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A distributed algorithm for weighted max-min fairness in MPLS networks

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‡‹& !&‡Œ‰‚# ‡‰ ‰‡ … Ž& !&‡Œ‰‚# &‚ˆ‰‚…!-&! ‘"’ *† !„ “’," ”•–— Œ‰*Š(

Š #&‡‰ ‰‡ … Ž& ‡‹&*‡ Š Ž‡ ‰! ‰ˆ ‡†$ ŠŠ&!&‡Œ‰‚#‚&†‰*‚-&†&‡Œ&&! ŠŠ‡‹&

,"’†‹&-‹‰ -&‰ˆ !&‡Œ‰‚#!(Œ (‡‹ †‹‚ !„‰Š -+…‰!„-‰…&‡ !„ ,"’†

†  #&+ &‚ˆ‰‚…!-& ††*&

˜‰!† (&‚ †&‡ ‰ˆ ,"’†™ &-‹ -‚‚+ !„ …!+˜’ -‰!!&-‡ ‰!†™-‚&‡ !„ -‰!š

„&†‡ ‰! › ‡‹‰*‡ &œŠ - ‡ ‰Š - !„™…‰‚& „„‚&†† $& ,"’† Œ ‡‹ …‰‚& ž‰Œ†Ÿ„&‡

…‰‚& ‡‹‡ ‡‹& ‚ ˆ ‚ †‹‚&™ !(&&!(&!‡Š+ ‰ˆ ‡‹& ‚ ‚&†&‚$‡ ‰!†

‹&‚‰Š&… ‰ˆˆ ‚†‹‚ !„‰ˆ‡‹&!&‡Œ‰‚#!(Œ (‡‹‹†&&!Œ (&Š+‡‚&‡&(

!‡‹&†‡!(&†&- ŠŠ+ !‡‹&“ -‰!‡&œ‡ ” —”¡—Ÿ‹&-Š†† -Š…œš… !‚‡&

ŠŠ‰-‡ ‰! ‹† &&! Œ (&Š+ --&‡&( † ! ‰‡ …Š !&‡Œ‰‚# !(Œ (‡‹ †‹‚ !„

-‚ ‡&‚ ‰! …‰!„ *†&‚ ‡‚ƒ- ž‰Œ† ”• — ‘!ˆ‰‚…ŠŠ+™ ‡‹& …œš… ! ‰Š -+ ‡‡&…‡†

‡‰ …œ … Ž&‡‹&!&‡Œ‰‚#*†&ŠŠ‰-‡&(‡‰ ‡‹&*†&‚†Œ ‡‹ ‡‹&… ! …*… ŠŠ‰-‡ ‰!

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‰ ‚‰$ (& ( £&‚&!‡ Š †&‚$ -& ‰‡ ‰!†™ ”¤— ‚‰‰†&†  „&!&‚ - Œ& „‹‡ ‚‰š

‰‚‡ ‰!Š !&‡Œ‰‚# !(Œ (‡‹ †‹‚ !„ ‰Š -+™ Š†‰ -ŠŠ&( ›& „‹‡š’‚‰‰‚‡ ‰!Š

“œš“ ! ›’““Ÿ ‹& ›’““ ‰Š -+ „&!&‚Š Ž&† ‡‹& -Š†† -Š …œš… ! +

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›& ˆ‰-*† ‰! ! &(„&š‡‰š&(„&‚‡&š†&( ˆ&&(-# -‰!‡‚‰Š †-‹&…&™Œ‹&‚& †&- Š

-‰!‡‚‰Š -#&‡† ‚&*†&( ! ‰‡‹ ˆ‰‚Œ‚( !( -#Œ‚( ‡‹†

‰*™ ! ”¤—™ ‚‰$ (&†  „‰‰( ( †‡‚ *‡&( †‰Š*‡ ‰! ˆ‰‚ -‰…*‡ !„ ‡‹& ›’““

‰Š -+ *‡ ‹† ‡Œ‰ …‰‚ (‚Œ-#†  ‚†‡Š+™ ‡‹ † †‰Š*‡ ‰! † …Š&…&!‡&( !

‡‹& “ -‰!‡&œ‡ !(*†&†  Š‰‡ ‰ˆ &†‰*‚-& “!„&…&!‡ “Ÿ “ -&ŠŠ†‡‹‡

-‚&‡&  †*†‡!‡ Š ‰$&‚‹&(!( †&-‰!(Š+™‡‹& -‰!$&‚„&!-& † *†*ŠŠ+ †Š‰Œ †

ŠŠ*†‡‚‡&( Š‡&‚

*‚ ‚‰‰†Š †‰Š$&† ‡‹&†& ‡Œ‰ Š … ‡‡ ‰!† + (( !„ ! &œŠ - ‡ ‰‡‡Š&!&-#

Š !# !ˆ‰‚…‡ ‰! ! &-‹ -‰!‡‚‰Š -#&‡ !( + *† !„  ( £&‚&!‡ -‰!‡‚‰Š -#&‡

*(‡&‚-‹ ‡&-‡*‚&‹&†&‡Œ‰(( ‡ ‰!†(( ‡ ‰! -‹!„&‚( -ŠŠ+‡‹&(+!… -†

‰ˆ ‡‹& ‚‰‡‰-‰Š !( …‚‰$& ‡‹& -‰!$&‚„&!-& ‡ …& +  ˆ-‡‰‚  ‰‚  (&&!( !„

‰! ‡‹& ‚&- † ‰! †&&† …*Š‡ ‰! ‚&†*Š‡†Ÿ

!‰‡‹&‚ -‰!‡‚ *‡ ‰! † ‡‹& (‡‡ ‰! ‰ˆ ‡‹& †‰Š*‡ ‰! ‡‰ ‡‹& -‡*Š “’,"

‚-‹ ‡&-‡*‚& *† !„ ‡‹& Œ (&†‚&( "’š ‚‰‡‰-‰Š ”— !†‡&( ‰ˆ “ -&ŠŠ†

› ‡‹ ‰*‚ ‚‰‰†&( !‡&„‚‡ ‰! ! "’™ Œ& …‚‰$&‡‹& †-Š Š ‡+ ‰ˆ ‡‹& ‚‰š

‡‰-‰Š + (&-‚&† !„ ‡‹&‰$&‚‹&( ((&( + -‰!‡‚‰Š -#&‡†

†  ‰†† Š& Š -‡ ‰! ‰ˆ ‡‹ † Œ‰‚#™‡‹& Œ& „‹‡&( …œš… ! ˆ ‚ ‚‡& ŠŠ‰š

-‡&( ‡‰ ! ,"’ -‰*Š( & *†&( +  …‚#&‚ ‡ ‡‹& !„‚&†† ‚‰*‡&‚ ‡‰ …‚# ‡‹&

‡‚ƒ-*† !„ ‡‹‚&&-‰Š‰*‚† „‚&&! *!(&‚‚&†&‚$&(‚‡&Ÿ™+&ŠŠ‰Œ &‡Œ&&!‚&†&‚$&(

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‚‰*‡&‚†( †-‚(‡‹& ‚&( -#&‡† ‚†‡!(‰†† Š+™(*‚ !„ ‡‚!† &!‡&‚ ‰(†™†‰…&

‰ˆ ‡‹& +&ŠŠ‰Œ -#&‡† + *† !„  › ‰Š -+ ˆ‰‚ &œ…Š& ‹& ‰&-‡ $& ‰ˆ

‡‹& Š„‰‚ ‡‹… † ‡‰ -‰…*‡& ‡‹& ˆ ‚ ‚‡& ‰ˆ &-‹ ,"’ ‡‰ ‰‡ !  !&‡Œ‰‚# !

Œ‹ -‹ ‰!Š+ ‡‹& ‚&( -#&‡† ‚& ( †-‚(&( !( ŠŠ ‡‹& „‚&&! !( +&ŠŠ‰Œ -#&‡†

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 ™‡‹&(& ! ‡ ‰! ‰ˆ‡‹& Œ& „‹‡&( …œš… ! ˆ ‚$&-‡‰‚ Œ ‡‹ ‚&†&-‡‡‰ †&‡

 † (& !&( † ˆ‰ŠŠ‰Œ†   ! " #$ % &'()*+ , -. /0 12 ' / 34 )'5 67896 /:; 7 / + * : 0 ')  1/; 7 : 5 * :<= /; >? , -@ A B C >DE A FGHIIIHJK B @L M .L N > DO A FGHIIIHJK B @P QP R .P QP S .L QL T 24 '+' QU /0 ) 4 ' 2 ' / 34 ) 7 00 *( / 7)'5 2/ ) 4 ) 4 ' ' < '6' : )V

›& -!*†&‡‹ †‡‹&‰‚&‡ -Š (& ! ‡ ‰! ‡‰ †‹‚&‡‹&ˆ‚&& !(Œ ‡‹ $ ŠŠ& !

! “’," !&‡Œ‰‚# *‚ Œ‰‚# † †&( ‰!  ‚&$ ‰*† Œ‰‚# ! ‡‹& “ -‰!‡&œ‡

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Š !#† X  †&‡ ‰ˆ ,"’† Y ‡‚$&‚†&†‰!& ‰‚ …‰‚& Š !#† ! X !( &-‹ ,"’ E

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Œ‹&‡‹&‚ ‰‚ !‰‡ ‡‰ --&‡  !&Œ ,"’ ‰!  Š !#

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‘! ‡‹& „&!&‚ - Œ& „‹‡š‚‰‰‚‡ ‰!Š …œš… ! ›’““Ÿ ‰Š -+™ Œ& ††‰- ‡&

&-‹ ,"’ E A Y Œ ‡‹  Œ& „‹‡ ‰‚ ‚ ‰‚ ‡+Ÿ QL  ‘!ˆ‰‚…ŠŠ+™ ‡‹ † ‰Š -+ ‚†‡

ŠŠ‰-‡&† ‡‰ &-‹ ,"’ ‡†  ‹&! ˆ‚‰… ‡‹& ‚&… ! !„ !&‡Œ‰‚# -- ‡+™ ‡

ŠŠ‰-‡&†(( ‡ ‰!Š!(Œ (‡‹ ˆ‰‚ &-‹ ,"’ *† !„  ‚‰‰‚‡ ‰!Š$&‚† ‰! ‰ˆ‡‹&

…œš… ! ‰Š -+ †&( ‰! &-‹ ,"’ † Œ& „‹‡Œ‹ Š& †‡ †ˆ+ !„ ‡† “ -‰!†‡‚ !‡

‹& !Š !(Œ (‡‹ ˆ‰‚&-‹ ,"’ † ‡† Š*†! (( ‡ ‰!Š & Œ& „‹‡&(&…œš

… ! ˆ ‚ †‹‚&  ‰‚…ŠŠ+™ ‡‹ † ‰Š -+ † (& !&( † ˆ‰ŠŠ‰Œ† !( ( ‚&-‡Š+ (&‚ $&†

ˆ‚‰… &ˆ •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

‹& ˆ‰ŠŠ‰Œ !„ ‚‰‰† ‡ ‰! Š !#† ‡‹& ‚&Š‡ ‰!†‹  &‡Œ&&! ‡‹& ‰$&›’““

‰Š -+!( ‡‹&›’““š‰‡‡Š&!&-# Š !# (& ! ‡ ‰!†

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‹&-&!‡‚Š Ž&( ›‡&‚  ŠŠ !„ Š„‰‚ ‡‹… -‰…*‡&†‡‹& ˆ ‚‚‡&ˆ‰‚&-‹,"’

--‰‚( !„ ‡‰ ‡‹ † ‰Š -+ ”¢— ‹ † -&!‡‚Š Ž&( Š„‰‚ ‡‹… ˆ‰‚ ‡‹& ›’““ ‚‡& Šš

Š‰-‡ ‰! ‚&¥* ‚&†„Š‰Š !ˆ‰‚…‡ ‰!Œ‹ -‹ †!‰‡†-ŠŠ&‡‰ ž‰‰(‘‡ † !‡&!(&(

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‘! ‡‹ † †&-‡ ‰!™ Œ& ‚‰‰†& ! Š„‰‚ ‡‹… ‡‹‡ -‰!$&‚„&† ‡‰ ‡‹& ›’““ ‰Š -+

¥* -#Š+ ‡‹‚‰*„‹ ( †‡‚ *‡&( !( †+!-‹‚‰!‰*† ‡&‚‡ ‰!†

'$# ./ 3 .1"4 !

*‚( †‡‚ *‡&(†‰Š*‡ ‰!*†&†‡‹&"’ † „!ŠŠ !„‚‰‡‰-‰Š‡‰-‰!$&+ !ˆ‰‚…‡ ‰!

‡‹‚‰*„‹ ‡‹& !&‡Œ‰‚#

‹& ’ !( " -#&‡† -‰!‡ ! ‡‹&ˆ‰ŠŠ‰Œ !„ ‚…&‡&‚† 

 &†&‚$&( ‡& Ÿ ‚‰$ (&( ‡ ‡‹& -‚&‡ ‰! ‰ˆ ‡‹& ,"’

› ›& „‹‡ Ÿ ‚‰$ (&( ‡ ‡‹& -‚&‡ ‰! ‰ˆ ‡‹& ,"’

 œ&-‡&( ‚ ‡&Ÿ ‡‹&ˆ ‚‚‡&‡‹‡‡‹& !&‡Œ‰‚#ŠŠ‰Œ†ˆ‰‚‡‹ †,"’





‰‡‡Š&



&-#Ÿ  ( ‰ˆ‡‹& ,"’ † ‰‡‡Š&!&-# Š !#

’&‚ ‰( -ŠŠ+™‡‹& !„‚&†† †&!(† ’ -#&‡-‹ ‚‰*‡&‚ ! ‡‹&‡‹ -‰…š

*‡&†Š‰-Šˆ ‚†‹‚&ˆ‰‚‡‹&,"’ !(*(‡&†‡‹& !(



&Š(† ˆ ‡†Š‰-Š

ˆ ‚‚‡& † Š&†† ‡‹! ‡‹&ˆ ‚ ‚‡& ‚&†&!‡ ! ‡‹&’ -#&‡‰! ‚&-& $ !„ 

’ -#&‡™‡‹&&„‚&†† ‚‰*‡&‚ †&!(†  " -#&‡™ ! ‡ Š Ž&( Œ ‡‹ ‡‹& (‡

‰‡ !&( ! ‡‹& ’ ™‡‰ ‡‹& !„‚&††‚‰*‡&‚

‘! ‡‹& -#Œ‚( ‡‹™ &-‹ ‚‰*‡&‚ *(‡&† ‡† !ˆ‰‚…‡ ‰! Œ ‡‹ ‡‹& "

‚…&‡&‚† ™



Ÿ ‰! ‚&-& $ !„  " -#&‡™ ‡‹& !„‚&†† ‰‡ !† ‡‹&

!ˆ‰‚…‡ ‰! ‰*‡ ‡† ŠŠ‰Œ&( ˆ ‚ †‹‚&

“!+ ‚ ‰‚ &£‰‚‡† ! “ !&‡Œ‰‚#† ‹$& &&! (‰!& ‰! ‡‹& (&† „! ‰ˆ 



Š„‰‚ ‡‹…† ‡‰ -‹ &$& ‡‹& -Š†† -Š …œš… ! ”  — ”¡—Ÿ ‹& Œ‰‚# ‰ˆ ˜‹‚!+ ”—

Œ† ‰!& ‰ˆ ‡‹& ˆ&Œ Š„‰‚ ‡‹…† ‡‹‡ Œ&‚& ‚‰$&! ‡‰ -‰!$&‚„& ‡‰ ‡‹& …œš… !

! ‡‹& “ -‰!‡&œ‡ ‰* &œ‡&!(† ˜‹‚!+ †‡&-‹! ¥*& ‡‰ †*‰‚‡‡‹&… ! …*…

‚‡&™ &# ‚‡& !( Œ& „‹‡ ˆ‰‚ &-‹ ž‰Œ *‚ ‚‰‰† ‡ ‰! *†&† ‰* † Œ‰‚# *‡

…‚‰$&† ‡‹& &‚ˆ‰‚…!-& + …‰( ˆ+ !„ ‡‹& *(‡& …&-‹! †…‘! ‰* † Š„‰š

‚ ‡‹…™‡‹&‚‰*‡&‚†*(‡&‡‹& ‚ !ˆ‰‚…‡ ‰! !‡‹&ˆ‰‚Œ‚(‡‹"‰™‡‹&+-!!‰‡

‹$& ‡‹& !ˆ‰‚…‡ ‰! -‰…*‡&( + ‡‹& (‰Œ!†‡‚&… ‚‰*‡&‚†‹&†& ‚‰*‡&‚† ‹$&

‡‹& -‰‚‚&-‡ !ˆ‰‚…‡ ‰! ! ‡‹&!&œ‡ -+-Š&‘! ‰*‚‚‰-‹™Œ&*(‡&‡‹&‚‰*‡&‚

!‡‹&-#Œ‚(‡‹™†‰ ‡‹&+‹$&‡‹&-‰‚‚&-‡ !ˆ‰‚…‡ ‰!Š #&‡‹& !„‚&††!‰(&

“‰‚&‰$&‚™Œ&((  !&Œ ‚…&‡&‚ 



Ÿ‡‹‡-‰!$&+†&œŠ - ‡Š+ ‡‹& ‰‡‡Š&!&-#

Š !# ! ‡‹& ‡‹‹ † !ˆ‰‚…‡ ‰! …‚‰$&†-‰!† (&‚Š+ ‡‹& -‰!$&‚„&!-&‡ …&

‹&ˆ‰ŠŠ‰Œ !„ ‚&‡‹&Š !# ‚…&‡&‚†!( $‚ Š&†*†&( + ‡‹& Š„‰‚ ‡‹…

X  "&‡ ‰ˆ Š !#† Y "&‡ ‰ˆ ,"’† ‡‚$&‚† !„ Š !#   U 

ˆ ‚ ‚‡& $Š*&‰ˆ‡‹&,"’ V

A

Y †#!‰Œ! ‡ Š !# 



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(5)

J   ‰‡‡Š&!&-# Š !# ( ‰ˆ ‡‹& ,"’ V A Y † #!‰Œ! ‡ Š !#  ‚&ˆ&‚ ‡‰ & ! ‡ ‰! ¢ Ÿ 

  "&‡ ‰ˆ,"’† ‰‡‡Š&!&-#&( ‡ Š !# ™ &

   F V  V A Y $ J U    K 

 "&‡‰ˆ,"’†!‰‡‰‡‡Š&!&-#&(‡Š !#™ &

   F V  V A Y $ J U    K ›& ‹$&      Y

‹& ˆ ‚ ‚‡&‰ˆ  ,"’ V † -‰…‰†&( ‰ˆ‡‹&‚&†&‚$&( ‚‡& ‰ˆ ‡‹& ,"’  U

Ÿ

!( ! (( ‡ ‰!Šˆ ‚†‹‚&ŠŠ‰-‡&(+‡‹&!&‡Œ‰‚#‹ †(( ‡ ‰!Šˆ ‚†‹‚&

† ‚‰‰‚‡ ‰!Š ‡‰ ‡‹& Œ& „‹‡ ‰ˆ ‡‹& ,"’ 

!  ‚‡ -*Š‚ Š !#™ Œ& -! -‰…*‡&

 $Š*&  ‡‹‡ „ $&† ‡‹& (( ‡ ‰!Š ˆ ‚ †‹‚& &‚ *! ‡ ‰ˆ Œ& „‹‡ ˆ‰‚ ‡‹& ,"’†

‰‡‡Š&!&-#&( ‰! ‡‹ †Š !#Š„‰‚ ‡‹… -‰…*‡&†   1"4 !  .13 1.! " "-  ( - Y   B    , U U , U > U  , U B                    C   D  !    "#$$ U  D   CDD  !   %&'( )  *  + •Ÿ

‹&† --†&  &

 



 Ÿ‰--*‚†Œ‹&! †‰…&,"’†‚& ‰‡‡Š&!&-#&(‰! Š !#

™Œ& -‰…*‡&‡‹&ˆ‚&& !(Œ (‡‹ ‰! Š !#   Ÿ+ ‡# !„ ‡‹& -- ‡+‰ˆ‡‹&

Š !# … !*† ‡‹& ‰ˆ ŠŠ ,"’† ‡‚$&‚† !„ ™ … !*† ‡‹& (( ‡ ‰!Š ˆ ‚ †‹‚& ‰ˆ

‡‹& ,"’† !‰‡ ‰‡‡Š&!&-#&( ‰!   † &¥*Š ‡‰ ‡‹& ˆ‚&& !(Œ (‡‹ ( $ (&( +

‡‹& †*… ‰ˆ‡‹& ,"’ † Œ& „‹‡† ˆ‰‚ &-‹ ,"’ ‰‡‡Š&!&-#&( ‰! ™ &  † ‡‹&ˆ‚&&

!(Œ (‡‹ Œ&Œ ŠŠ ŠŠ‰-‡& ‡‰ &-‹ V

A



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‹& †&-‰!( -†&  &



  Ÿ ‰--*‚† Œ‹&! ŠŠ ,"’† ‚& ‰‡‡Š&!&-#&( ‡ !š

‰‡‹&‚ Š !# ‡‹!  ‘! ‡‹ † -†&™ ‡‹& $Š*& ‰ˆ  † -‹‰†&! † ! ” ¤— !( ”— ‡‰

-‹ &$&-‰!$&‚„&!-& ‹&ˆ ‚ ‚‡&  U  Ÿ‰ˆ ,"’ V ‰‡‡Š&!&-#&( ‰! Š !#  † -‰…*‡&( +  >? V A  B  U   U "  , -U

 #&+ &Š&…&!‡ ‰ˆ ‡‹& Š„‰‚ ‡‹… † ‡‹& †‡‚‡&„+ ‡‰ †&‡ ‡‹&

J U



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‹ †(& ! ‡ ‰!(&‚ $&†( ‚&-‡Š+ˆ‚‰… & ! ‡ ‰! ¢!(…&!†‡‹‡ŠŠ,"’†!‰‡

‰‡‡Š&!&-#&( ‡  Š !#…*†‡ ‹$&  ‰‡‡Š&!&-# &Š†&Œ‹&‚&‰‚‚&-‹‡‹& ‚ …œ …Š

0

(6)

-*‚‚&!‡Š !#‘ˆ‡‹‡Œ&‚&!‰‡‡‹& -†&™†‰…&,"’† !



 Œ‰*Š( ‹$&‡‰ &…‰$&(

‡‰



  &Œ‰*Š( & ‰‡‡Š&!&-#&( ‡ Ÿ

*‚ Š„‰‚ ‡‹… &…Š‰+† &‚š,"’ --‰*!‡ !„ ‡ &-‹ ‰*‡*‡ ‰‚‡ ‰ˆ  !‰(&

‹‡ †™ Œ& … !‡ !  ‡Š& ‡‰ #&& ‡‚-# ‰ˆ ‡‹& †‡‡& !ˆ‰‚…‡ ‰! ‰ˆ &-‹

‡‚$&‚† !„ ,"’ ‰‚ &-‹ ,"’ V™ Œ& #&&  U  H U H -U H J U   †&( ‰! ‡‹ †

†‡‡& !ˆ‰‚…‡ ‰!™ Œ& -‰…*‡& ‡‹& &œŠ - ‡ ‚‡& ˆ‰‚ &-‹ ,"’ ‡‰ -‹ &$& ‡‹&

›’““ ‚‡&ŠŠ‰-‡ ‰!

‹& ˆ‰ŠŠ‰Œ !„ † ‡‹& !‰(& Š„‰‚ ‡‹…™ Œ ‡‹ &-‹ ‰*‡*‡ ‰‚‡ Š !# ! ‡ Š Ž&(

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F0/ vE=;2784G$8;31E3FE18/ 73

7@  %3 4 5 6 84/6  2/e/18

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F0/ vE=;2784G$8; 31E3FE18/ 73

  F687E  / 3  / . / 17( 3 ;2  %3 56  

‹& *(‡&Ÿ ‚‰-&(*‚& Œ ŠŠ ‚†‡ ‚&-‰…*‡&  †&( ‰! -*‚‚&!‡ $Š*&†

‰ˆ  U  !( J U 

(7)

 !‰‡ †‡ †ˆ+ !„ > U  , U B -U B S  †‹‰*Š( & …‰$&( ˆ‚‰…   ‡‰   ‡ &-‹ ‡&‚‡ ‰!™ ‰!& ,"’™ †+ ™ †*-‹ ‡‹‡ >   ,  B - B  # $$ U  > U  , U B -U

B † ‚&…‰$&( ‹& !&Œ

  !(



 †&‡† …+ Š&( ‡‰ ! !&Œ ™ Œ‹ -‹

!&&(† ‡‰ & ‚&-Š-*Š‡&( ‹ † ‚‰-&†† &!(† Œ‹&! & ‡‹&‚



 † &…‡+ ‰‚

  †

‰‡‡Š&!&-# -‰!† †‡&!‡

 !ŠŠ+™ ‡‹& &(„&&‹$ ‰*‚ † † …Š& ‹& !„‚&†† † ‚&†‰!† Š& ˆ‰‚ †&!( !„

’ -#&‡†!(ˆ‰‚*(‡ !„‡‹&,"’ ˆ ‚‚‡& !ˆ‰‚…‡ ‰!*‰!‡‹&‚&-&‡ ‰!

‰ˆ  " -#&‡ ‘! ‰*‚ …‰(&Š™ ‡‹& *†‡‚&… ‚‰*‡&‚ ‰ˆ  Š !# -‰…*‡&† ‡‹&

ˆ ‚‚‡&††‰- ‡&(Œ ‡‹‡‹‡Š !#‹& !„‚&††‡‹*†-‰…*‡&†‡‹&ˆ ‚‚‡&‰ˆ‡‹&

‚†‡ Š !# !( †‰ *†&† ‡‹& !‰(& Š„‰‚ ‡‹… (& !&( ! Š„‰‚ ‡‹…

‹& „‚&†† &‹$ ‰*‚ † † …Š&™ *‰! ‚&-& $ !„  ’ -#&‡™ ‡‹& &„‚&††

†&!(†  " -#&‡™ ! ‡ Š Ž&( Œ ‡‹ ‡‹& (‡ ‰‡ !&( ! ‡‹& ’ ™ ‡‰ ‡‹&

*†‡‚&… !‰(&

‹&†‡‚*-‡*‚&‰ˆ‡‹&Š„‰‚ ‡‹…„*‚!‡&&†‡‹‡ˆ‰‚&$&‚+,"’ V

A

Y™‡‹&ˆ ‚

‚‡& U



Ÿ !ˆ‰‚…‡ ‰! !&-‹!‰(& †“’,"šˆ&† Š&™ & U S  U  S U 

› ‡‹ ‰*‚ *(‡& ‚-‹ ‡&-‡*‚& ! ‡‹&-#Œ‚(‡‹ !( ‡‹& &œŠ - ‡ ‰‡‡Š&š

!&-#Š !# !ˆ‰‚…‡ ‰!™Œ& (&-‚&†&‡‹& -‰!$&‚„&!-&‡ …& †*†‡!‡ ŠŠ+ † †‹‰Œ!

! ‡‹& !&œ‡ †&-‡ ‰!

'$ O4"2 .1"4 ! !" .1  ! !  L* 4-4/ 3 . /

‹& "’ ‚‰‡‰-‰Š -‰…&† Œ ‡‹ ! ‰‡ … Ž&( …&-‹! †… ‡‰ … ! … Ž& ‡‹& ‚‰š

-&†† !„‡ …&‰ˆ‡‹&’ !(" -#&‡†‘ˆ‡Œ‰†*--&†† $&’ ‰‚"Ÿ

-#&‡†‚&‡‹&†…&™‡‹&*†‡‚&… !‰(&‰!Š+†&!(† ‚&ˆ‚&†‹’ ‹&(‰Œ!š

†‡‚&… !‰(&‚&ˆ‚&†‹&†‡‹&,"’ &!‡‚+*‡(‰&†! ‡‚‰-&††‡‹&Œ‹‰Š&’ -#&‡

*‚ †‰Š*‡ ‰! -! &† Š+& &œ‡&!(&( ‡‰ #&& ‡‹ † ‚‰&‚‡+›&…*†‡ (&$&Š‰

 †‡‚‡&„+ ‡‰ (&‡&‚… !& ˆ  !‰(& …*†‡ †&!(  ˆ*ŠŠ !&Œ ’ ‰‚*†‡  ‚&ˆ‚&†‹

’  ! &-‹ ‰*‡*‡ ‰‚‡™ Œ& ††‰- ‡& Œ ‡‹ &-‹ ,"’ V  †&- Š  ‡  U

Ÿ

‡‹‡ ††&‡ ˆ Œ&…*†‡†&!( ˆ*ŠŠ!&Œ ’ -#&‡›‹&! †‰…&$Š*&‰ˆ !,"’

-‹!„&† ! ‡‹& ‰*‡*‡ ‰‚‡ ‡Š&™ Œ& †&‡ ‡‹ †  ‡ ˆ‰‚ ŠŠ ‡‹& ,"’† ‰ˆ ‡‹ † ‡Š&

›‹&! Œ& ‚&-& $&  ’ !( Œ& ‚&ˆ‚&†‹ ‡‹& ,"’ &!‡‚+™Œ& -Š&‚ ‡‹ †  ‡ ˆ‰‚

…‰‚& !ˆ‰‚…‡ ‰! †&& ”—Ÿ

›‹&! ‡‹& †+†‡&… † †‡Š&™ ‰!Š+ ‚&ˆ‚&†‹ "’ -#&‡† ‚& *†&( › ‡‹ ‡‹ †

…‚‰$&(Š„‰‚ ‡‹…™Œ&-!#&&‡‹&($!‡„&†‰ˆ‡‹&"’ ‚&ˆ‚&†‹…&-‹! †…

‹& *†& ‰ˆ "’ !( ‡† ‚&ˆ‚&†‹ …&-‹! †… ‚&(*-& ‡‹& ‰$&‚‹&( !&&(&(

-‰…‚&(‡‰ “ “ -&ŠŠ†‘! “™‰!&“ -&ŠŠ ††&!( &$&‚+  -&ŠŠ†‹&‚&š

ˆ‰‚&™ ‡‹& ‰$&‚‹&( !‡‚‰(*-&( + “ (&&!(† ‰! ‡‹& ž‰Œ ‚‡& !

‡‹& ‰‡‹&‚

‹!(™ ‰*‚ †-‹&…& !‡‚‰(*-&†  œ&( ‰$&‚‹&(  !ŠŠ+™ ‡‹& *†& ‰ˆ ‡‹& ‚&ˆ‚&†‹

…&-‹! †… ‚&(*-&†‡‹&‚‰-&†† !„ !&&(&( +‡‹& -‰‚&‚‰*‡&‚†‡‹‡‰!Š+ !&&( ‡‰

‚&ˆ‚&†‹ ‡‹& ,"’ &!‡‚+ ! ‡‹& ‚ ‡Š&†

 €~z€|y {~z

‰ -‰…‚& ‰*‚ †‰Š*‡ ‰! Œ ‡‹ ‰* †™Œ&‹$&(&$&Š‰&(  †&- -‚‰‡‰‡+& ‡‰

(8)

‰! ‡‹ † ‡‰‰Š‰„+



&œ‡™ Œ& (( ‡‹& ,"’† ‰!& + ‰!& !( Œ& &œ&-*‡& ‡‹& ‡Œ‰

Š„‰‚ ‡‹…†‰! ‡‹ † ‡‰‰Š‰„+

 ‰‚† …*Š‡ !„‡‹&"’ ‚‰-&††™Œ&*†&‡‹&-‰!-&‡‰ˆ ‡&‚‡ ‰! !( "’š

-+-Š&  "’š-+-Š& -‰!† †‡† ‰ˆ ‡‹& ˆ‰‚Œ‚( !„ ‰ˆ  ’ -#&‡ Š‰!„ ŠŠ ‡‹&

!‰(&† !‡‹&‡‹ ˆ‚‰… ‡‹& !„‚&††‡‰ ‡‹&&„‚&††!‰(& !( ‡‹&-#Œ‚( !„ ‰ˆ 

" -#&‡ Š‰!„ ŠŠ !‰(&† ˆ‚‰… ‡‹& &„‚&††‡‰ ‡‹& !„‚&†† !‰(& ! ‡&‚‡ ‰!

†‡‹&&œ&-*‡ ‰! ‰ˆ!"’š-+-Š&ˆ‰‚ŠŠ ‡‹&,"’†‡ ‡‹&&!( ‰ˆ&-‹ ‡&‚‡ ‰!™

Œ&‹$& $&-‡‰‚‰ˆ,"’ ˆ ‚ ‚‡&†

›& ‹$& ‡Œ‰ ‰††  Š ‡ &† ‡‰ †‡‰ ‡‹& ‚‰-&†† ‹& ‚†‡ † ‡‰ &œ&-*‡& ‡‹&

‡&‚‡ $& ‚‰-&†† *!‡ Š ‡‹& …&! ‚&Š‡ $& &‚‚‰‚ &‡Œ&&! ‡‹& Š†‡ ‚‡& $&-‡‰‚ !(

‡‹& ›’““ ŠŠ‰-‡ ‰! $&-‡‰‚ -‰…*‡&( ‚ ‰‚ *† !„ ›‡&‚ ŠŠ !„ Š„‰‚ ‡‹…Ÿ

† *!(&‚  œ&( ‚&- † ‰! &„

G

CŸ !‰‡‹&‚ ‰††  Š ‡+ † ‡‰ †‡‰ ‡‹& ‚‰-&††

ˆ ‡‹& …&! ‚&Š‡ $& &‚‚‰‚ &‡Œ&&! ‡Œ‰ †*--&†† $& ‚‡& $&-‡‰‚† † *!(&‚  œ&(

‚&- † ‰! ‹& ‚†‡ ‰††  Š ‡+ †‹‰Œ† *† ‹‰Œ ‰*‚ †‰Š*‡ ‰! ‚&-‹&† ‡‹& ‰‡ …*…

!( Œ ‡‹ Œ‹ -‹ -‰!$&‚„&!-& †&&( ‹& †&-‰!( ‚‰-‹ †‹‰Œ† *† ‹‰Œ ‰*‚

Š„‰‚ ‡‹… -!&*†&( ! ‚-‡ -&Œ ‡‹‰*‡ !+ ‚ ‰‚ #!‰ŒŠ&(„&‰ˆ‡‹&›’““

‚‡&ŠŠ‰-‡ ‰!‹& † …*Š‡ ‰! ! ‡‹& †&¥*&Š ‚& †&( ‰! ‡‹& ‚†‡ ‚‰-‹

‹& † …Š& !&‡Œ‰‚# -‰! „*‚‡ ‰! Œ& *†& ! ‰*‚ † …*Š‡ ‰! † ‰‚„! Ž&( Š #&

‡‹& ‰Š+… - †+…‰Š ‹&‚& ‚& $& “’," ‚‰*‡&‚† -‰!!&-‡&( Œ ‡‹ ¤ Š !#† ‰ˆ

•––“W†!(   ,"’† ‡‚$&‚† !„ ‡‹&‡‰‰Š‰„+  „ ‚&ˆ‡‰‰

Link1

Link0

Link3

Link2

Link6

Link5

Link4

LSP4

LSP2

LSP3

LSP1

LSP0

C 1H of..l. d/7=48 . D9     j   $  $   $          j '  '  ' ''j ' ' CgH  cc         !

 „ • Š †‡† ‡‹& ,"’ ‚…&‡&‚†   ‚&¥* ‚&…&!‡™ “ -‰!†‡‚ !‡™ Œ& „‹‡

!(ˆ ‚‚‡&ŠŠ‰-‡ ‰!ˆ‰‚&-‹,"’ ‡‚$&‚† !„‡‹&!&‡Œ‰‚#-‰! „*‚‡ ‰! „ 

†‹‰Œ† ‡‹& &$‰Š*‡ ‰! ‰ˆ ‡‹& ˆ ‚ ‚‡&† *!(&‚ ‰* † ( †‡‚ *‡&( Š„‰‚ ‡‹… ‹&

‚‡&† -‰!$&‚„& ‡‰ ‡‹& ‚ ‰‡ …Š ›’““ ‚‡& Š †‡&( !  „ • ‹& Š„‰‚ ‡‹…

‡#&†

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&*-Š (&! ( †‡!-& *!(&‚ ––• "Ÿ  „  †‹‰Œ† ‡‹& &$‰Š*‡ ‰! ‰ˆ ‡‹& ˆ ‚ ‚‡&†

*!(&‚‰*‚ ‚‰‰†&( ( †‡‚ *‡&( Š„‰‚ ‡‹…*‚ †‰Š*‡ ‰! ‚&-‹&†‡‹& ‰‡ …*… !

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-‰…Š&œ‡‰‰Š‰„ &†›& -‚&‡&(  ‡‰‰Š‰„ &†Œ ‡‹ – ‡‰ •–– !‰(&†!(Œ ‡‹ –™

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(9)

0

20

40

60

80

100

0

5

10

15

20

25

30

35

40

45

Fair rate (MB/s)

Iterations

Evolution of the LSP’s fair rates using Hou’s solution

LSP0

LSP1

LSP2

LSP3

LSP4

0

20

40

60

80

100

0

5

10

15

20

25

30

35

Fair rate (MB/s)

Iterations

Evolution of the LSP’s fair rates using our solution

LSP0

LSP1

LSP2

LSP3

LSP4

C 1H CgH  c c     !      !       

ˆ†‡&‚ ‡‹! ‰* † †‰Š*‡ ‰! Š& • ‚&†&!‡†  †‹‰‚‡ ‚&‰‚‡ ‰ˆ ‡‹& † …*Š‡ ‰!

‚&†*Š‡† k2/3707;6 ;F0 1E=;2784G nF21E=;2784G 176 j 'j $j' $j  j  j  j $j  j j j $j j j j j'$ a]  c      ! !      ! 

‹& !*…&‚ ‰ˆ ‡&‚‡ ‰!†!&&(&( + ‰*‚ †‰Š*‡ ‰! ‰! Š‚„&‡‰‰Š‰„ &† † ‚&Šš

‡ $&Š+‹ „‹ *‡ ˆ Œ& Š‰‰# ‡‡‹&!*…&‚ ‰ˆ ,"’† ‡‹‡ ‚&-‹ ‡‹& ‚ ˆ ‚ †‹‚&™Œ&

†&&‡‹‡ ˆ‡&‚ ˆ&Œ ‡&‚‡ ‰!†–" ‰ˆ‡‹&,"’†‹$&‚&-‹&(‡‹&…"‰ ‰!Š+ ˆ&Œ

,"’†-‰!‡ !*&‡‰ -‹!„&‹& …‚‰$&(†‰Š*‡ ‰!(‡&(‡‰ "’ &-‰…&†$&‚+

*†&ˆ*Š !(Œ ŠŠ …‚‰$&‹*„&Š+‡‹&&‚ˆ‰‚…!-&!(†-Š Š ‡+‰ˆ‡‹&Š„‰‚ ‡‹…

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Iterations

Minimum and average number of LSPs frozen at each iterations

Min of Hou algo

Mean of Hou algo

Min of our algo

Mean of our algo

 cc!!            ! ! 

›&Š†‰&œ&-*‡&(‡‹&† …*Š‡ ‰!†‰! –‡‰‰Š‰„ &†Œ ‡‹ •––‡‰ •––– ,"’†!(

Œ& ‹$& Š‰‡‡&( ‡‹& … ! …*… !( $&‚„& &‚-&!‡„& ‰ˆ ,"’† ‡‹‡ ‹$& ‚&-‹

‡‹& ‚ ˆ ‚ †‹‚& ˆ‡&‚ &-‹ ‡&‚‡ ‰! ˆ‰‚ ‡‹& ‡Œ‰ Š„‰‚ ‡‹…†  „  ‹& $&‚„&

!(‡‹&… ! …*… ‚&-*…*Š‡ $&$Š*&†!(†‰‚‰(*-&…‰!‰‡‰! -ŠŠ+ !-‚&† !„

ˆ*!-‡ ‰!†



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‡&‚‡ ‰!† ‡‰ †‡ Š Ž&



– " ‰ˆ ‡‹& ,"’† Œ‹ Š&‰*‚ †‰Š*‡ ‰! ‡#&† ‰!Š+

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›& ‚&†&!‡&(  !‰$&Š ( †‡‚ *‡&( Š„‰‚ ‡‹… ‡‰ -‹ &$& ‡‹& ›’““ ‚‡& ŠŠ‰š

-‡ ‰!† ‹ † †‰Š*‡ ‰! ‚‰$ (&†  †-ŠŠ& ‚-‹ ‡&-‡*‚& ‡‰ †‹‚& ‡‹& $ ŠŠ&

!(Œ (‡‹ …‰!„ ŠŠ ‡‹&,"’† --‰‚( !„ ‡‰ ‡‹& ‚ Œ& „‹‡†‹ † ‰Š -+ ‰£&‚†!

‡‡‚-‡ $& ‚ - !„ …‰(&Š ‡‰ ‡‹& !&‡Œ‰‚# †&‚$ -& ‚‰$ (&‚† Œ †‹ !„ ‡‰ !‡‚‰(*-&

‚ ‰‚ ‡+ ‰‡ ‰!† ‡‰ *†&‚† !  *†„&š†&( ‚ - !„ …‰(&Š

*‚Š„‰‚ ‡‹… …‚‰$&†-‰!† (&‚Š+‡‹&&‚ˆ‰‚…!-&+*† !„!&Œ *(‡&

Š! ‰ˆ‡‹&-‰!‡‚‰Š -#&‡!( Œ ‡‹ ‡‹& !‡&„‚‡ ‰! ‰ˆ!&œŠ - ‡ ‰‡‡Š&!&-# Š !#

…‚# !„ †‡‚‡&„+ ‹& *‡ Š Ž‡ ‰! ‰ˆ ‡‹& "’ ‚‰‡‰-‰Š ‡‰ ‚&ˆ‚&†‹ &‚ ‰( -ŠŠ+

‡‹& -‰‚& !‰(& !ˆ‰‚…‡ ‰! (&-‚&†&† ‡‹& ‰$&‚‹&( ‚&†&!‡ ! ‡‹& “š



‚-‹ ‡&-‡*‚& ˆ‰‚ …‰‚& !ˆ‰‚…‡ ‰! †&& ”—Ÿ



*‚ ˆ*‡*‚& Œ‰‚# Œ ŠŠ ˆ‰-*† ‰! ‡‹& (&$&Š‰…&!‡ ‰ˆ  ‚ „‰*‚‰*† -‰!$&‚„&!-&

‚‰‰ˆ † Œ&ŠŠ † &œ&-*‡ !„ ‰*‚ Š„‰‚ ‡‹… ‰! &$&! …‰‚& -‰…Š&œ ‡‰‰Š‰„ &†!š

‰‡‹&‚-‹ŠŠ&!„ !„ ††*& †‡‰ …‚‰$&‡‹&Œ& „‹‡-‰!-&‡+*† !„*‡ Š ‡+ˆ*!-‡ ‰!†

‡‰ (&†-‚ & ‡‹& ‡‚ƒ-‹& ‰&-‡ $&† &-‰…& ‡‰ …œ … Ž&‡‹& „Š‰Š *‡ Š ‡+ ‰ˆ

‡‹& -Š &!‡†!( !ŠŠ+™Œ&Œ ŠŠ !$&†‡ „‡&‡‹& !‡&„‚‡ ‰!‰ˆ‡‹ † Š„‰‚ ‡‹… ! 

‚&Š £†&‚$ “’," !&‡Œ‰‚#

3J"1!/ ‹ † Œ‰‚# Œ† †*‰‚‡&( + ‡‹& *‚‰&! ‚‰&-‡

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š–¤ Ÿ!( ‡‹& &Š„ ! ‚&„ ‰!Š ‚‰&-‡ “

 {y j vsl;2FGs/346731E;GG78//j      ! !    !   /g2F12?  j vsl ;2FGp 'i.j

$j &jv:fF34/AjD/2=/2&j16sjA7rjq2767>101616f jq:1EE;:j 

  !        &/3/Gg/2$j . $ j

j qjD1E;6jDE1634?nj&/E3;F28jA/fF3AjlE;616fjq<7>/jq?684/07016f

2/3;GG/6f187;6 @;276821if;G176 8213 /6=76//276=j s/346731E ./e;28vs.o5l

&/E7>/21gE/ &j567>/2078? ;@ A7B=/9;>/Gg2/ $j

j &jD/280/<10 16f .j1EE1=/2j     j k2/6873/ 1EE $j 'j j416=9j;EG7/ 16f &jq7Fj        !   !    ! !   vF=F08 'j s:/E@84 o68/26187;61E;6@/2/63/ ;6 ;GeF8/2;GGF673187;6 o 'j

j vj 4126?j v6 1E=;2784G @;2 218/ 1EE;3187;6 76 1 e13</8i0:783476= 6/8:;2< :784

@//fg13<j s/346731E ./e;28 lospAqps.i  j

j j s4;G10 ;F /62? s/6= q47>/6f21 qj k16:12 16f r71? kj FG12j v =/6/273:/7=48ie2;e;287;61Eg16f:7f84 041276=e;E73?@;2 18G 1g20/2>73/j      ! e1=/0 $   j j .j 176 qj 1E?16121G16 .j ;?1E qj 14G? 16f .j r70:1618416j   !   !     !! vF=F08  j vsl ;2FGp i $j j jjA/D;Ff/316fDj.1fF6;>73jvF67/f@21G/:;2<@;2G1hiG7616fG76iG1h @1726/00:784 1eeE73187;60j n38;g/2 $$j  j tj .;0/6 16f 1Ej  !    !!  !   16F12? $j . j

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