HAL Id: inria-00070757https://hal.inria.fr/inria-00070757
Submitted on 19 May 2006
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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�
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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
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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��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′
±�$f+���+fq
ξm,m′ ≥ ξm,n + ξn,m′ .� �¡���+f¸���¹oq��� �¡sn��yq�Â
ξm,m+k = ξ0,k(θm) �¢s ©ºyq����y��{m¯©ºxvyqe "³�����Degoq�_mnp� �oq��� �¢sv�¡¤Df Ooq�#snp�oq�¡��t_mnp��®fUx£§
oD��� �¡sn��¤qf "¯fUxn�Dy � ���¸svc�f+yDxnfUe ªmnf+f ��h���� "�snc�o�s4snc�fUxnf³��m{o��¹o���egyDm£s{m£p�xvf³�¡��eg�¢s
limm→∞
1
mξ0,m = ξ .
ì�ì �/�ô�07ó�1
��� �$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
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
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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)
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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 = �� �.
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(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
��� �$Ì�, ,7Æ+ÔîÔ('��¸ÏÂÅ�Ì�'îÈ�ÃHÓnÆ7Ì������ '����#'ªÌ�)7Ê�Ò
�+yq�¨¤qfUxn�DfUmU[m£y¥snc®o�sg©ºxnyDe snc�f ��yDxnfU�¢§£}~o��Lsvf+���¡�~�¡fUegego�P (lim supm→∞ x(k,l, ��� � )/m ≤ x) = 1
��xny�¤¨�����/svc�f-xvfUmnp��¢sV±
Ù¾s$��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
��m~snc�f-e*o�r¨��e/p�e �4����� y��Xmn��µ+f º��yqsnf³svc�o�s$b$}�wÚ��e��AyDmnfUm~svc�o�sWmax ≤ B ���
"#±²$�Do����9�snc���m{©ºp����·sv�¡yD�&svoq�qfUm{�¡�LsnfU�qfUx{¤�o���p�fUm¸o����¹�¡s¸� fV�#xvfUoqmnfUm$musvxn���·sv�¡t¹o���yq���Joq�¨t_��o�snc&���&snc�f
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v′(k, l, h, m)���Jsvc�f���o�svc�m~��m~c�oqxv� ��t*o����mvm£p�fq oqm
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G±
b c�f_ego�r �¡e*o�����fU�¡�DcLs�y�©$o¹��o�snc�c�oDm� Af+f+�Ýmusvp�� ��fU�¿��xvf+¤¨��yqp�mn�¡t�©ºyqxÁxnfU�qp���o�x-�Dxvoq��c�m@ ª��o�s£sn���#f�4�¡snc»o��¨t������¡snf*� ��egf+��mn�¡yD� "·±-Ù¾s³� oqm�m£c�y��4���¡�¥��o�xnsn���#p���o�x¸snc®o�s�snc�f�e*o�r �¡e*oq���~f+���qcLs¸yq©�o��o�svc���¹m£p��7c¹oÁ��o�s£sn���#f³��m �¡�®�#xvfUoqmn������o�s~egyLmus4���¡��fVo�xv�¡t*����©ºp����#sn��yq��y�©��¡svm4mn��µ+f³��xvy�¤L��� fV�*snc�f��~f+���qcLs7m�yq©svc�f-��y � fVm4o�xvf�x7o��®� yqe ¤�o�xv��o� ���fUm~�4�¡snc������¡snf�egyDe�fU�Lsvm y�©:yqx7� f+x
2 + ε©ºyqx{mnyqegf
ε > 0±
å yDxnfÁ��xvfU�#��mnf+��tq®çÂf+ege*o��± ��¡� �ý�����"�Uo��¶ Af/p�mnfU�¥o���egyDm£s¸� ��xnfV�·sn��t�snyf#r¨snfU���¶b c�fUyqxvf+e.h�snysvc�f�Uoqmnf_yq©$o��7c�oq�¡�¿svyq�Ayq��yq�qt º�H± fq±�b$}�w��#yD����fV�·sn��yq�®m/����mnf+xv�¡fVm#"/o����×o��#yq�®mus7o��LsÁ �p�ÀAfUx�mn�¡µUfqp���� f+x svc�f-oqmvm£p�e���sn��yq�
E[σ2+ε] < +∞©ºyqx
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38
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0 200 400 600 800 1000
Gro
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hrou
ghpu
t (P
kts/
sec)
Group Size
TCP using SACK ; Overlay with 10 Routers ; Packet Loss Proba =0.01
Service Time : Exponentially distribued with mean = 10msBuffer = 10 000 Packets
Buffer = 1 000 PacketsBuffer = 100 Packets
Buffer = 50 Packets
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30
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0 200 400 600 800 1000
Gro
up T
hrou
ghpu
t (P
kts/
sec)
Group Size
TCP Reno ; Overlay with 10 Routers ; Buffer=100Pkts ; Packet Loss Proba =0.01
Service time with mean = 10msExponential law
Pareto law with parameter 2.1Pareto law with parameter 1.9
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Gro
up T
hrou
ghpu
t (P
kts/
sec)
Packet Loss Proba
TCP Reno ; Overlay with 10 Routers ; Buffer = 40 Pkts
Service Time : exponentially distribued with mean = 10ms
One connectionGroup size=126
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berkeley_2
uoregon_4
upenn_4
stanford_4
columbia_4
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columbia_2
dartmouth_4
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berkeley_5
uga_5
caltech_1
utah_1
uga_2
cornell_2
utah_5
cmu_6
umass_5
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Number of Fail-Join Iterations
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Number of Fail-Join Iterations
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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