Download - Heterogeneous Congestion Control Protocols
![Page 1: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/1.jpg)
Heterogeneous Congestion Control Protocols
Steven LowCS, EE
netlab.CALTECH.edu
with A. Tang, J. Wang, D. Wei, CaltechM. Chiang, Princeton
![Page 2: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/2.jpg)
Outline Review: homogeneous case Motivating experiments Model Equilibrium
Existence, uniqueness, local stability Slow timescale control
Tang, Wang, Low, Chiang. Infocom March 2005Tang, Wang, Hegde, Low. Telecommunications Systems, Dec 2005
![Page 3: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/3.jpg)
F1
FN
G1
GL
R
RT
TCP Network AQM
x y
q p
))( ),(( )1())( ),(( )1(
tRxtpGtptxtpRFtx T
Reno, Vegas
DT, RED, …
liRli link uses source if 1 IP routing
Network model
![Page 4: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/4.jpg)
for every RTT{ W += 1 }for every loss{ W := W/2 }
Reno: (AI)
(MD)Jacobson
1989
ililill
llli
i
ii
tptxRGtp
tpRxT
tx
)(),()1(
)(2
1)1(2
2
AI
MD
TailDrop
Network model: example
![Page 5: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/5.jpg)
FAST:Jin, Wei, Low
2004 W
RTTbaseRTT :W
periodically{
}
ilili
lll
llliii
i
iii
ctxRc
tptp
tpRtxT
txtx
)(1)()1(
)()()()1(
Network model: example
![Page 6: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/6.jpg)
Duality model of TCP/AQM TCP/AQM
Equilibrium (x*,p*) primal-dual optimal:
F determines utility function U G guarantees complementary slackness p* are Lagrange multipliers
) ,(
) ,( ***
***
RxpGp
xpRFx T
cRxxU iix
subject to )( max
0
Uniqueness of equilibrium x* is unique when U is strictly
concave p* is unique when R has full row rank
Kelly, Maloo, Tan 1998Low, Lapsley 1999
![Page 7: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/7.jpg)
Duality model of TCP/AQM TCP/AQM
Equilibrium (x*,p*) primal-dual optimal:
F determines utility function U G guarantees complementary slackness p* are Lagrange multipliers
) ,(
) ,( ***
***
RxpGp
xpRFx T
cRxxU iix
subject to )( max
0
Kelly, Maloo, Tan 1998Low, Lapsley 1999
The underlying concave program also leads to simple dynamic behavior
![Page 8: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/8.jpg)
Duality model of TCP/AQM Equilibrium (x*,p*) primal-dual optimal:
cRxxU iix
subject to )( max
0
Mo & Walrand 2000:
1 if )1(
1 if log)( 11
i
iii x
xxU
Vegas, FAST, STCP
HSTCP Reno XCP (single link
only)
Low 2003
![Page 9: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/9.jpg)
Some implications Equilibrium
Always exists, unique if R is full rank Bandwidth allocation independent of AQM or arrival Can predict macroscopic behavior of large scale
networks Counter-intuitive throughput behavior
Fair allocation is not always inefficient Increasing link capacities do not always raise
aggregate throughput [Tang, Wang, Low, ToN 2006]
FAST TCP Design, analysis, experiments[Jin, Wei, Low, ToN 2007]
![Page 10: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/10.jpg)
Some implications Equilibrium
Always exists, unique if R is full rank Bandwidth allocation independent of AQM or arrival Can predict macroscopic behavior of large scale
networks Counter-intuitive throughput behavior
Fair allocation is not always inefficient Increasing link capacities do not always raise
aggregate throughput [Tang, Wang, Low, ToN 2006]
FAST TCP Design, analysis, experiments[Jin, Wei, Low, ToN 2007]
![Page 11: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/11.jpg)
Outline Review: homogeneous case Motivating experiments Model Equilibrium
Existence, uniqueness, local stability Slow timescale control
![Page 12: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/12.jpg)
Throughputs depend on AQM
FAST and Reno share a single bottleneck router NS2 simulation Router: DropTail with variable buffer size With 10% heavy-tailed noise traffic
FAST throughput
buffer size = 80 pkts buffer size = 400 pkts
![Page 13: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/13.jpg)
Multiple equilibria: throughput depends on arrival
eq 1 eq 2Path 1 52M 13Mpath 2 61M 13Mpath 3 27M 93M
eq 1
eq 2
Tang, Wang, Hegde, Low, Telecom Systems, 2005
Dummynet experiment
![Page 14: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/14.jpg)
eq 1 eq 2Path 1 52M 13Mpath 2 61M 13Mpath 3 27M 93M
Tang, Wang, Hegde, Low, Telecom Systems, 2005
eq 1
eq 2 eq 3 (unstable)
Dummynet experiment
Multiple equilibria: throughput depends on arrival
![Page 15: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/15.jpg)
Some implications
homogeneous heterogeneous
equilibrium unique non-uniquebandwidthallocation on AQM
independent dependent
bandwidthallocationon arrival
independent dependent
![Page 16: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/16.jpg)
Duality model: , ***
lilliii xpRFxcRxxU iix
s.t. )( max0
llliii
i
iii pRx
TxF
Why can’t use Fi’s of FAST and Reno in duality model?
l
llii
ii pRx
TF
21 2
2
delay for FAST
loss for Reno
They use different prices!
![Page 17: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/17.jpg)
Duality model: , ***
lilliii xpRFxcRxxU iix
s.t. )( max0
llliii
i
iii pRx
TxF
Why can’t use Fi’s of FAST and Reno in duality model?
l
llii
ii pRx
TF
21 2
2
They use different prices!
ilili
ll ctxR
cp )(1
iililll txRtpgp )(),(
![Page 18: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/18.jpg)
F1
FN
G1
GL
R
RT
TCP Network AQM
x y
q p
iililll
il
lliii
txRtpGtp
txtpRFtx
)( ),( )1(
)( ,)( )1(same price
for all sources
Homogeneous protocol
![Page 19: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/19.jpg)
F1
FN
G1
GL
R
RT
TCP Network AQM
x y
q p
)( ,)( )1(
)( ,)( )1(
txtpmRFtx
txtpRFtx
ji
ll
jlli
ji
ji
il
lliii heterogeneousprices for
type j sources
Heterogeneous protocol
![Page 20: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/20.jpg)
Heterogeneous protocols Equilibrium: p that satisfies
i,j ll
lji
jlil
ll
jlli
ji
ji
pcc
pxRpy
pmRfpx
0 if
)( : )(
)( )(
Duality model no longer applies ! pl can no longer serve as Lagrange
multiplier
![Page 21: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/21.jpg)
Heterogeneous protocols Equilibrium: p that satisfies
i,j ll
lji
jlil
ll
jlli
ji
ji
pcc
pxRpy
pmRfpx
0 if
)( : )(
)( )(
Need to re-examine all issues Equilibrium: exists? unique? efficient? fair? Dynamics: stable? limit cycle? chaotic? Practical networks: typical behavior? design
guidelines?
![Page 22: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/22.jpg)
Heterogeneous protocols Equilibrium: p that satisfies
i,j ll
lji
jlil
ll
jlli
ji
ji
pcc
pxRpy
pmRfpx
0 if
)( : )(
)( )(
Dynamic: dual algorithm
llll
ll
jlli
ji
ji
ctpyp
tpmRftpx
))((
))(( ))((
![Page 23: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/23.jpg)
Notation Simpler notation: p is equilibrium if
on bottleneck links
Jacobian:
Linearized dual algorithm:
cpy )(
)( : )( ppyp
J
p(t)pp )( *J
See Simsek, Ozdaglar, Acemoglu 2005for generalization
![Page 24: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/24.jpg)
Outline Review: homogeneous case Motivating experiments Model Equilibrium
Existence, uniqueness, local stability Slow timescale control
Tang, Wang, Low, Chiang. Infocom 2005
![Page 25: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/25.jpg)
ExistenceTheoremEquilibrium p exists, despite lack of
underlying utility maximization
Generally non-unique There are networks with unique bottleneck
set but infinitely many equilibria There are networks with multiple bottleneck
set each with a unique (but distinct) equilibrium
![Page 26: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/26.jpg)
Regular networksDefinitionA regular network is a tuple (R, c, m, U) for
which all equilibria p are locally unique, i.e.,
Theorem Almost all networks are regular A regular network has finitely many and
odd number of equilibria (e.g. 1)
Proof: Sard’s theorem and index theorem
0 )(det : )(det
ppypJ
![Page 27: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/27.jpg)
Global uniqueness
Corollary If price mapping functions ml
j are linear and link-independent, then equilibrium is globally unique
e.g. a network of RED routers almost always has globally unique equilibrium
Theorem If price heterogeneity is small, then equilibrium is
globally unique
0any for ]2,[
0any for ]2,[/1
/1
jjLjj
l
llL
lj
l
aaam
aaam
![Page 28: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/28.jpg)
Local stability:`uniqueness’ stability
Theorem If price heterogeneity is small, then the unique
equilibrium p is locally stable
0any for ]2,[
0any for ]2,[/1
/1
jjLjj
l
llL
lj
l
aaam
aaam
p(t)pp )( *JLinearized dual algorithm:Equilibrium p is locally stable if
0 )( Re pJ
![Page 29: Heterogeneous Congestion Control Protocols](https://reader035.vdocuments.net/reader035/viewer/2022062521/5681584e550346895dc5aaea/html5/thumbnails/29.jpg)
Local stability:`converse’
Theorem If all equilibria p are locally stable, then it is globally
unique
LpI )1( )( Proof idea: For all equilibrium p: Index theorem:
L
p
pI )1( )( eq