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Asymptotic Analysis of Congested Communication Networks

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(1)Asymptotic Analysis of Congested Communication Networks J. Frederic Bonnans, Mounir Haddou. To cite this version: J. Frederic Bonnans, Mounir Haddou. Asymptotic Analysis of Congested Communication Networks. [Research Report] RR-3133, INRIA. 1997. �inria-00073556�. HAL Id: inria-00073556 https://hal.inria.fr/inria-00073556 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Asymptotic analysis of congested communication networks J. Frédéric Bonnans , Mounir Haddou. N˚ 3133 Mars 1997. THÈME 4. ISSN 0249-6399. apport de recherche.

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(116) 5 6. WaAX. 8 . W W 2  X I. . *. 5. 2 I&. . . 2 I  'X . . *. 3Wa X f & K. $ & $ & 2 I& R I2 !  h E & & 8. TU X z X§i z XW 2  'X ? DW >X¡eŠk s f U XIk DW X P eŠk s W P 4 J i z a>b =n£na†po_`kd‘0f U X s X)QDkn_jfh_ji9kiŠ DW+^ _ XÑ£nŸ¡X U eG›ŠX f W 2 WaAXG3WaAX X ? e z ‘mY\_`k  h E $& h 2 I a  & h &  m & . a. 2 I& K. 2 I&. & E 8. $ &ih W =  X I& h. &. /. . a. & E 8 .  .  . N WaAX f & / a 2 I& WaAX . h 8LW /. aAXGR. XÑ¥. $& h / a  f & R =W  X I&  h / a 2 I&. . W¤Z†¥ 9X.

(117) ‹5=. - 5"  -   . ]u_rknvX.  W2. WaAXG3WaAX X. z XIemv U. XK§i. ? (gW. bc=†£nŸ{X U eG›mX $& h / a  3WaAX f `2 I& WaAX. . Y   

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(119) f U eŠf =. $&. . e. R. W¤Z†¥æŽAX. WaAXG 3WaAX X*foi W W 2  X I'. 3WaAX K   h. 8   . a 3 WaAX f & W = X I& K / a $H& 25I& WaAX P =bR RW =  X I& h / a 2 I& WaA+X T . 2 I&. & E 8'. ? (gW >X. =! 3 WaAX. a  3WaAX f & / a i2 I& WaAX /. X;YwY\ew‰†¥ C\eŠk s e IX YwYwew‰n¥ ‰n£dŸ¡X(‘mXf. Ñ£DeŠlnldcrtu_rkn‘ NW Z†¥ 9X4§i z. W 2 WaAXG 3WaAX X. /. 8LW !. . /. egf /. . h 8LW /. a XGR. a XGR. aAXGR. DW >X{eŠp¡§imcjcri˜Ÿ!p 2 I P W2  XI *DW >X P W 2 'X ? (gW XG eŠk s  P    R '    ¦ X ¢ kdi˜ŸLf U egfW 2 Wa jX 3Wa X/X ? (gW X¡eŠk s W 25I WaAXj3WaAX/X*vi9km›mX z ‘ŠX;pKfhi W W 2  X I    X ¥ B*t e XIYwYwew‰n¥r‹Š£u§i z X;eŠv U a b \ = f U X z XXxd_`pofqpW 2  _   X ? *DW >X¡pha†v U f U egf  2  _ K 2 WaA X ?! ( W/W 2  X I' K 2 I' WaAXG   K 3WaAX/X P 8LWo‹Xj UŸ~ X z z X ( _`pe\lDimph_rfo_r›Šz X

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(121) ‹m‹. 

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(132) cr_rY\_rf€l z imždcrXIY $ &

(133)  ` _ k  = D & &E . V = 4 PkW 4  ; P ‹QSR[R[R[]\  Sp R Sf R = &4 P = I&  O P Z‹ [RSR[R 0  $H& K E .  8   . W^ X. 4 4 _ fj=`F  FL = # = I P =. . ;. P =bSR[R[R[]\.  . f U egf!pqegfh_rp Q†X;p EW ^ X = P DW X¥  %Y  # 2ZI WaAX _  I  %& "A#"  K«

(134) $

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(157)    !#"$# %&'  () +*, -/.. T U U _rp_`kdX 9a†egcr_jf tPz l z i˜›mXIU p¡f U eŠ2 f 2 žDXcrimkd‘9p*foi DW@XÑ¥  U i9i9Tpo_`U kd‘ 2 z ? DW@XKU _rkZWNZ†¥ C =X{emk s lneŠpqpo_`kd25‘wI foi fvimXwknphcjfh_`emY\k9_jf¡fI£"i˜Ÿ¡›mX X0z l DW i˜ ›mXbXÑ¥f egf _`p

(158) egc`poieQpoimcradfh_ji9k„iŠ  f ¥ X0l iuigK_`pf X;k³vimYwldcrXfhX0po_`knvX _rp J _`knegcrcrtœŸ{XvIeŠkQpofqegf fhXe\f U Xi z X;Y f U eŠf!pha†Y\Ywe z _IHX;pegcrc z X;phadcrfhp*iŠf U _`p!poX;vfh_ji9kt  «u¯ – «

(159)    -& W = WaAXj N WaAX/X %Y  W = WaAXj N WaAX/X ? EW^ _ X   +d) . *. *. *. . $. . N WaAX P.   /. WaAX P. = I.  . a. $. h 8LWa Xj. aLW 2  X I /. WaAX P W = X I    X ? *DW X " 2. = I. * $ )- W = / =LX. ? *DW. X W2 .  .   . *. $. h . . WaAXG . DW  f [X .  ?. h 8LW. aLW 2  X I . h/. a XG /. *. 

(160)   Q  \ b P  Æ Š%'¡   s U s U §}ž†ikweŠz phf X as _`pKvimcrphimkXIphvX0XIfhld_jfoi9_`iZvk\i9=nŸ{km¥ ›mX!X Tphz fh‘ŠU a X;XkntvY\X‚f XIf f U X€U XIi vi si9z po(t f f U ¤Ræ‹a†egT»fkn¥ vŸ¡ fh_jX0X i9z k §Xim6'cj_rcrpW i˜Ÿ-fXU t X _ra p z  XIz X;pqadY6 cjf*W_`kd^ iŠ_`_ phX vf XIemU k9k _rfp!iŠph‘ŠXI€_rv›ŠeŠfoX_rphimtuemkWYwk0¥ ldeŠfhphiŠtufhYw_rv0ldfhfhiŠXIfhv _rU v*knX_#uxdadl†X;eŠp knphR _j‹Ii9‰k T d#%&  +  6 W X ) D  «u¯ – «

(161) . . $. . . "%$. '6 W+^ _ X P * $ )- W    2  X 1 -  ¥ e XIf. /. . a 6 W X. a 6 W  X h. h. & E 8' . ? *DW GX R.  f & W 2  X I&. h 8LWa Xj . ±ž†Xewkdi9knkdXI‘megfh_j›mX z X;egcWkuanYbžDX z f U egfŸ¡XŸ_jcrc"v U iui9poXkdXxdfI¥{]9Xf

(162) = N a _ E W =  N X ? gW!d^ X  s _W  X K = N P s _ W  Xut  W

(163) :d¥ Cd‹X a Xc`poXm¥ h TU _rp z X;phvIegcrU _rkn‘_rpY\iŠzfo_r›˜egfhX s U ž9t(f U X¡Xxnlnemknpo_rimkiŠdf U X*poz ims cranz fo_rimknpI¥ }k s XIzX s £˜f U XKi9ždfheŠ_rknX s zQXxdz l†eŠknph_ji9k fhpq XIadž†v ‘mU po‘ŠX knXIz _#ph›mufhXadpPX;Q f p*z ph_reŠkQf¡ff imU e lnegfof!ln_`Ye phiŠI_ fPHIcr›ueŠiŠ_rfoknb_rim‘ fkQW+^lX z_ imvX{i9žd_rpocrp*XIfPXYw¤9anpankn_jv›˜vemegfhk_jcri9XIžDkk9Xf*_`§fopQimiœa†igphkiŠs cri ›Š_rX(k fX RU ‹IX‰ T=ph¥ vIaueg¥ crX s li z Xi9ždeŠcrXIžDY imad fQ_`pok _`Y\s _j_c`e W =  N XIXGpqR vegcr_`kd‘ Dnm o e Xf W = Wa 4 XG N Wa 4 X X W W = Wa 4 XG N Wa 4 X X ? DW+^ _ X XžDXPe'ph X uadXIk†vXœvi9k9›ŠX z ‘Š_`kd‘Qfoi?e'lne z fh_reŠcKvX;k9foX z W = =X{eŠk s vimknph_ s X z eŠkiŠf U X z ph X uadX;knvX W = _ 5 N _ ;X vi9km›mX z ‘Š_`kd‘\foi W = /=LXKemk ss )X QBkdX s ž9t = _ P = h a 4 2  eŠk s N _ P   a 4  /. . (. . /. . . *. . . ÅÅØnmIÒ;ÌoÒÒ. /. . /. .

(164) ‹59. - 5"  -   . Y   

(165) Z %&'j- A";"A %. Ÿ U X z XW   2  X{_`p!eQnxdX s phiŠc`adfo_rimkigDW >X¥ J i z X;eŠv U a 4 n£ Ÿ¡X ¢ kdi˜Ÿ f U Še f. W

(166) :d¥ CACX. W = Wa 4 XG N W a 4 X/X1! s _ W = _  N _ XGR. s  _. Xf!anpphfha s tPf U X

(167) f Ÿ{i U emk s © po_ s X;p{iŠf U _`p_rknX uanegcr_rf tŠ¥ ¦ŒX U eG›ŠX@Q z pof e. s  _ W =. N Wa 4 jX  N Wa 4 /X X P Wa a 4 X 4 P. ‹. /. a. K /. a. . h 4. = I&. & E 8 . Wa. $H&. 3Wa 4 X h 2 I 4 & E 8' &. N Wa X f & h = I 4 h & Wa 4 X  4 X & E 8 . $H&. Wa 4 X. K  h. . ' . 3Wa 4 X f & h 2 I. & E 8'. sC2 z U s ]u_rknvX X kdW im2 fhWega fh4 _jXji9kn3p Wa 34 WX a X 4 ?X¡emgk W >XW£†a v 4 U X¡iueimphX‚_`kdf ‘ imphX

(168) P an6 phWX >X{_rk?Ÿ¡X]9X;U veGfh›Š_ji9X k‰†¥ X W TU. $H& h. . &. Wa 4 X. & E 8  . =iI&. N Wa 4 X f Wa 4 X &. . N Wa 4 X f & Wa 4 X  $& h N Wa 4 X f = I& Wa 4 X. = I&. & R. (. Wa 4 Xj N Wa 4 X/X. s  _ W =. _. &E. 3Wa 4 X f & h 2 I &. 8. $ &ih. Wa 4 X. &E . 8 . y emphph_`kd‘bfhi0f U X

(169) cr_rY\_rf!eŠk s anph_`kd‘ e XIYwYwe\Z†¥ Cd£dŸ¡Ximždfqeg_`k cr_rY_ _`Iku s_ W = Wa 4 XG N Wa 4 X X _ & E W 2  f X I&& h & E  i˜Ÿ

(170) £nvi9knpo_ s X z f U X z _j‘ U f U emk s © po_ s XigEW+:n¥ C;C;X dŸ{X U eG›mX . . #. 8   . 8' . . $H& =  I&. P. &E. 8.  f W2  X. . W

(171) :d¥ Cg‰LX R. &. I& h 6 W. W

(172) :d¥ C˜Z&X. XGR . . s  _ W =. N _  N _ X P a _ 4 P. ]u_rknvX. W   2  X ? s  _ W =. /. ‹. a. EW >X£df U *. _  N _ X P. N _ f & 6 W >X $& h h = = I I h E / a 4 & E 8  W _ AX & & E 8 W _ AX & &8      f &  $ &    h 6 W >X h 2  I K ' 2  I h 4 & E 8 W X & & E 8 ' W X &   K. X

(173) c`eŠphfX uaneŠcj_rf tPžDXIvi9Y\XIp. & E 8'.  jf W2  X. I& h. N _ f & P W _ X I&. $ &ih. &. & E 8  . =. y emphph_`kd‘bfhi0f U X

(174) cr_rY\_rf!eŠk s na ph_`kd‘ W+:d¥ CACXÑ£dŸ¡X U eG›ŠX cr_`Y _ pqanI l s_ W = Wa 4 XG N Wa 4 X X J i z Ybadc`eŠ@p W

(175) :d¥ C˜LZ X{emk s W

(176) :d¥ CA:X4_`YwldcrtPf U X z X;phadcrf¥ . #. . . !. &E. & E 8' . 8 .  f W2  X. N _ f W = _ AX I&. $& h. I& h 6'W . . N _ f W = _ AX I&. $& h &E . 8.  jf &  W 2  X I& h 6'W. &. &. XGR. . X. & R. h‹. h 8LW XGR. W

(177) :d¥ CA:X. çÄÅÆÃ¶Ç.

(178) ‹˜Ž. 

(179)    !#"$# %&'  () +*, -/.. P €  \. . s X z e0pql†X;v_`egc"U l†egf U W = WaAXG U N WaAX X ? DW^ _ X vimk9›mX z ‘Š_`kd‘fhiPphimY\z Xlne z fo_`egc _ ` € p h p I X  v h f j _ 9 i Q k { Ÿ  X  v m i n k h p _ z = J z U vvXIimk9YwfolDX imkdW X;kmfq=p*XÑ¥igf U i Xphf iŠc`_`adpfo_rpqimlDk†XIpv_`eŠegk c

(180) s lnkdeŠimf f¡i9Ÿ¡kdX„crt X;= pofqI eŠ¥ ždcr_`p emkLempotdYwldfoimfo_`v?Xxdl†eŠknph_ji9k kdXIe = igbeŠcjc e XfOW = WaAXj N WaAX/X*ž†X s X)QDkdX s žmt N WaAX t P e z ‘mY\_rk N W+9n¥ C 9LX  DQm o emk s = WaAX t P e z_ ‘mY\_`k  W = X t P E E W = &4 K = &4 X R W+9n¥ CmŽ;X }kf U. . . . *. . . . . . 5. 5. 6 8;: 5=. DQm o  5 66 8;: 5= 6. . 6. . 4 FL & 8. fjF. . K«

(181) $

(182) ±— W = WaAXj N WaAX/X O*, j "; , #"   " cr_ _`Y I = WaAX P =  -  ¥±]u_rknvX DW+^`_5X

(183) _`pvcji9poX s eŠk s ž†i9ank s X s N WaAXXxd_rphfhp\eŠk s _rp\ankd_uadXŠ¥ TU XIkÆ£4f U XQpof z _`vf v\ imk9X›Š_rXYwxdld_jf crt¨_rXIp0iŠ>f U s X„_ anW z kdX;_ phulWan¥ kdX;php0XŸiŠ _rf = U WaAz XXI¥ pqlDJXIva f0z f fhU i¨X z f Y\U XŒi z vXŠim£¡Yw_jf0lDim_`pPkdX;eŠk9c zfhXIpwe _`s k t±l I z i˜W ›Šz XX;sphlÆ_`¥k _rcjk X;YwY\4 £e¨‹ ‰†¥ ‰³f; U egf W = WaAXG N WaAX _`pž†i9ank s X s eŠk s egcrcW_rfhpcr_rY\_rf}© l†im_rk9fqp!ž†XIcji9kd‘wfoi DW X¥ e XfOW = /=LX P cj_`Y W = Wa 4 XG N Wa 4 X X{§i z poi9Y\XphX uadX;knvX a 4 vimk9›ŠX z ‘m_rkd‘foi =n¥ 4  TU XIk W = =X P 4 cr_rY W = Wa 4 XG N Wa 4 X X{Ÿ U X z  X W = Wa 4 XG N Wa 4 X X P W = Wa 4 Xj N Wa 4 X X h W = /=LX K W = =XGR  ]u_rknvX = I P = I £dŸ{X U eG›ŠX N Wa 4 X P N Wa 4 X eŠk s = I Wa 4 X P = I Wa 4 XG f U XIZk W+9n¥ CmAŽ XK‘m_j›mXIp. .

(184). 1. . . . . $#. . . *. . . . . . . . . . . . . !. !. . *. . . . . fjF. E. 4 FML. E&. 8. W =  &4  Wa 4 X. .  K =  &4  X !. E. fGF. 4 FML. E&. W = &4  W a 4 X. 8.  K =  &4  X R. y emphph_`kd‘bfhi0f U X

(185) cr_rY\_rfŸ{Xi9ždfheŠ_rkQf U egf = P = ¥ z{XIi9pqknadphcrfh_ sp*X iŠz  cjT_ ¢ U X*XI_`i kbz XIf UY X!ZDl ¥jz ‹mX£ ›uN _ji9WaAanX p P poX;vfha_ji9 k  2 h WaA8LX WaAt P X¡ema k e3s f = WW aA= X{Wa _`p!X K ew= poimXcremank fos_rimk3WigaA Xuf t PU Xa lDe3X f z fha N z WaAžDXÑX ¥ s JBl z zi9i9Y ždcrfXIU Y X

(186)  _`k W = &4 K = &4 X  D E E& fjF 4 FML  Sp R Sf R V = 4 PqW 4  ; P ‹Q[RSR[R"\ W ^ _ X 4 I = & PN= &  $ &Jh N Wa X f &K O P Z‹ [R[RSR 0  E  .  . . . . . /. . . 8. *. . = _ ` =. ÅÅØnmIÒ;ÌoÒÒ. = I. 4 P fj= F I Fh L. 2 I. /. a. h 8LW. /. a XGR.   .

(187) ‹5h. - 5"  -   . TU Xvi zqz X;phlDimk s _rkn‘cr_rY\_rf!l z imždcrXIY _rp

(188)  _`k D E E. fGF 4 FML &  Sp R Sf R V = 4 PkW 4  W ^ kX *. . 8.  &4 K =  &4 X . ;. = &4 X P = I& . $& K. . W=. E 4 F L = _ =bjf  F M = I. P. P. O. Y   

(189) Z %&'j- A";"A %. ‹Z[RSR[R"\ ‹QSR[RSR 0  P. = I R. P«

(190) ±—d–  (gW ^*kX = P D* W  X¡eŠk s *DW ^*kX P =   ¥ po_`kd‘ e X;YwY\e0‰†¥j‹m£uf U X z X

(191) Xxu_`phfhp =  WaAX ? (g W ^* _ X¡pqanv U f U eŠf s _`pof W =  WaAXG =  X P  W / a X. . emk s _rklne z fh_rvIadcre z. fjF. TU _rp*i9žm›u_rimanphcrt0_rYwldcr_jX;pf U egf. emk s vimk†poX 9anXIk9focrt i z X uad_r›GeŠcjX;k9focrtŠ£nf U XpoX uadXIknvX K«

(192) $

(193) ±—  = Wa X P = h 2  ,+. . . /. E&. 4 FL. _X P. 6 W ^*.

(194). E. W =  &4 WaAX. 8. E. E&. fjF 4 F L. W =  &4 WaAX. 8. . . Wa X .  K =  &4 X. s _rphf W = Wa Xj = X P W 2 WaAX _  I _`pž†i9ank s X s ¥ . P.  K =  &4 X. P . WaAXG. aAXG /. a. . h 8LW 2  . a X d*nj $. /. e z ‘9Y_` k. P. 8 : 5 ;. fGF. 6. E. E&. 4 FML. 8. W 2 &4 X . W+9n¥ C hLX R. ƒ nl ldcrtu_rkn‘ TU Xi z X;Y ZD¥j‹m£"cr_ _rY I 3WaAX P   eŠk s emk9t„cr_rY\_rf}© l†im_rk9fiŠ 2 WaAX žDXcrimkd‘9p

(195) foi 2 s2 e U U z z §Ei W z  pof i9XY\¥ XXIphfX uadX;ž†knXvimX kdXa 4f X;vpoimXbk9cj›Š_`Y\X z _j‘mfo_r©=kdlD‘iŠ_`fok9iUfhp=†£deŠkf U eŠf žDX\emkiŠf X XcrXIY\XIk9f€iŠ DW  f X¥€¦ X\vIeŠkŸ _jfhXŠ£ 2 P cj_`Y 2 Wa 4 X^emk s 2 P 4

(196) cj_`Y 2 Wa 4 X Ÿ_rf U 2 Wa 4 X P 2 Wa 4 X K 2 h 2 R 4  ~ XŸ z _jfh_rkd‘wf U X s )X QBkd_jfh_ji9kQig = Wa 4 X{Ÿ¡Ximždfqeg_`kW£nph_rknvX 2ZI Wa 4 X P 2 I Wa 4 X£nf U eŠf 1. -  ¥. $#. . *. . . . *. . . . . . . E. fjF 4 F L. . E&. 8. W 2  &4 Wa 4 X/X . y emphph_`kd‘bfhi0f U X

(197) cr_rY\_rff U X z XIpqadcrf U iŠc s pI¥. !. E. fGF 4 MF L. E&. 8. W 2 &4 W a 4 X X . R. çÄÅÆÃ¶Ç.

(198) ‹;. 

(199)    !#"$# %&'  () +*, -/.. €g€. €!. Ræ‹ T 

(200)   £ - + `# )-  # ) -n#%&  +  O"U # )- ! -# £ y _rfhYweŠk ƒ s ›˜emknvX s y anždcr_rp U _`kd‘ y{z iŠ‘ z emY hW ‹I;gh ZLX ¥ R CT 

(201) !#"$ &%(')* +'-,/.0,

(202) 

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(205) 32#54- ,6,7

(206) 89

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(208) 32#4- +,B,

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(210) ¥ R :TJ"63KL",+ † £ "%$  -/# !" -/)

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(212)   +-/j#"g

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(214) SJ

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(216) ,VUWU-+X,YZK£*[ )-#  --)- ) $ 5"n) -j#-D'-/  A-/ j ~ XIlDi z f  iœ†‹©mC Žu£ J emvadcrf tPiŠ  egf U XIYwegfh_rvIp!eŠk s } ku§i z Ywegfh_rvIp£  XIc §f  kd_r›ŠX z ph_jf t. iŠ T XIv U kdiŠcriŠ‘mtŠ£ XIc §fI£ iŠcrcremk s ¥ R h TJ"6)"6]\W ^_,T,£ y egf U §imcjcri˜Ÿ_rkn‘LY\Xf U i s p §i z cj_`kdXIe z l z im‘ z eŠYwY\_rkd‘D£a`*[  cb 5 * ‰ŠZYWh‹IŠC;X£ul†lW¥Æ‹ 9uŽd@ CACmŽd¥ R  TB :;:*%, £  #;'-/ Aj  !  %& + Y U# !) Uj )%Y j + ¥>†¥ ~ X;p¥  egfh_ji9knegc Ba z XIema]ufheŠk s e z s pI£D]9XIvf¥ BZ9 Wo‹I:ACXÑ£dlnlW¥'CA9Љe@&C 9:d¥ R ‹ = T-fgahW')

(217) YKj £ i %&'   G Dk " `

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(219) pn6?%') o&Y&'»F£ [ '

(220) - -  j' E&     -j  j #    -/  -/ j ¥,]  ƒ! y anžncj_`veŠfo_rimknpI£D]  ƒ! £ y{U _rc`e s XIcrl U _re†£W‹IŠŠZ†¥ R 5‹ CT-=E,G') £ - %&-  +  # -  Yj#- '-/  - *   )   ' +jg ` %  + YO  gjA[

(221)  £ ƒ lnldcr_jX s  egf U XIYwegfh_rvI(p q  lnfo_`YI_ HIeŠfo_rimkŒ‹ h Wo‹;A h;hXÑ£dlnlÆ'¥ Cd‹ :G© CACg†¥ R ‹I ‰ T-RPrU

(222) + ,Xst,5£ u   [ A ) )' >

(223)  %Y)@ %&- I k)  5"A) "v A'- ;jg  Z& 5 # j ` ¥ ‚¥ ~ ¥ ƒ ve s ¥B]uv_N¥ y e z _`pI£mfI¥†;‰ C;Cd£D] X  z _rX Wo‹;Š; 9XÑ£nl†lW¥Æ5‹ CŠM‰ @,5‹ C h†¥ . -. . &. . $. '&. $. 1. $. . $. ÅÅØnmIÒ;ÌoÒÒ. $. $. &.

(224) Unité de recherche INRIA Lorraine, Technopôle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LÈS NANCY Unité de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unité de recherche INRIA Rhône-Alpes, 655, avenue de l'Europe, 38330 MONTBONNOT ST MARTIN Unité de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex Unité de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex. Éditeur INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) ISSN 0249-6399.

(225)

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