszX~koqd\£s¦ ¤³egJN ¯ ~W!X~\#Z\cSSfZfZ[#eyL \knm!qusg¤³sz¤¼q0}kdfZX~b{\Ckdsg['[Ee\chiqob0['koXZegVekl|ZezkdsE\cq kdX~sgqo¤ sg~fZsig|iX\ ~si|i\ iª g s s § Y (n) ¯ i≥1 n. Y (i) (n) n. α i. n c. (i). X (i),n ≥ σ (i),n (1, Y (i) (n)). ¯rº Z|¤³sgq~si|i\ 0 ª X. (0),n c. (0),n. ≥ σc. (1, n(0),n ). ¯ u9 \2}\gªZ§ \9XZe±¢z\. X (i),n σ (i),n (1, Y (i) (n)) πiα , ≥ lim = (i) n→∞ n→∞ n n µ. lim inf. (0),n. (0),n. Xc n→∞ n. ≥ lim. lim inf. n→∞. σc. (1, n(0),n ) cn(0) = . n λ. Z}\ X = max X ∨ X ªikoX~\4s§0\qLsgf~2|j¤³sz s§ub ¯ ½Ã Ç È½ \#}sgZbd |~\qukoX~\szqd g Zey%zez}¦bdsg+~\kn§ szqd¦ ¯ WXZeyZ¦bkdsprqosgxL\qdknm+ZªZ§0\¦~s§ kdX2ekukoX~\!}sgqoqd\ bdxLsgZ|i ~Zx~fikley2|jszfikdx~f~kxZqdsi}\Nbdbd\cb eyZ| }sz¢g\qog\lkds'e#¨Zf~©|j [Ek eyZ| qo\cbdxL\c}B ko¢z\ m ¯ \k T = inf{t > 0, Aˆ (t) =ADˆ (t)}Dª T = max T eyZ| MAˆ= T ∨Dˆcn /λ ¯ \4XZe±¢z\ 4. n c. (0),n c. (i),n. 1≤i≤K. n. (i). (i). n. (i). i∈[1,K]. Aˆ(i) (t) = n(i) + p0,i n(0) +. ∀t ≥ M,. XZ\Z}\gªi§0\4XZe±¢g\. K X. pj,i Aˆ(j) (t),. (i). πiα0 µ(i0 ). ∀t ≥ M,. 1{]NM3. ˆ (i) (t) = πiα . Aˆ(i) (t) = D. \'|i\Zsykd\ i = arg max{T } egZ|P§0\XZe±¢g\ Aˆ (T ) = Dˆ ª~XZ\Z}\ T = ¯ 8sgqo\s¢g\cqcªSv½SfZeykd sg 1n]N%3 [Ex~ \Nb Aˆ (i0 ). (0). j=1. ∀t ≥ M,. 0. (i). (i0 ). (i0 ). (T ) = µ(i0 ) T. mP}szZ}e±¢knm+sg¤. Y (i) (n) − D(i),n (nt) n→∞ −−−−→ 0, n. §XZ\qo\ ©bukoX~\#kosykey5SfZ[#L\q9sy¤½}fZbnkosg[E\qblkdXZeyk£gsjkdX~qosgfZgX~si|i\ i ¯ 4 Z}\ §0\9¦~sY§ koXZ(n) ek¤³szqueym t ª (i). Xcn. ≤ nt +. K X i=1. Ì%Ì@å NZÓBÞ>O>P. Xcn < +∞. σ (i),n (D(i),n (nt), Y (i) (n)) + σc(0),n (Σ(0),n (nt), n(0),n ),. ª.
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