Heterogeneous Congestion Control Protocols
Steven Low
CS, EEnetlab.CALTECH.edu
with A. Tang, J. Wang, D. Wei, CaltechM. Chiang, Princeton
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
F1
FN
G1
GL
R
RT
TCP Network AQM
x y
q p
))( ),(( )1(
))( ),(( )1(
tRxtpGtp
txtpRFtx T
Reno, Vegas
DT, RED, …
liRli link uses source if 1 IP routing
Network model
for every RTT{ W += 1 }for every loss{ W := W/2 }
Reno: (AI)
(MD)Jacobson
1989
ililill
llli
i
ii
tptxRGtp
tpRx
Ttx
)(),()1(
)(2
1)1(
2
2
AI
MD
TailDrop
Network model: example
FAST:Jin, Wei, Low
2004 W
RTT
baseRTT :W
periodically{
}
ilili
lll
llliii
i
iii
ctxRc
tptp
tpRtxT
txtx
)(1
)()1(
)()()()1(
Network model: example
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 )( max0
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
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 )( max0
Kelly, Maloo, Tan 1998Low, Lapsley 1999
The underlying concave program also leads to simple dynamic behavior
Duality model of TCP/AQM
Equilibrium (x*,p*) primal-dual optimal:
cRxxU iix
subject to )( max0
Mo & Walrand 2000:
1 if )1(
1 if log)(
11
i
i
iix
xxU
Vegas, FAST, STCP
HSTCP Reno XCP (single link
only)
Low 2003
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]
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]
Outline
Review: homogeneous case Motivating experiments Model Equilibrium
Existence, uniqueness, local stability Slow timescale control
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
Multiple equilibria: throughput depends on arrival
eq 1 eq 2
Path 1 52M 13M
path 2 61M 13M
path 3 27M 93M
eq 1
eq 2
Tang, Wang, Hegde, Low, Telecom Systems, 2005
Dummynet experiment
eq 1 eq 2
Path 1 52M 13M
path 2 61M 13M
path 3 27M 93MTang, Wang, Hegde, Low, Telecom Systems, 2005
eq 1
eq 2 eq 3 (unstable)
Dummynet experiment
Multiple equilibria: throughput depends on arrival
Some implications
homogeneous heterogeneous
equilibrium unique non-unique
bandwidthallocation on AQM
independent dependent
bandwidthallocationon arrival
independent dependent
Duality model:
, ***
lilliii xpRFxcRxxU ii
x
s.t. )( max
0
llliii
i
iii pRx
TxF
Why can’t use Fi’s of FAST and Reno in duality model?
l
llii
ii pR
x
TF
2
1 2
2
delay for FAST
loss for Reno
They use different prices!
Duality model:
, ***
lilliii xpRFxcRxxU ii
x
s.t. )( max
0
llliii
i
iii pRx
TxF
Why can’t use Fi’s of FAST and Reno in duality model?
l
llii
ii pR
x
TF
2
1 2
2
They use different prices!
ilili
ll ctxR
cp )(
1
iililll txRtpgp )(),(
F1
FN
G1
GL
R
RT
TCP Network AQM
x y
q p
iililll
il
lliii
txRtpGtp
txtpRFtx
)( ),( )1(
)( ,)( )1(
same pricefor all sources
Homogeneous protocol
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
Heterogeneous protocols
Equilibrium: p that satisfies
i,j ll
lji
jlil
ll
jlli
ji
ji
pc
cpxRpy
pmRfpx
0 if
)( : )(
)( )(
Duality model no longer applies ! pl can no longer serve as Lagrange
multiplier
Heterogeneous protocols
Equilibrium: p that satisfies
i,j ll
lji
jlil
ll
jlli
ji
ji
pc
cpxRpy
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?
Heterogeneous protocols
Equilibrium: p that satisfies
i,j ll
lji
jlil
ll
jlli
ji
ji
pc
cpxRpy
pmRfpx
0 if
)( : )(
)( )(
Dynamic: dual algorithm
llll
ll
jlli
ji
ji
ctpyp
tpmRftpx
))((
))(( ))((
Notation
Simpler notation: p is equilibrium if
on bottleneck links
Jacobian:
Linearized dual algorithm:
cpy )(
)( : )( pp
yp
J
p(t)pp )( *J
See Simsek, Ozdaglar, Acemoglu 2005for generalization
Outline
Review: homogeneous case Motivating experiments Model Equilibrium
Existence, uniqueness, local stability Slow timescale control
Tang, Wang, Low, Chiang. Infocom 2005
Existence
Theorem
Equilibrium 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
Regular networks
Definition
A 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
pp
ypJ
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
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