the one-to-many tcp overlay: a scalable and reliable ... · issn 0249-6399 isrn...

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HAL Id: inria-00070757 https://hal.inria.fr/inria-00070757 Submitted on 19 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. The One-to-Many TCP Overlay: A Scalable and Reliable Multicast Architecture François Baccelli, Augustin Chaintreau, Zhen Liu, Anton Riabov To cite this version: François Baccelli, Augustin Chaintreau, Zhen Liu, Anton Riabov. The One-to-Many TCP Overlay: A Scalable and Reliable Multicast Architecture. [Research Report] RR-5241, INRIA. 2004, pp.41. inria-00070757

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Page 1: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE

HAL Id: inria-00070757https://hal.inria.fr/inria-00070757

Submitted on 19 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

The One-to-Many TCP Overlay: A Scalable andReliable Multicast Architecture

François Baccelli, Augustin Chaintreau, Zhen Liu, Anton Riabov

To cite this version:François Baccelli, Augustin Chaintreau, Zhen Liu, Anton Riabov. The One-to-Many TCP Overlay:A Scalable and Reliable Multicast Architecture. [Research Report] RR-5241, INRIA. 2004, pp.41.�inria-00070757�

Page 2: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--52

41--

FR

+E

NG

ap por t de r ech er ch e

THÈME 1

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

The One-to-Many TCP Overlay:A Scalable and Reliable Multicast Architecture

François Baccelli — Augustin Chaintreau — Zhen Liu — Anton Riabov

N° 5241

Juin 2004

Page 3: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE
Page 4: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE

Unité de recherche INRIA RocquencourtDomaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

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

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Page 5: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE

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Page 6: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE

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Page 18: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE

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b$}�wØ�+yq����fV�·sv�¡yD�(k, l)

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)

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x(k,l,H(k,l))m =

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)

+ σ(k,l,H(k,l))m

�8"

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x(k+1,l′,H(k+1,l′))

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,(k+1,l′)

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∪ {(k, l, h, m), k ≥ 1, l ≥ 0, h ∈ { ���� , 1, 2, . . . , H(k,l), �� � }, m ∈ Z}

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Page 19: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE

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E1 ∪ E2 ∪ E3 ∪ E4 ∪ E5�4�¡snc¿¦

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∪{

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∪{

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h = 2, . . . , H(k,l)

o��®�∀k ≥ 1, l ≥ 0, m ∈ Z

}

∪ {(k, l, ���� , m)→ (k − 1, � (k, l), �� � , m), ∀k ≥ 1, l ≥ 0, m ∈ Z} ,

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hoq���

∀k ≥ 1, l ≥ 0, m ∈ Z}

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}

E4 ={

(k, l, ����� , m)→ (k, l, �� � , m−B(k,l)��� ), ∀k ≥ 0, l ≥ 0, m ∈ Z

}

E5 ={

(k, l, �� � , m)→ (k + 1, l′, H, m−B(k,l)����� ,(k+1,l′)), ∀l

′ ∈�(k, l)

oq���∀k ≥ 0, l ≥ 0, m ∈ Z} .

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k, l, h, mx(k,l,h)

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Page 20: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE

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. . . . . .

. . .. . .

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. . .. . .

. . . . . .. . .. . .

. . . . . .

h = �����h = 1

h = 2h = 3

h = 4h = �����

h = �����h = 1

h = 2h = 3

h = �����

. . .

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. . .

. . .

k = 0

h = �����

+,����-��(�m

+,����-��(�m − 1

. . .

+,����-��(�m − B

. . .

+ ����-��(�m − Wm

H1 = 4

k = 1

H2 = 3

k = 2

E3

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Page 24: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE

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±�b c�fgoq�q�DxnfU�Do�svfU�&mnf+xv¤¨���+fsv�¡egfVm+­¨o����*snc�f¸�¡yLmnmnfUm¯�4c����7cJ�+xnfVo�svf$m£yDegf{y�©Äsvc�f$fV� �qfVm[���

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m±:Û¥f ©ºp�xnsnc�f+x[oqmvmnp�egf�svc�o�s=©ºyDx=fUoD�7c�b$}�w��#yD����fV�·sn��yq�­�snc�f4�4����� y��Ý��xvy¨�+fUmvm

{W(k,l)m }m

��m"�¡�g�¡svmm£svo�sn��yq��oqxnt¹m£svo�svfq±-|¯oD�7c�yq©[snc�fVm£f���xvy �#fUmvmnfUm¸��m³o��¥f+xv�qy � ���� ª�¡xvxnfV� p��+�¡ ���fq­Äoq�®fUxn��y � ����o��®�&�®yLm£�¡sn��¤qfxvfU�+p�xvxnfU�Ds�" å o�xv�qy�¤&�7c�oq�¡�9­��4c����7c»��m-eg�¡r¨�����®±_b c�f*��xvy � p��·s�yq©�svc�fUmnf*����� f+�Af+�®� f+�Ls/o��®�¥eg�¡r �¡���å o�xv�qy�¤�7c®o�����m{��m$eg�¡r �¡���_sny¨y�±¸Û¥f�� f��®��f

θsny_ AfÁsnc�f�m£c��¡©îs³y�©[����� f#r

m�¡��snc�fÁxvoq��� yqe°�qx7o���c

G± ��yqx �¡�®mus7o����+f

σ(k,l,h)m (θ(ω)) = σ

(k,l,h)m+1 (ω)

­W

(k,l)m (θ(ω)) = W

(k,l)m+1 (ω)

o��®�Jm£��eg�¡��o�xv��tÁ©ºyDx~snc�fyqsnc�fUx4¤�o�xv��oq ��¡fVm oqmvm£y �#��o�svfU�_�4�¢svc�fV� �qfVm+±=b c�f³��o?�Õyq©

G��m ���¨¤?oqxn��o��Ls  ¨tJsnc���m4mnc��¢©îsV­�o��®�_svc���m4m£c��¡©îs

��m f+xv�qy � ���. º�¡s{��m fU¤qfU�eg�¡r¨�����&"·±��yDx$o����

n ∈ Z­®�¡f+s

G[n]  Af-svc�f��qx7o���c�yD  svoq�¡��fU�©ºxnyDe G�4c�fU�¹xnfU����oD�#�����gsnc�f���fU�¡�DcLsvm4yq©"oq�¡�

¤Df+xnsn���#fVm(k, l, h, m)

�4�¢svcm < n

 ¨t−∞

±�è{yqsnf*svc�o�s-svc���mÁ� f��®���¢sv�¡yD��f#r¨snfU����m³svc�o�sÁy�©G[0] ���

� fU�·sv�¡yD� ��± ��±Û»f-��y��Õ���Lsnxvy � p��#f�svc�f�m£f %Lp�f+���+fg¦

ξm,m′ = maxπo���o�snc¹���

G[m]

(0, 0, ���� , m′) (0, 0, ���� , m)

{� ��� ����� (π)} , m < m′.

��yDx{o����m < n < m′

­ξm,n + ξn,m′

�Uo��� ®f-xvf+�4xv�¢snsnfU��oDm

maxπog��o�svc���

G[m]

(0, 0, ���� , n) (0, 0, � �� , m)

{� ��� ���� (π)}+ max

πo���o�snc¹�¡�

G[n]

(0, 0, � �� , m′) (0, 0, ���� , n)

{� � � ���� (π)} .

²¸mn > m

­Aoq�Lt�¤qfUx£svf#r¹�¡�G[n] c�oqm¸o_mnegoq�¡��f+x¸��fU�¡�DcLs{snc�oq�&snc�f��+yqxvxnfVm£�Ayq�����¡���*¤Df+xnsnf+r���� G[m] ­mny�snc�o�s4snc�f-oq ®y�¤Df³mnp�e���m{mne*o����¡fUx�svc�o��

maxπog��o�svc���

G[m]

(0, 0, ���� , n) (0, 0, � �� , m)

{� ��� ���� (π)}+ max

πo���o�snc¹���

G[m]

(0, 0, � �� , m′) (0, 0, ���� , n)

{� � � ���� (π)} ,

�4c�f+xvfG[n] ��m[xvf+����oq�+fU�� ¨t G[m] ���gsnc�f¸mnfU�#yD���ÁsnfUxne¹±=b c���m[��oqm£s�m£p�e´��m¯oq��mny³snc�f$ego�r �¡e*o��®��fU�¡�DcLs

y�¤Df+x�o����¸��o�snc�m����G[m] �¡fVoq� �����שºxvyqe (0, 0, ���� , m′)

sny(0, 0, ���� , m)

o���� snc�o�s��Dy×svc�xnyDp��qc(0, 0, ���� , n)

îsvc���m���m[svxnp�f$ ®fV�+o�p®m£fσ

(0,0,� � � )m = 0

©ºyqx o����m"[oq���*c�fU���#f¸�¢s~��m~mnegoq�¡��f+x[snc�oq�

ξm,m′

±�$f+���+fq­

ξm,m′ ≥ ξm,n + ξn,m′ .� �¡���+f¸���¹oq��� �¡sn��yq�­

ξm,m+k = ξ0,k(θm)­ �¢s ©ºyq����y��{m¯©ºxvyqe "³�����Degoq�_mnp� �oq��� �¢sv�¡¤Df Ooq�#snp�oq�¡��t_mnp��®fUx£§

oD��� �¡sn��¤qf "¯fUxn�Dy � ���¸svc�f+yDxnfUe ªmnf+f ��h���� "�snc�o�s4snc�fUxnf³��m{o��¹o���egyDm£s{m£p�xvf³�¡��eg�¢s

limm→∞

1

mξ0,m = ξ .

ì�ì �/�ô�07ó�1

Page 27: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE

��� �$Ì�, ,7Æ+ÔîÔ('��¸ÏÂÅ�Ì�'îÈ�ÃHÓnÆ7Ì������ '����#'ªÌ�)7Ê�Ò

� �¡���+fθ��m4fUxn�Dy¨�����q­

ξ��m$�#yq�®mus7o��LsU±"Û»f�� f�����f

γ = 1ξ

oq���snc¨p�m �~f-c�o?¤qf�¦

limm→∞

m

x(0,0,� ��� ) = γo�± mU±

.

é¸��f-� fU��p��#fVm ©ºxnyDe snc�f³©ºyq���¡y��4�����g�#yqeg��oqxn��mnyq��¦

x(0,0,� ��� )m ≤ x(k,l,h)

m ≤ x(0,0, � ��� )m+B � � +k×(B � � +B � ��� ).

svc�o�s¸snc�fÁmnoqe�f/�¡��eg�¢s³c�yq����m{©ºyqx¸oq�¨t(k, l, h)

±{Û¥f��+oq�¶m£c�y��Xsnc�f$��x7mus¸�¡��f %Lp�o����¢sut¹oqm¸o��¨t���o�svc¶�¡�G©ºxvyqe

(0, 0, ���� , m)sny

(0, 0, ����� , 0)�+o��¿ Af_fVoqmn�¡��t¥�+yqeg����f#snfV�¿�¡�Lsvy¶o&��o�svc¿©ºxnyDe

(k, l, h, m)svy(0, 0, ���� , 0)

±�b c�f¥mnfU�+yq���Ø����f %Dp®o����¢sutØ�#yDegfUmJ ¨tØ�#yDeg���¡f+sn�����Ýo���o�svcØ©ºxvyqe(k, l, h, m)

svy(0, 0, ���� , 0)

�4�¢svcsvc�f�o�x7�

(0, 0, ����� , m + B ��� + k × (B ��� + B ����� )) (0, 0, � � , m + k × (B ��� + B ����� ))

oq���_snc�fU��p®m£������snc�o�s4©ºyqx{oq�Lt(k′, l′)

oq�����o�p��DcDsvf+x4��y � f(k′ + 1, l′′)

svc�f+xvf-f#r ��mus7m4oÁ�®o�snc¿¦

(k′, l′, 0, �� � , n) (k′ + 1, l′′, �� � , n− (B ��� + B ����� )) .

�b c�f¥�¨p�e/ Af+xγ� fU�®fU����myq�Øsvc�f¥mn��µ+f�oq���Úsnc�f¶snyD�®yD�¡yD�qtÝy�©-snc�f¶snxvf+fD­{yq�Úsvc�f¥�¨p�eÁ ®fUx_yq©

xvyqp�snf+x7m ���&fUoq�7c&y�¤Df+xv��o?t mU­¨yq��snc�fÁf+¤Dyq��p sn��yq�¹y�©"svc�f��4����� y��-­®oq����yq�¹svc�f���o?�]yq©�svc�fÁoq�q�DxnfU�Do�svfU�mnf+xv¤¨���+fJsv�¡egfUm�egy � f+�������»�+xnyLmnm�snx7o�ÜJ��­=yD�×svc�f��yDmvm���xvy¨�+fUmvm����×svc�f&b$}�w´�#yD����fV�·sn��yq�®m+­[oq���×yD�svc�f¹��o�x7o�egf+snf+x7m�y�©¸snc�f¹ �oD�7�D§¾��xvfUmvm£p�xnfe�fV�7c�o�����mne*m+±ÝÙ¾sJ�4�¡���4 ®f¶�Uo����¡fV��svc�f�Ì�+·Í � �AüÊ�à ' ,��qÓnÊ�� �úŨÓvÊ�� �?Å�� � Ã~y�©:snc�f-e/p��¡sn���+oDmus4y�¤qfUxn��o?tq±

��yqx snc�f-��yq� §¾c�yDe�yD�qfU��f+yDp�m��Uoqmnfq­

lim infm→∞

m

x(k,l,h)m

≥ γ, uhV�8"

�4c�f+xvf¸o��Lo����*��m~� f+snf+xveg�¡����m£sn���¸oq���*��m¯����� fU�®fU��� fU�Ds4yq©(k, l, h)

±"Ù¼�*svc���m �Uoqmnfq­γ��m¯svc�f³oqmntLeg� svy�sv���

�DxnyDp��Jsvc�xvyqp��Dc���p s4y�©=mnyqegf³�¡y��~f+x  Ayqp����¹mnt musvf+e��4c����7c&m£c�yqp����_ Af��#��fUoqxU±

� ��� �� �D�2Cª� � % C�%º� à � ' ä ß ��� % � � � � � %,C ß äØ�_�V����� �:� % ã ' �b c�f/oqmvm£p�e���sn��yq��m$y�©=snc���m�m£fV�·sv�¡yD�¶o�xvf-snc�fÁ��yq��§Hc�yDegyq�qfU��f+yDp�m yq��fUm{yq©"snc�fÁ��oDmus³m£fV�·sn��yq�±4Û¥fÁp�m£f�®o�snc¹fU�Lp�e�fUxvo�sn��yq��©ºyqx¸musvp�� t¨�¡���*snc�xvyqp��qc���p�s{�¡�¹svc�f/�+oDm£f-�4c�f+��snc�f/xvoq��� yqe3¤�o�xv��oq ��¡f

σ��m4���¡�DcDs

s7o����¡fV�Ä­ �H± fD±[m£p��7c�svc�o�s snc�f+xvf-f#r ��m£svm4o�xvfUoq�Ä�Lp�e/ Af+xτ > 0

�4�¡snc

E[etσ ] ≤ A(t) < +∞, uhqh "

©ºyDx{o����0 ≤ t ≤ τ

±

êºë:ì9êîí

Page 28: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE

ÎFÅ�ÆJÇ ÈAÆ+ɼüÊ�ɾË�Ì�ȮͥÎ[ÏÂÐ Ç~Ò�Æ+Ó·Ô¡Ì�Í �D�

� ��ߨã � ß � ��Ï[Ê�È +�' $qÆ+Ó_Ì�È�Ê�Ò�Æ+Ó·Ô¡Ì�Í � � Ô Ã ' ,vÌ�+·Ã{ÃOÓvÆvÆ � 'îÃîÅ 'îÈ��[È 'îþÆ/Å�Æ '�?ŨÃk = 0, 1, 2, . . . ��� È%$DÆ#ÓúÅ�Æ�Ì�+ + � � �Aà 'ªÊ�È +-úÅ�Ì�ïúÅ�Æ/Ô¡Ì��XÊ��

σ' +-Ô('�VŠà þÌ�'îÔ¡Æ $ � �¯Æ³Å�Ì�Ò?Æ

lim supm→∞

m

x(k,l, ���� )m

≥ ��� ���� (H, D) > 0 a.s. .

� È ' �#Ê�Ó ��Ô Í�'îÈ(k, l)

� )7Ê�úŠ�#Ê�ÓÃîÅ�Æ���Ï�� Ì�È%$¥ÃîÅ�ÆÔ¡Ê�+ + 3ÓnÆ+É ++Æ�� ��Æ+È%,*'îÈ � ,7Ì�++Æ + � ��Å�Æ#ÓnÆD' +_úÅ�Æ

)7Ê�� È%$�Ê�È�ÃîÅ�Æ�$DÆ���ÓnÆ7Æ-Ì�È%$HúÅ�Ì�Ã[Ê�È�úÅ�Æ4Å�Ê �¶È � ��)7Æ#Ó ��� È�ÃîÅ�Æ ��Ì�Ó·Ã ' , � Ô¡Ì�Ó ,vÌ�++Æ-Ê �¸Ì³Å�Ê�gÊ �LÆ#ÈFÆvÊ���+ÈFÆ#à �¯Ê�Ó����ÂúÅ�Æ ��ÓvÊ�� � ,7Ê � � � È ' ,7Ì�à 'ªÊ�ÈÃîÅ ÓnÊ�� �?Å���� Ã

γ' + )7Ê�� È&$qÆ $ �7ÓnÊ� )7Æ+Ô¡Ê��")#Í/Ì ��Ê�+ 'îà 'îÒ�Æ�,7Ê�È +·Ã¾Ì�È�Ã

úÅ�Ì�à $DÊUÆ +�ÈAÊ�à $qÆ ��Æ#È&$Ê�È�úÅ�Æ + '��VÆ�Ê �³ÃºÅ�Æ/ÃOÓvÆvÆ �Ð ÓnÊUÊ�� ��b c�f¿x7o���� yDe ¤�o�xv��oq ��¡f

x(k,l, ��� � )m

��m�snc�f¿�~f+���qcLs�yq©/svc�f¿ego�r �¡eÁp�e �~f+���qcLs��®o�sncX©ºxvyqe(k, l, �� � , m)

svy(0, 0, ����� , 0)

�¡��svc�f_x7o��®� yqe.�qx7o���c���xnfU¤L��yqp®m£��t����Lsnxvy¨��p��#fV� yqx�|~}~è ��oq�7�Df#se*oqxn�¨�����Á��� � fU�#sn��yq� ��± ��­¨©ºyDx4��oq�7�Df#s4�¡yLmnmnfUm~��� � fU�#sn��yq� ��± � "·±

ç9f+s{p�m���x7mus4svxnfVo�s snc�f��Uoqmnf¸yq©:| }~èÑ��oq�7�Df#s4e*o�xv�L�����®±�Û»f-�¡�Lsvxny � p��+f�snc�f³©ºp��®�·sn��yq�φ¦

φ(k, l, h, m) = (H + 2)k + 2(H + 2)m + v(h)

�4�¡sncv(h) =

0©ºyDx

h = ����h

©ºyDx{o��¨t1 ≤ h ≤ H

H + 1©ºyDx

h = �� �.

Ù¾s4s7o��DfUm~�¡�Lsvf+�qfUx4¤�o���p�fUm4oq���m£snxv���·sn��t��fU�#xvfUoDm£fVm o���yq���go��¨tJ�AyDmvm£�� ���f³��o�svc����snc�f-x7o�����yqe��Dxvoq��c9±²¸m¯o³�#yD��m£f %Lp�f+���+fq­Lo¸�®o�snc*����snc���m[�Dxvoq��cÁ©ºxnyDe

(k, l, h, m)sny

(0, 0, ���� , 0)�+oq����y�s¯�+yqegf4 �oD�7�

svy»snc�f&mvo�egf¤Df+xnsnf+rF­~o����Ýo���m£y¥snc�f¶m£f+sgyq©$¤qfUx£sv���+fUmg�#yD�Lsvo�����fV�×���Úmnp��7c½o¥�®o�sncÚ�+oq����yqs�f+r �+f+fV�(H + 2)k + 2(H + 2)m + H + 1

±²¸m"snc�f4e*o�r¨��e*o�� �¨p�eÁ ®fUx=y�©®��f+���qc¨ ®yDxvm:yq©Fo���y � f ���Ásvc�f4�qx7o���c/��m

max(3, D+1)­?snc�f4�¨p�eÁ ®fUx

yq©��AyDmvm£�� ���f³��o�svc�m ©ºxvyqe(k, l, h, m)

svy(0, 0, ���� , 0)

��m  Ayqp���� fU�� ¨t¶¦

(max(3, D + 1))(H+2)k+2(H+2)m+H+1 .

�$f+���+fq­�p�mn����� å o�xv�qy�¤ ' m¯����f�%Lp�oq�¡�¡sutq­ �~f-�qf+s snc�o�s4©ºyqx{o����πoqm4� fVmn�+xn�� ®fV��¦

P (� ��� ���� (π) ≥ xm) ≤ e−txm

E[et � ��� �! �" (π)]

≤ e−txmA(t)(H+2)k+2(H+2)m+H+1

²$���p�m£�����/snc�o�s~svc�f³��xvyq �oq ��¡���¡sut�yq©�mnyqegf¸p����¡yD�_yq©Âf+¤Df+�Lsvm���m p����®fUx  ®yDp���� fV�J ¨tgsnc�f-mnp�e y�©�snc�f��xnyD �o� ������¢sv�¡fVm+­¨�~f�m£fUf¸svc�o�s4svc���m4�¡eg������fUm~snc®o�s4©ºyqx

m ≥ k ≥ 1o����©ºyDx

D ≥ 2­

P(

x(k,l, ��� � )m ≥ xm

)

≤ (D + 1)5(H+2)me−txmA(t)5(H+2)m.

Ù¾©x��m4�7c�yDmnf+����oqxn�Df¸fU��yqp��qc� º���©ªoD�·s$mnp��7csnc�o�s

etX ≥ (A(t)(2D + 1))5(H+2) "·­ svc�f�mnf+xv�¡fVm∑

m

P(

x(k,l, ��� � )m ≥ xm

)

ì�ì �/�ô�07ó�1

Page 29: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE

��� �$Ì�, ,7Æ+ÔîÔ('��¸ÏÂÅ�Ì�'îÈ�ÃHÓnÆ7Ì������ '����#'ªÌ�)7Ê�Ò

�+yq�¨¤qfUxn�DfUmU­[m£y¥snc®o�sg©ºxnyDe snc�f ��yDxnfU�¢§£}~o��Lsvf+���¡�~�¡fUegego�­P (lim supm→∞ x(k,l, ��� � )/m ≤ x) = 1

­��xny�¤¨�����/svc�f-xvfUmnp��¢sV±

Ù¾s$��m{��yqs¸� �¡ÜJ�#p��¡s{snym£fUf³snc�o�s{snc�fÁmnoqe�f�oqxn�Dp�egf+�Ls{c�yD����m4©ºyqx{��oD�7�qf+s{�¡yLmnmnfUmU­����¹snc�f�x7o��®� yqe�Dxvoq��cgxnfU��xvfUmnf+�Lsn������xvf#svxvoq��m£eg��mnmn��yq��m=oq���J�#yD��mnfU�#p�sn��¤qf{xvf+yDxv� fUxn�����®±Âb c�f{©ºp����·sv�¡yD�

φ��m���y��Ú�q��¤qfU�

 ¨tÁ¦φ(k, l, h, m) = Wmax(H + 2) k + (H + 2)m + v(h),

�4�¢svcv(h) =

0©ºyDx

h = ����h

©ºyDx{o��¨t1 ≤ h ≤ H

H + 1 + (H + 2)(Wmax − 1)©ºyDx

h = �� �,

�4c�f+xvfWmax

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Page 45: The One-to-Many TCP Overlay: A Scalable and Reliable ... · ISSN 0249-6399 ISRN INRIA/RR--5241--FR+ENG apport de recherche THÈME 1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE

Unité de recherche INRIA RocquencourtDomaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France)

Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

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

ISSN 0249-6399