• Aucun résultat trouvé

Many TCP User Asymptotic Analysis of the AIMD Model

N/A
N/A
Protected

Academic year: 2021

Partager "Many TCP User Asymptotic Analysis of the AIMD Model"

Copied!
18
0
0

Texte intégral

(1)

HAL Id: inria-00072390

https://hal.inria.fr/inria-00072390

Submitted on 24 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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.

Many TCP User Asymptotic Analysis of the AIMD Model

Dohy Hong, Dmitri Lebedev

To cite this version:

Dohy Hong, Dmitri Lebedev. Many TCP User Asymptotic Analysis of the AIMD Model. [Research Report] RR-4229, INRIA. 2001. �inria-00072390�

(2)

ISSN 0249-6399 ISRN INRIA/RR--4229--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 1

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Many TCP User Asymptotic Analysis of the AIMD Model

Dohy Hong & Dmitri Lebedev

N° 4229

July, 2001

(3)
(4)

Unité de recherche INRIA Rocquencourt

Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

3$4/57698:4/;=<+>3$?A@CBEDE@GFIHKJLHKM=HKNO

PQRTSVUVWYX[Z]\_^)Ua`cb/deUTf]^&g/^&f)RTSVU_^

h2i)jcklUTfYPmZ]npo

ZY`cq7q=jri&fs7U'i)U_t"QUTi)t)QU#uvLwyxzxz{|X[}zb7~g€(xz‚z‚/WX WTwVq(`cƒU_^

„†…L‡_ˆa‰cŠ7‹zˆzŒŽ ^&Sq7~‘U’b7‘seSVj“s7UT~=”•jrif&QU–kljru“fmf&Q7i)jrb7ƒzQ7q7b7f]UT—jr~b7f&‘jruAjc”2`˜^)UTfmjc”GP]o–h^"U_^)^&‘jru^

^&Q(`ci&u7ƒ†`˜t_jrSSVjruA™šjrf&f&~‘UTuU_t"›œi)jrb7f)UTi]Q(`r^m™šU_UTu|u“f&i)j“s/bt_U_sAu9žxcŸ ™(`r^)U_s|jru¡q7i)j¢s/btTf)^jc”i"`cus7jrS

S£`cf&i&‘t_U_^a¤œPQ7‘^q(`cq=UTi:^&f&bs/‘U_^:f&QU†`r^&g¢Sq7f)jrf&‘t_^:jc”mf&QU£`cSVs¥SVj“s7UT~I¦mQUTu§f&QUuyb7S™šUTi$jc”P]o–h

^)U_^)^)‘jru^'¨©ƒj¢U_^ªf)j|u/«u7f¬g¤V­9U^&Qj®¦f&Q(`cff&Q7‘^'~Sfq7i)j¢t_U_^)^ta`cu¥™šUs7UE«uU_s¯™“g9`œi)U_tTb7i&i)UTut_U

i)UT~°`cf&‘jru¯u²±`cus+¦–U^&Qja¦Qj®¦³f)jœt"Q(`ci"`rtTf)UTi&´aUf)^ª^&f"`cf&‘jru(`ci&gµ™šUTQ(`a—“‘jriª—UTi&g+^)Sq7~gA™(`r^)U_s²jru

f&QUei)U_tTb7i&i)UTut_U|i)UT~°`cf&‘jru ¤¶­9U|^&f&bs/g§f&QU¡`rt_tTb7i"`rtTg¶jc”ªf&Q7‘^˜~SfSVj“s7UT~]`cus·f&QUeSq(`rtTfVjc”f&QU

^&g¢ut)Q7i)jru7´®`cf&‘jruµUE¸=U_tTfujrfYjru7~gœjru¡f&QUªSVUa`cuAf&Q7i)jrb7ƒzQ7q7b7f]™7b7fY`c~‘^)jjruAf)^Y¹jyºš¤

»|¼y½=¾)¿À ‰cÁG‡zŒ P]o–hpÂcÃÄhL€“`rs7s/f&—UmutTi)Ua`r^)US:b7~f&q7~‘ta`cf&—UYs7U_tTi)Ua`r^)Uz€¢¹jyºš€z^)g“ut"Q7i)jru7´®`cf&‘jru €¢`r^&g¢S˜Å

f)jrf&‘t:`cu(`c~g/^&‘^a€/i)U_tTb7i&i)UTut_U:i)UT~°`cf&‘jru

Æ¥ÇÉÈLÊLÇÉËÍÌCÎ(ÈρÐYÑǬÐÒ_Ó·ÔÖÕc×§ØyÙÚG۞ܪІÝÞÓEßÞàÞáâ7ã)ÔÖäæå¯çš×&ØK×ÄèéáTÓaМêKÔëã"ìcíl×Eî Ḭ̂Üã"äæÛ†ïðÑñEò_ó_îôGñÞìcõ_öG×Äìcålî÷øÔ&Ð

ÑÜmäúù•ÔÖäîû“×lüz×&Ø®×lýKöG×lìcålî÷þÔ

(5)

!L]$2! š )$

‡ Œ u SVj¢s7RT~‘Uµ’b7‘s7U²^&Sq7~‘U²q=jrb7iœ~ú\T—jr~b7f&‘jru·k&jruyf)U¯s7U_^|s7\T™7f)^|sb7uéUTu^)UTS™7~‘U9s7U

^)jrb7i)t_U_^P]o–h q(`ci&f"`cƒUa`cu“fmb7uœi)jrb7f)UTb7it_jrSSb7uA`:\Tf)\u“f&i)j“s/b7fs`cu^ªžxcŸ=™(`r^)\^&b7ims7U_^pq7i)j¢s/b7f)^ms7U

S£`cf&i&‘t_U_^'`c~‘\a`cf)jri)U_^a¤šo–UTf'`ci&f&‘tT~‘U:\Tf&bs/‘U~‘U_^Y`r^)g“Sq7f)jrf&“bU_^ªs/bASVj¢s7RT~‘U  Ã~‘jri)^“bU'~‘UujrS™7i)U

s7U$^)jrb7i)t_U_^ªP]o–hé¨f)UTus+—UTi)^~u/«u7Ö¤jrb^]SVjru“f&i)jru^ybU$t_Uq7i)j“t_U_^)^)b^]~Sf)U$q=UTb7fTf&i)U$s7\E«u7

q(`cib7uUi)UT~°`cf&‘jru¶s7Ui)\_tTb7i&i)UTut_Uœs`cu^± UTfujrb^ta`ci"`rtTf)\Ti&‘^)jru^$^)jru9t_jrSqšjri&f)UTSVUTu“f$^)f"`cf&‘jru7u(`ci)U

f&i)R_^'^&Sq7~‘UTSVUTu“f£q(`ci&f&is7U˜t_UTf&f)U$i)\_tTb7i&i)UTut_Uz¤ jrb^ª\Tf&bs/‘jru^'~°`†q7i)\_tT‘^&‘jru9s7U˜t_USVj¢s7RT~‘U$~Sf)U

UTfm~Sq(`rtTfYs7U~°`˜^&g“ut"Q7i)jru7‘^"`cf&‘jruAujru+^)UTb7~‘UTSVUTuyf^&b7im~‘U's7\T™7fSVjagUTu+S£`c‘^]`cb^)^& ^)b7i~°`V¹jyºš¤

!.À ˆa‡ ¾ ‹#" ‡¯Œ P]o–hpÂcÃÄhG€Í`rt_tTi)jr‘^)^)UTSVUTu“f)^:`rs7s/f&”^:UTfªs7\_tTi)jr‘^"^"`cut_U˜S:b7~f&q7~‘ta`cf&—Uz€¹jyºš€ ^&g“ut"Q7i)jcÅ

u7‘^"`cf&‘jru €(`cu(`c~g/^)U:`r^)g“Sq7f)jrf&“bUz€7i)UT~°`cf&‘jruAs7U'i)\_tTb7i&i)UTut_Uz¤

(6)

8 -/Y(='1

! š )

ìu žxcŸë€2`+^&Sq7~‘U£SVj¢s7UT~p” jri$f&QUk&jruyf˜UT—jr~b7f&‘jru jc”YP]o–ht_jruyf&i)jr~~‘U_s ¨:9Pmh³^)U_^)^)‘jru^^&Q(`ci&u7ƒ¯`

^&u7ƒz~‘Um™=jrf&f&~‘UTuU_t)›£i)jrb7f)UTi ¦–`r^ ƒz—UTu£™(`r^)U_s†jruVq7i)j¢s/btTf)^ jc” i"`cus7jrS S£`cf&i&‘t_U_^a¤2ìuVf&Q7‘^ SVj“s7UT~<;•`r^Iu

žxKŸ=¦pU¦m~~šta`c~~=f–f&QU.0=1Å  s7s/f&—UìutTi)Ua`r^)Uz€ +b7~f&q7~‘ta`cf&—U U_tTi)Ua`r^"UÅSVj“s7UT~?>E€/P]o–h·‘^pujrf

i)UTq7i)U_^)UTu“f)U_s¥`cf'q(`rt)›UTf~‘UT—UT~ր ™7b7f'—“°`œ^&Sq7~‘U$’b7‘s¯U “b(`cf&‘jru^ªf&Q(`cfs7U_^)tTi&™=U˜f&QUkljru“fUT—jr~b7f&‘jru^

jc” f&QUY¦mus7ja¦^–u†f&QUYt_jru7ƒU_^&f&‘jru¡`_—jr‘s`cut_Uq7Q(`r^)Uz¤2PQU`r^)^&b7Sq7f&‘jru^jc” f&QU  à SVj¢s7UT~¦–UTi)U

SVjrf&—z`cf)U_s ™yg·f&QU|^&S£`c~~f&SVUA^)ta`c~‘U¡`cu(`c~g¢^)‘^£`cus u q(`ci&f&‘tTb7~°`ci£™“g·f&QUœ”•`rtTfVf&Q(`cf†UT—UTu u f&Q7‘^

^&Sq7~«(U_s|SVj¢s7UT~ ” i"`rtTf"`c~ ™=UTQ(`_—¢‘jri]jc”2`cƒzƒzi)UTƒy`cf)U_sAf&i"`A@£t'¦–`r^`c~i)Ua`rs/geq7i)U_^)UTuyfa¤

PQUq7i)U_^"UTuyfq(`cq=UTi:^&f&bs/‘U_^:f&QUV`r^&g“Sq7f)jrf&‘t  à SVj¢s7UT~I¦mQUTu9f&QU˜u“b7S:™=UTijc”m^)U_^)^&‘jru^¨

ƒj¢U_^mf)j˜u/«u7fëg¤pPQU'S£`cu+`rs/—z`cuyf"`cƒU:jc”2^&bt"Q+`˜t_jru^&‘s7UTi"`cf&‘jruA‘^f&Q(`cff&Q7‘^m~Sfq7i)j¢t_U_^)^m‘^m—UTi&g

Ua`r^&~gœs7UE«uU_s†™“gœ`$jruUªs/SVUTu^&‘jru(`c~ i)U_tTb7i&i)UTut_U'i)UT~°`cf&‘jru €7UT—UTu|¦mQUTuef&QU~‘jz^)^–q7i)jr™(`c™7~fëge‘^–i"`cf)U

s7UTq=UTus7UTuyfa¤PQ7‘^UTu(`c™7~‘U_^b^Yf)j†^)f&bs/g|u+q(`ci&f&‘tTb7~°`ciªQja¦.f&QU$^&f"`cf&‘jru(`ci&g¡~°`_¦jc” f&QU:¦mus7ja¦³s7UEÅ

q=UTus7^jru¡f&QU'^&g¢ut)Q7i)jru7´®`cf&‘jruµi"`cf)Uz¤

ìuCBÞx/€š¦–U:i)U_ta`c~~™7i&‘UE’g¡f&QU  ÃSVj“s7UT~Ö¤ªÃ¬uCB7/€¦pU$q7i)U_^)UTu“fª~Sff&QU_jri)UTSV^¦mQUTu²f&QU~‘jz^)^

q7i)jr™(`c™7~fëg|‘^i"`cf)U'us7UTqšUTus7UTu“f`cus|uDB)w£¦pUt_jru^&‘s7UTi]f&QU'i"`cf)Us7UTq=UTus7UTuyf]ta`r^)Uz¤

E , .-¢0 '1$2%

PQU  à SVj¢s7UT~'‘^¡`§’b7‘s `cq7q7i)j_d/S£`cf&‘jrujc”$¦mus7ja¦ UT—jr~b7f&‘jru^Au t_jru7ƒU_^&f&‘jru`a—jr‘s`cut_U

q7Q(`r^)UA”jriœ¨F9Pmh ^"U_^)^&‘jru^a¤é­9U+i)U_ta`c~~f&Q(`cfœ^&~‘j®¦^)f"`ci&fe`cus f&SVU_jrb7fe`ci)U+ujrfœf"`c›UTu `rt_t_jrb7u“f

u¶f&Q7‘^˜SVj“s7UT~Ö¤HGja¦–UT—UTia€If&SVU_jrb7f˜‘^$us/i)U_tTf&~g§q7i)U_^)UTu“f˜™yg¥f&QU£”Ö`rtTf$f&Q(`cf˜f&QU†f&Q7i)jrb7ƒzQ7q7b7fVta`cu

™=Ue`ci&™7f&i"`c~g tT~‘jz^)Uœf)j9‚I; tE”&¤ žxKŸ,>E¤KJU_^&‘s7U_^MLNONð‘^V`r^"^&b7SVU_s·f)jµ™šU|t_jru^&f"`cu“fVu·f&SVU¡`cus·f&QU

^"`cSVU¡”jri†`c~~^)U_^)^&‘jru^;ÉQjrSVjrƒUTuUTfëgP>E¤ Ãë”f&QUA¦mus7ja¦ ^&´aU¡jc”^)U_^)^)‘jruRQ˜`cf†f&SVUTS˜‘^VUXW,YZ([,S]\E€

f)^u^&f"`cuyf"`cuU_jrb^:f&Q7i)jrb7ƒzQ7q7b7fjriª~‘j“ta`c~f&Q7i)jrb7ƒzQ7q7b7f`cfªf&SVU^S‘^`cq7q7i)j_d/S£`cf)U_s²™yg+f&QU yb(`cu“f&fëg

_`W,YZa[,Sb\dcXUeW,YZ([,S]\bf&LNON¤

PQUhg=ÅÖf&Qi"]kj6! #C! ,l‘^s7UE«uU_sµ`r^]f&QU(g=ÅÖf&QµUTqšj¢t)Q²`cf]¦mQ7‘t)Q²jruU~‘jz^"^Yjri^)UT—UTi"`c~L^&Sb7~Å

f"`cuU_jrb^~‘jz^"^)U_^mS£`_gej¢t_tTb7i]jru¡f&QU'^&Q(`ci)U_s|i)jrb7f)UTia¤I­9U'b^)Uªf&QUª”•jr~~‘ja¦mu7ƒ†ujrf"`cf&‘jrum

n ¨ ‘^mf&QU'u“b7S:™=UTiYjc”q(`ci"`c~~‘UT~9Pmh^)U_^"^&‘jru^a€/¦mQ7‘t)QA¦pU`r^"^&b7SVUªf)j˜™šU't_jru^)f"`cuyf]uAf&SVUo

nIp

[¨D\qcsrT¨ ‘^mf&QU:ta`cq(`rtTfëg|jc”Gf&QU™šjrf&f&~‘UTuU_t"›œi)jrb7f)UTi5o

n

LN/NscsLt;t_jru^&f"`cu“fu¡f&SVU:`cus¡U “b(`c~=”•jri]`c~~Í^)U_^)^&‘jru^>o

n

Nvuœ‘^f&QUwg=ÅÖf&Q¡t_jru7ƒU_^&f&‘jruAf&SVUo

nix uy{z

c|N uy{zd}

N u

‘^f&QU'f&SVU'™=UTfë¦pU_UTu+f&QUg=ÅÖf&QA`cus|f&QUl[,g~s€\ëÅë^)f]t_jru7ƒU_^&f&‘jru¡f&SVU_^5o

n _ W,YZ

u c‚_ƒW,YZh[?N

y

u

\–‘^mf&QUªf&Q7i)jrb7ƒzQ7q7b7fjc”^"U_^)^&‘jru2Q klb^&f]`K” f)UTimf&QUwg=ÅÖf&Q¡t_jru7ƒU_^&f&‘jru¡f&SVUo

nI„

Zu ce…

Z

W‡†

z _ W?YZ

u o

niˆ

W,YZ

u c‚_ W,YZ [?N(‰

u

\p‘^mf&QUf&Q7i)jrb7ƒzQ7q7b7fYjc”2^)U_^)^&‘jru2Q klb^&fm™=UE” jri)Uªf&QU(g=ÅÖf&Q¡t_jru7ƒU_^&f&‘jruAf&SVUo

n

Š"+6bP Œ‹5! #Šb6b6Ž ?$ ,!mq Wuv‘

’6“

‘^w”6€f•k–5€&—_Å֗z`c~bU_s²i"`cus7jrS —z`ci&°`c™7~‘Uz¤–ìf‘^YU yb(`c~ f)jƒ€f•

‘^ªf&QU^"U_^)^&‘jruQ]UEd¢q=UTi&‘UTut_U_^`†~‘jz^)^`cfNvu¤:ìf)^~°`_¦³‘^`r^)^)b7SVU_s+ƒz—UTu¥`cus¯s7UTq=UTus9`†q7i&‘jri&

jru

’

¦mQ7‘t"Q¡‘^mf&QU'f&Q7i)jrb7ƒzQ7q7b7fjc”Gf&QU^"U_^)^&‘jru2Q klb^&fm™=UE” jri)UNvu¤

­9Uœ^)UTf™˜š[

’

\(ct›œ[?ŠWu

‘’6“

c€f•ž\E¤µÃë” _ W,YZ

u

‘^$^&f"`cf&‘jru(`ci&g€f&QUœ^&f"`cf&‘jru(`ci&g¯q7i)jr™(`c™7~fëg§f&Q(`cf

W

u ‘Ÿ“

c €f•Vs7UE«uU_^f&QU^Š"+6bP Œ‹5! #*ž¡f&Q(`cf¦–Us7UTujrf)U'™yg=›¢ ‘˜ “¤

(7)

GUTi)UV¦–Uœ`c~~‘j®¦ f&Q(`cf

x u c 7¤¡PQ7‘^Sq7~‘U_^$f&Q(`cf¦–U£S£`ag¯Q(`_—UV_

W,YZ

uy{z

c _

W,YZ

u €G™7b7f:¦mQ(`cf$‘^

uyf)UTi)U_^)f&u7ƒ:‘^f&QUq7i)jrq=UTi&fëg˜f&Q(`cfIƒz—UTu

’

€ Wuv‘

’6“

s7j¢U_^2ujrfIs7UTq=UTusjruMQ"¤Ã֔š¦pUmSq=jz^)U

x

u7€zƒz—UTu

’

€kŠWu

‘’ “

s7j¢U_^mujrf]s7UTq=UTus¡jru2Q t_jrus/f&‘jru(`c~LjruAf&QU”•`rtTff&Q(`cfmf&QUTi)U‘^]`cf~‘Ua`r^&fjruU'~‘jz^"^a¤  ~~‘j®¦mu7ƒ

x

u™c^&Sq7~«(U_^mSVjrSVUTuyf)^]ta`c~‘tTb7~°`cf&‘jruµ`cuseQ(`r^uj˜Sq(`rtTfjru|f&QU`r^&g¢Sq7f)jrf&‘t`cu(`c~g¢^)‘^5mus7U_U_s €

¦pU¦m~~Í^)U_Uªf&Q(`cfu|f&QU'~Sf]SVj“s7UT~

x

u˜¦mf&Q¡q7i)jr™(`c™7~f¬g9Wz¤

PQUTu €7¦–U'Q(`_—U

_

W?YZ

uy{z c|

W

u ‘ˆ W,YZ

u

“

ˆ

W,YZ

u

;lW>

¦mf&Q

ˆ

W,YZ

u

c‚_ W?YZ

u

~Kr } „ Z

u

¨

9jri^)Sq7~‘tTfëg€ ¦–UV`r^)^&b7SVU$f&Q(`cfªf&QUu7f&°`c~Is/‘^)f&i&™7b7f&‘jru9jc” _ W,YZ

‘

-–]r

“

‘^ƒz—UTu¥`cusµf"`c›UTu

us7UTq=UTus7UTuyf&~gµ”jri`c~~šQ"€ ^)bt)Q¯f&Q(`cfw›¢ ‘_ W,YZ

“

s7j“U_^ujrfs7UTq=UTus¯jruI[,Q0–)¨D\E¤  ^Y¦–U˜¦m~~I^)U_U$f&Q7‘^ª‘^

ujrfi)U_^&f&i&‘tTf&—U:^&ut_Uª”•jri]`c~~ ¨§€ _ W,YZ

u

‘^UTi&ƒj¢s/‘tz¤

-/$1œ21$2Y %)(2

! "$#&%'#)(* +%-,!.0/1*32$465)4'*7#8.053(:9;4<#)(:%=

GUTi)Uª¦pU:`r^)^)b7SVU'f&Q(`cf ŠW

uv‘’6“

‘^us7UTqšUTus7UTu“fYjc”

’

`cus|f&Q(`cf?> [?ŠW

u c €f•ž\qcA@2c‚›¢ ‘˜ “ ¤

BDC ¼yÀ ‰ ¼ FEHGJI (+k8&^,+-q)$,$8Kš , jDL z™Š4NMOMO$ ,^ !P

QSRUT

ZWVYX

„ Z

u

¨

c‚›¢ ‘_ W?YX

uF“

–

K<+-]_

W,YX

u

'hH3M‡ZMO$% ,™ , *) _

W,YZ

u M

PQUmq7i)j¢jc” jc”=f&QUY`c™šj®—Umf&QU_jri)UTS ‘^2ƒz—UTu£uVf&QUmuUEd/fI^)U_tTf&‘jruVu†`ªSVjri)UƒUTuUTi"`c~(t_jruyf)UEd/fa¤2PQU

`c™=ja—U'f&QU_jri)UTS Sq7~‘U_^f&Q(`cf QSRST u

VYX QSRST

ZWVYX\[3]

^

Z

c [*€ } @ f•ž\ rc¤

BDC ¼yÀ ‰ ¼ `_'G?a<b

dcWe a [gf#z5–

ŸŸŸ

–hfji\

dc i e

ݢlk

i

m

n † z [,_

WSoYZ

u

\qp or

} i

m

n † z ›¢ ‘ [,_

WSoYZ

u

\qp o “

cs [*€fE¨H\– ;•xž>

K<+-]Qbz–

ŸŸŸ

–bQqi ]1t 4 ubŽO$‡Ž Ž,a*)

c M

b5*)OMYPQU'q7i)j“jc”‘^m™“g†us/btTf&‘jruµjru¡f&QU's7UTƒzi)U_UDfc … in † z fn

¤d9jrivfc •/€¦pU'Q(`a—Um

›¢ ‘_ z YZ

uy{z _xw

YZ

uy{z

“ c ›¢ ‘ z uy{z

 wuy{z “zy

›¢ ‘_ z YZ

u _ w YZ

u “

~ir

w

~iݢ|{

[„ Z

u \ w

¨

w }

} •&›¢ ‘_ w YZ

u „ Z

u

¨

“U~

c [,›¢ ‘ “\ w y r w ~ €

¨

[,›¢ ‘_ z YZ

u _ w YZ

u “ } ›¢ ‘ [,_

z YZ

u \ w “ \ ~

c [,›¢ ‘ [,_

z YZ

u \ “\ w

~sl[¨ ‰ z \

(8)

ja¦é”•jri]`$ƒz—UTu [gf#z5–

ŸŸŸ

–hfiw\m

c‚›¢ k i

m

n † z [,_

n YZ

uy{z

\ p oZr

c ¢ k i

m

n † z y [,_

n YZ

u

~Kr } „ Z

u

¨ \ po ~ r

c ¢ i

m

n † z po

y fnn ~ [,_

n YZ

u } „ Z

u

¨ \ o r po ‰

o –

¦mQUTi)U c

in † z ›¢ ‘

[?

nz \ po “

¤ƒ9jri˜f&QUœq7i)j¢s/btTf$jc”]f&QUœSVjrSVUTu“f)^a€2f&QUœUEd¢q7i)U_^)^)‘jru^:”•jr~~‘ja¦ u¶f&QU

^"`cSVUª¦`_g

c i

m

n † z ›¢¡[,_

n YZ

u \ p

o

c i

m

n † z ›¢ k y _ n YZ

u

~Kr } „

Zu

¨ ~ po r

c i

m

n † z po

o0†

y n

fn ~ ›¢ ‘ [,_

n YZ

u } „ Z

u

¨ \ o “r po ‰ o

ja¦ jruU:ta`cu¡Ua`r^&~geq7i)ja—U'f&QU'i)UT~°`cf&‘jrum

›¢ ‘ [,_

z YZ

u

\

ZZ [,_

n YZ

u

\

y „ Z

u

¨

~O“

c‚›¢ ‘ [,_

z YZ

u

\

ZZ [,_

n YZ

u

\k_

n

y{z

YZ

u

ZZ

_ n y YZ

u “

~s [¨ ‰ z

”•jri'`c~~!

W

–"H–$#

c

¤|º“ut_Uf&QUt_j¢U@£tT‘UTuyf)^:jc”pf&QUUEd/q(`cu^&‘jru¯jc”

`cus% `ci)U`c~~IU yb(`c~ր

}

ƒz—U_^mf)UTi&SV^Yjc”Gf&QUª”•jri&STm

ݢ'&

i)(

m

n † z [,_

n YZ

u \u o+*

}

i)(

m

n † z ›¢-,6[,_

n YZ

u \ u o.

–

¦mf&Q

b0/21|b

`cusg

n3

fn ”4#

3 b € … n g

n53

fl¤9(jrif&QUf)UTi&SV^ª^&bt)Q²f&Q(`cfh…

n g n56

fl€¦pU$b^)U

us/btTf&‘jruµ`r^"^&b7Sq7f&‘jru¡f)j™=jrb7usef™yg sl[*€fE¨D\E¤IPQUf)UTi&S ^&bt)QAf&Q(`cf … n g n c f2`cus b /Pc b ‘^

U “b(`c~Lf)j7 [ } \m¦mf&Q8

6

€z€(QUTut_U$f&Q7‘^Yf)UTi&S ta`cuµ™=USVja—U_sµf)j†f&QU:~‘UE” fQ(`cus+^&‘s7Uz¤9jrif&QU

f)UTi&SV^]^)bt)Q¡f&Q(`cf

… n g n

cf2`cus b /6 b €7¦pUi)Ua`cq7q7~gœf&QU`c™šj®—U:UEd/q(`cu^&‘jru¡SVUTf&Qj¢s ¤ ìuA`$«u7f)U

uyb7S™šUTijc”^)f)UTq^a€/¦pUª«usAjru|f&QU'i&ƒzQyfYQ(`cu^^&‘s7U'jru7~gef)UTi&SV^jc”f&QUª”jri&S

ݢ & p

m

n † z [,_

n YZ

u \

u9(

o * } p

m

n † z ›¢-, [,_

n YZ

u \

u9(

o .

¦mf&Q…

n g /n 6

fĤ :;

< À ‰ À ""Ɋ7‰ ½ EHG .?)4=C4>= / ]wgKq@?BA b44 'DCŽ ,{FEP5Ž,wb)/”5_

W?YZ

u —

WΠ

zGHG

Z e

,+-,+-l](&/JI

! #"]$,$%l ?Š4 ¡ Š4 M

< À ‰ À ""Ɋ7‰ ½ _'G +-Ž! #PŠM$ ?™ !vŠb5"0h ?("Ž+kbž"¡ ‡‹Ab4`3P

_ X

uy{z

c| uy{z [,_

X

u ~

@Šr

• \ ;#7ž>

aKLE jA$Œ!$`3P

_ X

X c

@r

• X

u † z u

m

WΠ

z  ‰ W

(9)

^&u7ƒœf&QU:i)U_tTb7i&i)UTut_U$i)UT~°`cf&‘jruR;lW>]`cusK;#7ž>T€šjruµta`cu²Ua`r^&~g¡ƒUTf'`c~~GSVjrSVUTuyf)^'jc”d_JX

Z

`cusƒ_JX

X

™ygeus/btTf&‘jru ¤dGUTi)U:`ci)U'f&QU(7˜«i)^&fmSVjrSVUTuyf)^5m

›¢ ‘_ X

X “ c ›¢ ‘_

W,YZ

X “

c [*€ } @• \ r

›¢ ‘ [,_ W?YX

X \ w “ c š[*€ } @

\Ž[*€ } @ \ r w

›¢ ‘ [,_ W,YZ

X \

c ›¢ ‘

[,_ W,YX

u \

[*€ ~ € } @ ¨ \ ‰ z

›¢ ‘ [,_

W?YX

X

\

“ c r

[ } @v\Ž[#• } @v\Ž[ } @v\

›¢ ‘ [,_

W,YZ

X

\

“ c r

[ } @v\Ž[#• } @ \Ž[

} @v\ ~

Z

[*€ } @ \Ž[

} @v\Ž[ } @M~x@

w \

•-€ ~

Z

[*€ } @v\ ~

Z

[ } @v\Ž[*€ } @v\ ~ w

Z

[*€ } • @v\Ž[*€ } @v\

w

z#&4<#)(:%=$465& ,$( #85N(!#)(:%=|" /:% #&

9Gƒzb7i)UeW^&Qj®¦^f&QUi)UT~°`cf&—U˜s/¸ UTi)UTut_U$™šUTf¬¦pU_UTuD_

W,YX

X

`cusƒ_

W,YZ

X

¦mf&Qµ¨ c€Z-–5€Z&œ”•jriYf&QU˜xcus

`cus7ci)s£SVjrSVUTuyf)^–b^&u7ƒf&QU`cu(`c~g¢f&‘tm”•jri&S:b7~°`r^–ƒz—UTuœ`c™=ja—Uª`cus£^)S:b7~°`cf&‘jruœi)U_^&b7~f)^a¤­9U]^"U_Umf&Q(`cf

”•jri f&QUªxcusœ`cus=7ci)s£SVjrSVUTu“f)^a€¢f&QU]~Sf¨c ƒz—U_^ b7q7q=UTip™=jrb7usœU_^&f&S£`cf&‘jruœ”jri`c~~—z`c~bU_^pjc”

@L¤pPQU:s/¸ UTi)UTut_U:s7U_tTi)Ua`r^)U_^`c~SVjz^&f]q7i)jrq=jri&f&‘jru(`c~~g|f)jT€

}

@²`cus €fE¨ `cus @`cVƒz—U_^Y`˜™=jrb7us

jru9f&QUi)UT~°`cf&—U†s/¸=UTi)UTut_Um”jri'f&QU†xcus9SVjrSVUTuyfa€Lf&QU™=jrb7us¯‘^

z

Z

~-[*€fE¨H\E€2`cus²”•jri'f&QUl7ci)s

SVjrSVUTuyfa€(f&QU'™=jrb7us|‘^

zGz w

Z

~ [*€fE¨H\E¤

PQUpt_jrSq(`ci&‘^)jru˜¦mf&Q^&Sb7~°`cf&‘jru$^&Qj®¦^Gf&Q(`cf €Z2f)UTi"`cf&‘jru^Gƒz—U–` “b7f)UIƒj¢j“s:`cq7q7i)jad¢S£`cf&‘jru ¤

PQ7‘^$ƒz—U_^`cu§‘s7Ua`+jc”f&QU£u“b7S:™=UTiVjc”f)UTi"`cf&‘jru^˜f&Q(`cf‘^$uU_U_s7U_s§f)jAQ(`a—Uœi)UT~‘UT—r`cu“f$i)U_^)b7~f)^:”i)jrS

^&Sb7~°`cf&‘jru^]u|f&QU'i"`cf)U:s7UTqšUTus7UTu“fY^&g¢ut)Q7i)jru7´®`cf&‘jruµta`r^)Uz¤

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

0 0.2 0.4 0.6 0.8 1

x=p

Rel. error, N=10 M2 Rel. error, N=100 M2 Rel. error, N=1000 M2 Rel. error, N=10 M3 Rel. error, N=100 M3 Rel. error, N=1000 M3

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x=p

Rel. error(theory), M2 Rel. error(theory), M3 Rel. error (simulation 1M), M2 Rel. error (simulation 1M), M1 Rel. error (simulation 10M), M2 Rel. error (simulation 10M), M3

9Gƒzb7i)UeWm]PQU_jri)UTf&‘ta`c~2i)UT~°`cf&—UUTi&i)jriªq7~‘jrf;É~‘UE”Éf> o–jrSq(`ci&‘^)jru¯¦mf&Q¯^&Sb7~°`cf&‘jru ;Éi&ƒzQyfa€d€Z ˜`cus

€Zf)UTi"`cf&‘jru^>

9Gƒzb7i)U'x:^)Qja¦^f&QUUTSq7i&‘ta`c~ s/‘^&f&i&™7b7f&‘jruC;n (> q7~‘jrf`cusœf&QUªtTb7S:b7~°`cf&—U's/‘^&f&i&™7b7f&‘jrue” b7utEÅ

f&‘jru2;Öo 9d> q7~‘jrfGjc”7f&QU–t_jruyf&uybjrb^f&SVU–^&f"`cf&‘jru(`ci&g^&f"`cf)Upjc”f&QU f&Q7i)jrb7ƒzQ7q7b7fGf&Q(`cfG¦pU`cq7q7i)j_d/S£`cf)U

™ygM€Z f)UTi"`cf&‘jru^Ljc” ;#7ž>¦mf&Qhr/c €Z]`cus'ƒzi"`cu“b7~°`ci&fëgjru"!2c ‚/¤ž‚/W”•jri @ ”

€–h

–h

k–h

–h

$#

–5€&—“¤

ìfLi)jrb7ƒzQ7~g:^)Qja¦^f&Q7i)U_Upfëg¢q7‘ta`c~/^)Q(`cqšU_^jc”(f&QU–n ”b7utTf&‘jru^2s7UTq=UTus/u7ƒªjru @{m @ 3 /€&

3 @ 6 €

`cus @=c €z¤

(10)

0 0.0002 0.0004 0.0006 0.0008 0.001

0 5 10 15 20 25

x granularity 0.01

p=0.1 p=0.3 p=0.5 p=0.7 p=0.9 p=1.0

0 0.2 0.4 0.6 0.8 1

0 2 4 6 8 10 12 14 16 18 20

x granularity 0.01

p=0.1 p=0.3 p=0.5 p=0.7 p=0.9 p=1.0

9Gƒzb7i)U:x-m2n `cus+o 9§q7~‘jrf

9Gƒzb7i)U(7˜^&Qja¦^f&QU:¹jyºœtTb7i&—U'q7~‘jrf]`r^`”Éb7utTf&‘jru+jc”0@L¤2PQU's7UE«u7f&‘jruAjc” ¹jyº7#‘^ms7UE«uU_sA`r^

f&QUS£`Kd¢Sb7S u^&f"`cuyf"`cuU_jrb^ªf&Q7i)jrb7ƒzQ7q7b7fl;C”•jri]f&QU˜t_jruyf&u“bjrb^'f&SVUq7i)j“t_U_^)^>mf&Q(`cfªjruU:ƒUTf)^

jc”f&SVUm

¹jyº7 c ” !8> [,_K[,Sb\ !v\ 1 <fk€Z&k—

;tE”&¤–7rŸ(”•jri SVjri)Us7UTf"`c~‘^pjruœf&Q7‘^ps7UE«u7f&‘jruŠ>E¤ Ãëfp‘^p^&b7i&q7i&‘^)u7ƒf&Q(`cfp¦mQUTu†f&QU^&g¢ut)Q7i)jru7´®`cf&‘jruei"`cf)U

utTi)Ua`r^)U_^a€/f&QU:¹jyº tTb7i&—U_^”•jri–Q7ƒzQ“fm—r`c~bU'jc” 9‘^utTi)Ua`r^&u7ƒ£¦mQUTi)Ua`r^f&QUªSVUa`cu|f&Q7i)jrb7ƒzQ7q7b7f

‘^s7U_tTi)Ua`r^&u7ƒ¤IPQ7‘^mta`cue™=UYq(`ci&f&°`c~~gœUEd¢q7~°`cuU_se™“g£f&QUY”Ö`rtTf–f&Q(`cff&QU—z`ci&°`cut_U'jc”Íf&QUf&Q7i)jrb7ƒzQ7q7b7f

‘^'`£s7U_tTi)Ua`r^)u7ƒe”Éb7utTf&‘jru¯jc” @9`cusµf&Q7‘^YQ(`r^`£S£`Kk&jriSq(`rtTf'jruµQ7ƒzQUTi¹jyºš¤Ã¬u²^&bt)Q9`œt_jruyf)UEd/fa€

f&QU:^)g“ut"Q7i)jru7´®`cf&‘jruµQ(`r^ª`q=jz^&f&—UUE¸=U_tTfªjruAf&QU:Q7ƒzQ¯¹jyºYuU:ta`cu²`c~‘^)jVujrf)Uf&Q(`cfYf&QU:i)UTƒz‘jru

$‘^]`c~SVjz^)fu^)UTu^&f&—U'f)j˜f&QU:^&g“ut"Q7i)jru7´®`cf&‘jru+UE¸ U_tTfa¤

2 4 6 8 10 12 14 16 18

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x=synchronization rate

QoS 10%

QoS 20%

QoS 30%

QoS 40%

QoS 50%

QoS 60%

QoS 70%

QoS 80%

QoS 90%

QoS 95%

mean

3 4 5 6 7 8 9 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x=synchronization rate

QoS 50%

QoS 60%

QoS 70%

QoS 80%

QoS 90%

QoS 95%

mean

9Gƒzb7i)Uh7-m ¹jyºœq7~‘jrf

ìuV9Gƒzb7i)Uw7/€¢¦pU'`c~‘^)j˜^)U_UYf&Q(`cf–” jrim`c~~=—r`c~bU_^jc”0@G€yf&QUSVUa`cuef&Q7i)jrb7ƒzQ7q7b7fm‘^–™šUTf¬¦pU_UTu+¹jyº£wy‚

`cus9¹jyº8z‚œ¤­9U`c~i)Ua`rs/gµ›“uj®¦.f&Q(`cf'f'ta`cu7ujrf'™=U$™šUTf&f)UTiªf&Q(`cu²f&QUs7UTf)UTi&Su7‘^&f&‘tta`r^)U @Hc €

¦mQ7‘t)Qet_jri&i)U_^&q=jrus7^ f)j˜¹jyºz‚œ¤<9jriz@

6 €¢f&QU]SVUa`cu£f&Q7i)jrb7ƒzQ7q7b7f‘^Iƒzb(`ci"`cu“f)U_U_s†wy‚ jc” f&SVUz¤

(11)

0 †2$12Y %)(2

! ! "$#&%'#)(* +%-,!.0/1*32$465)4'*7#8.053(:9;4<#)(:%=

ìu¶f&Q7‘^^"U_tTf&‘jru €2¦pUeƒz—Uei)U_^&b7~f)^˜™(`r^"U_s¶jru tT~°`r^)^)‘ta`c~m`cu(`c~g/^&‘^$”•jri µ`ci&›j®—¥q7i)j¢t_U_^)^)U_^jru·ƒUTuUTi"`c~

^&f"`cf)UY^&q(`rt_UmGf&QU]™(`r^&‘t]ujrf&‘jru^jru£i&i)U_s/btT™7~fëg|`cuslGª`ci&i&‘^pi)U_tTb7i&i)UTut_Uta`cu†™šU”jrb7us†u;tE”&¤ Ÿ,>E¤

BDC ¼yÀ ‰ ¼ G Zu

c [,_

z YZ

u –

ŸŸŸ

–b_

ZdYZ

u \ ?hMD8Šb5"0ZM

.?) R

˜š[

’

\šc ˜ Š4

˜š[

’

\šc ˜ 6 € e

,+- a ¨

dcWe

Z c ” Zu —5u

?q & !!

( 'h4l,+-Žb,)]^ !d ?(#jž54 !" M

b5*)OMYPQU ²`ci&›ja—q7i)jrq=UTi&fëg:‘^GtT~‘Ua`cia¤

Z¶‘^(Å֏i&i)U_s/btT™7~‘U(;tE”&¤  Ÿë€cq ¤ >E€r¦mQUTi)U‘^͔jriLu^&f"`cut_U

f&QU UT™=U_^&ƒzbU˜SVUa`r^&b7i)Ujru²±

y

¤$PÍjeq7i)ja—U˜f&Q7‘^®€ fª‘^'^&b-@£tT‘UTuyff)j|^)Qja¦³f&Q(`cfª”i)jrS `cu“gµ^)f"`cf)Ua_¶€

Zu

t_jrSVU_^ ™(`rt)›˜f)j:`cu“g$ƒz—UTu†jrqšUTuVu“f)UTi&—r`c~t_jruyf"`cu7u7ƒ2[!rf•k–

ŸŸŸ

–]r5f•ž\¦mf&Q£qšjz^&f&—Uq7i)jr™(`c™7~fëgvm

”I`c~~ ^)jrb7i)t_U_^]UEd/qšUTi&‘UTut_U`$~‘jz^)^]`cf N u €/f&QUTu

„

Zu fE¨csr5f•/€(`cus

_ W,YZ

uy{z } r5f•

c €

• _ W,YZ

u } r5f•

PQ7‘^UT—UTuyf'j“t_tTb7i)^Y¦mf&QAq7i)jr™(`c™7~f¬gA~°`ci&ƒUTif&Q(`cuD˜

Z

`cusµta`cuA™=Ui)UTq=Ua`cf)U_sµ`cu“g|«u7f)Uu“b7S:™=UTiªjc”

f&SVU_^¦mf&Q|q=jz^&f&—U'q7i)jr™(`c™7~f¬g¤

9Gu(`c~~g€

Z

‘^qšjz^&f&—UwGª`ci&i&‘^™“ge`cq7q7~‘ta`cf&‘jruµjc”<9(jz^&f)UTi ú^]tTi&f)UTi&‘jruAjru

Zu

;tE”l¤] Ÿë€/q ¤žx xž>E¤

:;

BDC ¼yÀ

‰ ¼ G

.?) R

˜š[

’

\ac ˜ e

˜š[

’

\ac ˜ 6 € Š4 )^˜X ?V"]! , E E I

46b"bb , j e ,+-<P

QSRUT

ZWVYX

„

Zu

¨

cuP– , LOz^Š4=3M‡ZM

K<+-]u™c|›¢ ‘_ X

u “ c QSRST

ZWVYX

›¢ ‘_ W,YZ

u “ M

a g

dc e _ W,YZ

u _

W,YX

u 3M‡ZMŠ4V,+6 ?$ ,^ !v*5"0(! ?? (,+-h]b" Ek]Š"M]$‡! !<P

_

W,YX

uy{z c|

W

uy{z‘

_

W?YX

u

~ir } u “

[,_

W,YX

u

~Kr } u \

”5_

W,YX

u —

W!

bl :M‡ :M?43MŠ4

X c ”5_ X

u

—5u

"

?^Ž! #Š#jž54 #"

e

+-"0 ,,™Ž! !ŠT$&KX ?

"+k*ž"5¡ Œ‹]4`,+ w]$‡! !

_ X

uy{z

c‚ uy{z

‘_ X

u

~$#

“

[,_

X

u

~$#

K<+-]#Mcsr } ›¢ ‘ _JX

X “ M

b5*)OM]­9U¡«i)^&fe^&Qj®¦~Sf)^ejc”(_

W?YZ

u `cus

[ ]

^

Z

™yg us/btTf&‘jru ¤ 9jriVg c _ W,YZ

c _

W?YX

™yg

t_jru^&f&i&btTf&‘jru§`cus²™“g²`r^"^&b7Sq7f&‘jruK”5_

W,YZ

—

W‡†

z Y GHG

YZ

‘^ªÖ¤Ö¤ús ¤€ QUTut_U

[ ]

%

Z ›¢ ‘_

W,YX

“

`/¤ú^a¤™ygµ~°`_¦jc”

~°`ci&ƒU'u“b7S:™=UTi)^a¤  ^)^&b7SVUªf&Q(`cfO_

W,YZ

u _

W?YX

u `cus'&

]

(*) ,+- (./

^ ‰ - (0.

]

^1+

Z £`/¤ú^a¤­9U'Q(`a—U

_

W?YZ

uy{z c|

W

uy{z ‘ˆ W?YZ

u “ [ˆ W,YZ

u \

º“ut_U

ˆ

W,YZ

u _

W,YX

u

~er } u `/¤ú^®¤ ¦–U¡Q(`a—U`_

W?YZ

uy{z

_

W,YX

uy{z

`/¤ú^a¤ ;Éf&QU+`/¤ú^®¤ t_jru“—UTi&ƒUTut_UA”•jri

ŠW

uy{z ‘ˆ W,YZ

u “

‘^s/bUªf)jf&QUt_jru“f&uyb7f¬g¡jc”š˜>E¤

9jrif&QU`/¤ú^a¤~Sf]jc”

[ ]

^

Z

€7¦–U'b^)Uªf&QUªuU “b(`c~fëgvm

_ W,YX

uy{z } _

W,YZ

uy{z

3  W

uy{z ‘ˆ W,YX

u “32

ˆ

W,YX

u } ˆ

W,YZ

u 4 ~  W

uy{z ‘ˆ W?YX

u “ }  W

uy{z ‘ˆ W,YZ

u “ ˆ W,YZ

u

3 ˆ

W,YX

u } ˆ

W,YZ

u ~  W

uy{z ‘ˆ W?YX

u “ }  W

uy{z ‘ˆ W,YZ

u “ ˆ W,YZ

u

Références

Documents relatifs

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des