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On the Stability of a Bandwidth Packing Algorithm

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(1)On the Stability of a Bandwidth Packing Algorithm Jean-François Dantzer, Mostafa Haddani, Philippe Robert. To cite this version: Jean-François Dantzer, Mostafa Haddani, Philippe Robert. On the Stability of a Bandwidth Packing Algorithm. [Research Report] RR-3605, INRIA. 1999. �inria-00073075�. HAL Id: inria-00073075 https://hal.inria.fr/inria-00073075 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. On the stability of a bandwidth packing algorithm Jean-Franc¸ois Dantzer, Mostafa Haddani, and Philippe Robert. N ˚ 3605 ` THEME 2. ISSN 0249-6399. apport de recherche.

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(168) - ( u $<-2W& 2 Fir F= H H  G- ) "&v  - W v \ o†tˆ"ˆ"mrx ~ƒ o}ˆ'o}wklsklo jCQ4«2žƒ_;st ¯ _. tcw_a|}v"_9j"qJ_ƒoœ{kujklskut2m'wsts_9w Fib0 H!G F5Iw -

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(170) -

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(172) FQg b g r q0g F * H g H  ‘#  ml ku`½w¥v" ˆ 4 g b

(173) [ F g* H  @F‘ H œAo  wo}`O_cˆ{o wklskl¬_†qFo}j"ws¥tj}sw @ tj'‰  j"os)m ‰_aˆ'_aj"‰kujp§o2œs ] _ew_9|4v_aj"qFC_ FibP HÚ¦ ‚j›_9mr_a`O_9j}s FF5P " HQ- F5I K HHo2'œ a  wtskuw+­{_9w P "  @ Fày}x›‰_J­{jklsklo <j Hnt j"‰ FJI K Hkuw†t¤ws  kujp´o2œ " « I K G2"OœAo  t2mlVm Z+FLwkuj"qJ_nt j£kls_9`¡‘žƒo vmu‰±­' s HÚ¦ \ žƒoCq9tw_9w€oqJqJv  «‰_aˆ'_aj"‰kujp±o j§s ] _c¬Gtmlv"_oœ  #S¦  t w_  # 0 ¦ \ ] _OqFo}j"‰klsklo j F‘ab H;klw  F‘ # H  -6¦nÁ}kuj"qF_Cs ] _  _Ot  _zt2mlžƒtGxw 0 ˆmutqJ_9w)œAo  s ] _‘ kuj´s ] _zykuj6«{s ] _zj4v"`Cy'_  o2œ;] ‘ekuj´s ] _C wxws _9` FiF  6 Fi= HHƒ kuwy{o v"j'‰_9‰ªy}x¤s ] _zj4v"`Cy'_  oœ qJmrkl\_9j}] s¥w kljª t wst yml_   0 |4v_9v_Cž krs kuj"ˆ"vs t2s_ #Q«6w_ ¬4kuqF_ ts_‘Otj"‰ @klj"krskltmq9v"wso `O_ w9¦ _e­ ws skl`O _c] ž ] _a j£s ] klw 2   ] 0 |4v_a] v_kuwƒ_9`±ˆ sx£kuwƒklj}s_Jp ]  ty"mr_ «4qJo j"w_9|4v_aj}smlxOs ] _†2 wt `O] _nˆ   o ˆ{_  sx ] omu‰"w œAo s _)­ wss] kl`O_  ž _9js ] _cj4v"`zy{_ o2œ,] ‘kljs _nwx] ws_a` klwƒŽ"¦‚;~€s€ skl`O_  «4s _ 2 _ž krmlm -§kls_9`±] w o œ•wkr¢9_ "£2 kljs _†yku>j J'v'wkujpOs _e] w¥t`O_c`C_9s o‰¤tw;] kujs  _†ˆ o}o] œSo2œ o ˆ{] o w krskro} j QF  ˆmutGx}kujpOs _ oml_eoœ 6 HÚ«·krsklw_9t wklmlx¤w_J_ajªs tsžƒ_Ct ww¥v"`O_es t2s)œAo tmrm•s _ bC« s _ _Ot _ -¤krs_9`±wno2œ,wkr¢9_ " kuj¤s ] _ey"klj6¦ y_z 7 T y {.

(174)  

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(180) F* H ‘  mrku`  g b[g G - y}x¤s ] _zmltGž o2œmut  p_ej4] v"`zy{_  w9¦ » _aqJtv' w_†oœs ] _zwst ykrmlkl sxoœs ] _   ]  0 `C_aj}sklo  j_a‰ªty{o>¬_«{s ] _ _J°"ˆ'_aqFs_a‰ ¬>t2muv_´o2œ* s _´so stm;j4v"`Cy'_ o2œkls_a`Ow´‘¤kuw F

(181) g b  g H׫€t j"‰ s _ªj}v'`zy{_ o2œkls_a`±w "Ÿklw y{o v'j"‰_9‰´y}x 8%#+ 3Ž - /Ì«'qFo}j"w_a|4v_9j}smlx m lkl`   4 A

(182) g b Fr  g F * HH G  F‘ - '# H  ‘ '  t w_  # ^ 0 ¦ \ ] _±j4v"`zy{_  oœ,krs_9`± w "¤klj›s ] _±ykujªœAo  s ] _±kl jklskut2m]wsts_ b  G FJP -jIP H&N a  klw ] ‰"_9jo s_9‰©y}x - z¦±‚;snskl`O_§Ž¤mr_awwns tj @ - 3"ikrs_9`±w±‘Ot _OkujNs _§ykljŸt j"‰ I*kuw†tws kujpoœ kls_a`±w  " ¦ D̜ #  @ - 4  "†s ] _9jCž krs ] kujOt jzkuj}s_9p ] tyml_€t `Co}v"j}so2œ{s kl`O_ t2mlms ] _ƒkls_a`±w;‘ƒ] ž klmlm ] ] tG¬_ƒmr_JœAss ] _ ] y" klj6« _9j'qF3_ -Okls_9] `±w "zž krml m'oq9qJv"] ˆ}xeklsJ¦¨ kls o}vs,mlo \ w¥w] o2œ· p_9j"_ t2mlkr sxežƒ_)tw¥w¥v"`O_ƒs t2ss _ kuj] _9|4v"tmrklsx  # ^ @ $-  " omu‰"wƒœAo tmrmQs _†kujkrskltmSwst2s_9w9¦ _ _9mltsklo j # ^ @ -  "kl`±ˆmlkl_9w;s t2s)t>œAs_ t£­{jkls_±sku`O_«6s ] _±krs_9`±wO‘Cž klmrm] w¥t2s¥v   ts_Cs ] _±ˆmutqF_awn mr_JœAsny}xªs ] _Ckls_a`O] w "ªt j"‰it qJqFo  ‰"kljpso¤s ] 2 _ qJo j"‰"krskro}j F‘Gb H,krs;ku`±ˆmrkl_9w tmlwoOs ts F‘ #H4 -  ¦ o}j"w_9|4v_9j}smrx§s _†ykljkuwœîv"mrmQv"ˆso±skl`O_ ž ] _9j¤s ] _†kujklskut2mQws  klj"pOoœ•kls_a`±w " ] t w y'_9_9j´w_  ¬ _9‰6«"qJmr_at  mrx m lku`  4 g b

(183)  F 2 g H G " - ‘  X ‘`Oo  _9o>¬_  4

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(186) ‘Gb . 

(187) 4-2

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(199) - ( u $<-2W& 2 Fir F= H H  0-2

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(201) #*( 04, - I  .  0   ‘   kx  x   Fi= H!G klj4œ  $ %I" F= H!G ‘   r Fi= HdG FF5I"Fi= HH

(202) -WFJPQ"2Fi= HHH A

(203)  +!  &u    " kuj4œf  G \ 5 v   ,I-[ 8u

(204) - . "

(205) S8 *   G + kx ,I -x    .  8 

(206) +  .     ‘  S

(207)   

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(209) # 

(210) +.  G

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(212) #  G kuj4œ  =   F= H G \5 '  [u

(213) #- "

(214) ' O * 

(215)  Fi= H '  # C-2 $2&   u     [u

(216) - "

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(221) : 

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(223) ‘aŒ.  

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(225) 

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(232) Fi= H ^ Ž

(233) .   (  -.  - *)  A  .G F576WH E S 

(234) #   =d&  . . -. ". )'- #E 

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(236). Fi= H G @ F‘#!#H$-  4 F576 H = # ' (  - ^ @ F‘ # H  4 F 6 H P  - ,)  .L " *-  (  -  & ( Qx & "      O

(237) #-),Q  u

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(239)   *

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(241)  .  

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(244) G - 4 F 6 H # @ F‘#!#H ' l   - W v \ ] _ˆ  o4o2œ klweœît2k  mlxUwku`±ˆml_ ? twzmro}jpNtwzs ] _­  wsCkrs_9` ‘ ] t wzjo s ] klszs ] _ ] _at‰Uoœ)s ] _ |4v_av_«4klsw ]  tj ¯ ‰_aq]  _9t w_aw€y}xª‘nž ] _9j¤tj krs_9` "±ml_9 tG¬ _9] w€s ] _†ykuj6] «kL¦ _ ¦ts  ts_ -  ¦•¨ ] _9] j¤tj£kls] _a` ‘cml_9tG¬ _9w;s _eykljQ«'s _†krs_9` ‘nmloqJts_9‰´ts F= Hƒ_aj}s_ w;s _zykuj>J"s _ekঠkL¦ ‰6¦€tw¥wv"`±ˆskro}j´w o>ž)w;s t2s s ] _ j_J°skrs_9` ‘ ] klj±s ]  _|4v_9v" _)klw,mroqJt\ s_a] ‰Ots   Fi= H \  ž ] _ *  _  klw ]  t j"‰] o}` ¬>t  klt ymr_  p _Jo}`C_9s  klq9t2mlmrx ‰"klws kly'vs_a‰ž kls ˆ"t t `O_Js_ #‘ !#S¦ _†ˆ oqJ_9w¥Pw F Fi=

(245) H H t w;s _ew¥t`O_†‰kuws kuy"vskro}jt w.

(246) R   

(247) U Ž ->= *

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