Optimization Models for Heterogeneous Protocols
Steven Low
CS, EEnetlab.CALTECH.edu
with J. Doyle, S. Hegde, L. Li, A. Tang, J. Wang, ClatechM. Chiang, Princeton
oInternet protocolsoHorizontal decompositionn TCP-AQMnSome implications
oVertical decompositionn TCP/IP, HTTP/TCP, TCP/wireless, …
oHeterogeneous protocols
Outline
TCP/AQM
IP
Link
Application
cRx
xUi
iix
≤
∑≥
tosubj
)( max0
Internet Protocols
TCP/AQM
IP
xi(t)
pl(t)
Link
Application o Protocols determines network behavioro Critical, yet difficult, to understand and
optimizeo Local algorithms, distributed spatially
and vertically à global behavioro Designed separately, deployed
asynchronously, evolves independently
Internet Protocols
TCP/AQM
IP
xi(t)
pl(t)
Link
Application o Protocols determines network behavioro Critical, yet difficult, to understand and
optimizeo Local algorithms, distributed spatially
and vertically à global behavioro Designed separately, deployed
asynchronously, evolves independently
Need to reverse engineer
… to understand stack and network as whole
Internet Protocols
TCP/AQM
IP
xi(t)
pl(t)
Link
Application o Protocols determines network behavioro Critical, yet difficult, to understand and
optimizeo Local algorithms, distributed spatially
and vertically à global behavioro Designed separately, deployed
asynchronously, evolves independently
Need to reverse engineer
… to forward engineer new large networks
Internet Protocols
Minimize path costs (IP)
TCP/AQM
IP
xi(t)
pl(t)
Maximize utility (TCP/AQM)
Link Minimize SIR, max capacities, …
Application Minimize response time (web layout)
o Each layer is abstracted as an optimization problem
oOperation of a layer is a distributed solutiono Results of one problem (layer) are parameters of
othersoOperate at different timescales
Internet Protocols
TCP/AQM
IP
Link
Application
cRx
xUi
iix
≤
∑≥
tosubj
)( max0
o Each layer is abstracted as an optimization problem
oOperation of a layer is a distributed solutiono Results of one problem (layer) are parameters of
othersoOperate at different timescales
TCP/AQM
IP
Link
Application
1) Understand each layer in isolation, assumingother layers are designed nearly optimally2) Understand interactions across layers3) Incorporate additional layers4) Ultimate goal: entire protocol stack as solving one giant optimization problem, where individual layers are solving parts of it
Protocol Decomposition
Layering as Optimization Decomposition
o Layering as optimization decompositionn Network NUM problemn Layers Subproblemsn Layering Decomposition methodsn Interface Primal or dual variables
o Enables a systematic study of: n Network protocols as distributed solutions to global
optimization problemsn Inherent tradeoffs of layeringn Vertical vs. horizontal decomposition
(M. Chiang, Princeton)
oInternet protocolsoHorizontal decompositionn TCP-AQMnSome implications
oVertical decompositionn TCP/IP, HTTP/TCP, TCP/wireless, …
oHeterogeneous protocols
Outline
TCP/AQM
IP
Link
Application
cRx
xUi
iix
≤
∑≥
tosubj
)( max0
Network model
F1
FN
G1
GL
R
RT
TCP Network AQM
x y
q p
))( ),(( )1())( ),(( )1(
tRxtpGtptxtpRFtx T
=+=+ Reno, Vegas
DT, RED, …
liR li link uses source if 1= IP routing
Network model: example
))( ),(( )1())( ),(( )1(
tRxtpGtptxtpRFtx T
=+=+ Reno, Vegas
DT, RED, …
liR li link uses source if 1= IP routing
=+
−=+
∑
∑
ililill
llli
i
ii
tptxRGtp
tpRx
Ttx
)(),()1(
)(2
1)1(
2
2
TCP Reno: currently deployed TCP
AI
MD
TailDrop
Network model: example
))( ),(( )1())( ),(( )1(
tRxtpGtptxtpRFtx T
=+=+ Reno, Vegas
DT, RED, …
liR li link uses source if 1= IP routing
−+=+
−+=+
∑
∑
ilili
lll
llliii
i
iii
ctxRc
tptp
tpRtxT
txtx
)(1
)()1(
)()()()1( αγ
TCP FAST: high speed version of Vegas
n TCP-AQM:
),(
),( ***
***
RxpGp
pRxFx T
=
=
Duality model
n Equilibrium (x*,p*) primal-dual optimal:
n F determines utility function UnG determines complementary slackness conditionn p* are Lagrange multipliers
Uniqueness of equilibriumn x* is unique when U is strictly concaven p* is unique when R has full row rank
cRxxU ii
ix
≤∑≥
t.s. )( max0
n TCP-AQM:
Duality model
n Equilibrium (x*,p*) primal-dual optimal:
n F determines utility function UnG determines complementary slackness conditionn p* are Lagrange multipliers
The underlying concave program also leads to simple dynamic behavior
cRxxU ii
ix
≤∑≥
t.s. )( max0
),(
),( ***
***
RxpGp
pRxFx T
=
=
Duality modeln Global stability in absence of feedback delayn Lyapunov functionn Kelly, Maulloo & Tan (1988)
n Gradient projectionn Low & Lapsley (1999)
n Singular perturbationsn Kunniyur & Srikant (2002)
n Passivity approachnWen & Arcat (2004)
n Linear stability in presence of feedback delayn Nyquist criterian Paganini, Doyle, Low (2001), Vinnicombe (2002), Kunniyur
& Srikant (2003)
n Global stability in presence of feedback delayn Lyapunov-Krasovskii, SoSTooln Papachristodoulou (2005)
n Global nonlinear invariance theoryn Ranjan, La & Abed (2004, delay-independent)
Duality model
n Equilibrium (x*,p*) primal-dual optimal:
cRxxU ii
ix
≤∑≥
t.s. )( max0
n α = 1 : Vegas, FAST, STCP n α = 1.2: HSTCP (homogeneous sources)
n α = 2 : Reno (homogeneous sources)
n α = infinity: XCP (single link only)
=≠−
=−
1 if log1 if )1(
)(1
ααα α
i
iii x
xxU (Mo & Walrand 00)
oInternet protocolsoHorizontal decompositionn TCP-AQMnSome implications
oVertical decompositionn TCP/IP, HTTP/TCP, TCP/wireless, …
oHeterogeneous protocols
Outline
TCP/AQM
IP
Link
Application
cRx
xUi
iix
≤
∑≥
tosubj
)( max0
FAST Architecture
Each componento designed independentlyo upgraded asynchronously
Is large queue necessary for high throughput?
Efficient and Friendly
o Fully utilizing bandwidth
o Does not disrupt interactive applications
One-way delay Requirement for VoIP
o DSL upload (max upload capacity 512kbps)o Latency: 10ms
Resilient to Packet Loss
o Lossy link, 10Mbpso Latency: 50ms
Current TCP collapsesat packet loss rate bigger than a few %!
FAST
max
Implications
n Is fair allocation always inefficient ?
n Does raising capacity always increase throughput ?
Intricate and surprising interactions in large-scalenetworks … unlike at single link
Implications
n Is fair allocation always inefficient ?
n Does raising capacity always increase throughput ?
Intricate and surprising interactions in large-scalenetworks … unlike at single link
Daulity model
n α = 1 : Vegas, FAST, STCP n α = 1.2: HSTCP (homogeneous sources)
n α = 2 : Reno (homogeneous sources)
n α = infinity: XCP (single link only)
cRx
xU ii
ix
≤
∑≥
t.s.
)( max0
=≠−
=−
1 if log1 if )1(
)(1
ααα α
i
iii x
xxU
(Mo, Walrand 00)
Fairness
n α = 0: maximum throughput n α = 1: proportional fairness n α = 2: min delay fairness n α = infinity: maxmin fairness
cRx
xU ii
ix
≤
∑≥
t.s.
)( max0
=≠−
=−
1 if log1 if )1(
)(1
ααα α
i
iii x
xxU
(Mo, Walrand 00)
Fairness
n Identify allocation with αn An allocation is fairer if its α is larger
cRx
xU ii
ix
≤
∑≥
t.s.
)( max0
(Mo, Walrand 00)
=≠−
=−
1 if log1 if )1(
)(1
ααα α
i
iii x
xxU
Efficiency: aggregate throughput
n Unique optimal rate x(α) n An allocation is efficient if T(α) is large
cRx
xU ii
ix
≤
∑≥
t.s.
)( max0
=≠−
=−
1 if log1 if )1(
)(1
ααα α
i
iii x
xxU
∑=i
ixT )(:)( throughput αα
Conjecture
ConjectureT(α) is nonincreasing
i.e. a fair allocation is always inefficient
Example 1
ConjectureT(α) is nonincreasing
i.e. a fair allocation is always inefficient
1=lc0
max throughput
1
1/(L+1)
proportionalfairness
L/L(L+1)
)()1( )0( ∞>> TTT
1/2
maxminfairness
1/2
Example 1
ConjectureT(α) is nonincreasing
i.e. a fair allocation is always inefficient
1/1
/1
+α
α
LL
11
/1 +αL1=lc
Example 2
21 cc ≤
−+++= 21
22
21213
2)1( ccccccT
2)( 2
1
ccT +=∞
)()1( ∞>⇒ TT
ConjectureT(α) is nonincreasing
i.e. a fair allocation is always inefficient
Example 3
↓)(αT
ConjectureT(α) is nonincreasing
i.e. a fair allocation is always inefficient
Intuition
“The fundamental conflict between achieving flow fairness and maximizing overall system throughput….. The basic issue is thus the trade-off between these two conflicting criteria.”
Luo,etc.(2003), ACM MONET
Results
oTheorem: Necessary & sufficient condition for general networks (R, c) provided every link has a 1-link flow
oCorollary 1: true if N(R)=1
1/1
/1
+α
α
LL
11
/1 +αL1=lc
Results
oTheorem: Necessary & sufficient condition for general networks (R, c) provided every link has a 1-link flow
oCorollary 1: true if N(R)=1
21 cc ≤)()1( ∞>⇒ TT
Results
oTheorem: Necessary & sufficient condition for general networks (R, c) provided every link has a 1-link flow
oCorollary 2: true ifnN(R)=2n2 long flows pass through same# links
Counter-example
o Theorem: Given any α0>0, there exists network where
oCompact example
0α
0 allfor 0 ααα
>>ddT
Counter-example
oThere exists a network such thatdT/dα > 0 for almost all α>0
oIntuitionnLarge α favors expensive flowsnLong flows may not be expensive
oMax-min may be more efficient than proportional fairness
expensive long
Efficiency: aggregate throughput
n Unique optimal rate x(α; c) n An allocation is efficient if T(α ; c) is large
cRx
xU ii
ix
≤
∑≥
t.s.
)( max0
=≠−
=−
1 if log1 if )1(
)(1
ααα α
i
iii x
xxU
∑=i
i cxcT );(:);( throughput αα
Throughput & capacity
oIntuition: Increasing link capacities always raises throughput T
oTheorem: Necessary & sufficient condition for general networks (R, c)
oCorollary: For all α, increasingn a link’s capacity can reduce Tn all links’ capacities equally can reduce Tn all links’ capacities proportionally raises T
Throughput & capacity
n
n
cRx
xU ii
ix
≤
∑≥
t.s.
)( max0
=≠−
=−
1 if log1 if )1(
)(1
ααα α
i
iii x
xxU
∑=i
i cxcT );(:);( Throughput αα
δε
εδααδα
ε cxcTcT
DT T
∂∂
=+−
=→
1 ),();(
lim:);(0
Throughput & capacity
n
n
∑=i
i cxcT );(:);( Throughput αα
δε
εδααδα
ε cxcTcT
DT T
∂∂
=+−
=→
1 ),();(
lim:);(0
Corollary:
oGiven any α0>0, there exists network (R, c) s.t. for all α>α0
nDT(α; e1) < 0 for some lnDT(α; 1) < 0
oFor all networks (R, c) and for all α>0nDT(α; c) > 0
oInternet protocolsoHorizontal decompositionn TCP-AQMnSome implications
oVertical decompositionn TCP/IP, HTTP/TCP, TCP/wireless, …
oHeterogeneous protocols
Outline
TCP/AQM
IP
Link
Application
cRx
xUi
iix
≤
∑≥
tosubj
)( max0
Protocol decomposition
TCP-AQM:n TCP algorithms maximize utility with different
utility functions
Congestion prices coordinate across protocol layers
BccRx
xU
T
iii
xRc
≤≤
∑≥
α , subject to
)( maxmax max0
TCP-AQM
IP
TCP/AQM
Applications
Link
Protocol decomposition
TCP-AQMIP
BccRx
xU
T
iii
xRc
≤≤
∑≥
α , subject to
)( maxmax max0
TCP-AQM
IP
TCP/AQM
Applications
Link
TCP/IP:n TCP algorithms maximize utility with different
utility functionsn Shortest-path routing is optimal using congestion
prices as link costs ……
Congestion prices coordinate across protocol layers
Two timescales
n Instant convergence of TCP/IPn Link cost = a pl(t) + b dl
n Shortest path routing R(t)price
static
TCP/AQM
IPR(0)
a p(0)
R(1)
a p(1)
… R(t), R(t+1) ,…
TCP-AQM/IP Model
TCP
cxtR
xUtxi
iix
≤
= ∑≥
)( subject to
)( max arg )(0
∑
∑ ∑
+
−=
≥≥
lll
i llliiiixp
pc
ptRxxUtpi
)()(max min arg )(00
AQM
))((minarg)1( lll
liRi bdtapRtRli
+=+ ∑IP
Link cost
Questions
nDoes equilibrium routing Ra exist ?nWhat is utility at Ra?nIs Ra stable ? nCan it be stabilized?
TCP/AQM
IPR(0)
a p(0)
R(1)
a p(1)
… R(t), R(t+1) ,…
Equilibrium routing
Theorem1. If b=0, Ra exists iff zero duality gapn Shortest-path routing is optimal with
congestion pricesn No penalty for not splitting
+
−
≤
∑∑ ∑
∑
≥≥
≥
ll
li l
lliRiiixp
iii
xR
cppRxxU
cRxxU
ii
max)( max min
subject to )( maxmax
00
0
:Dual
:Primal
Protocol decomposition
TCP-AQMIP
TCP/IP (with fixed c):n Equilibrium of TCP/IP exists iff zero duality gapn NP-hard, but subclass with zero duality gap is P n Equilibrium, if exists, can be unstablen Can stabilize, but with reduced utility
Inevitable tradeoff bw utility max & routing stability
BccRx
xU
T
iii
xRc
≤≤
∑≥
α , subject to
)( maxmax max0
TCP-AQM
IP
TCP/AQM
Applications
Link
Protocol decomposition
IPLink
TCP/IP with optimal c: n With optimal provisioning, static routing is optimal using
provisioning cost α as link costs
TCP/IP with static routing in well-designed network
TCP-AQMIP
BccRx
xU
T
iii
xRc
≤≤
∑≥
α , subject to
)( maxmax max0
TCP-AQM
IP
TCP/AQM
Applications
Link
Summary
BccRx
xU
T
iii
xRc
≤≤
∑≥
α , subject to
)( maxmax max0
TCP-AQMIPLink
n TCP algorithms maximize utility with different utility functions
n IP shortest path routing is optimal using congestion prices as link costs, with given link capacities c
n With optimal provisioning, static routing is optimal using provisioning cost α as link costs
Congestion prices coordinate across protocol layers
oInternet protocolsoHorizontal decompositionn TCP-AQMnSome implications
oVertical decompositionn TCP/IP, HTTP/TCP, TCP/wireless, …
oHeterogeneous protocols
Outline
TCP/AQM
IP
Link
Application
cRx
xUi
iix
≤
∑≥
tosubj
)( max0
Congestion control
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
Heterogeneous protocols
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 protocols
13M61Mpath 293M27Mpath 3
13M52MPath 1
eq 2eq 1eq 1
eq 2
Tang, Wang, Hegde, Low, Telecom Systems, 2005
Heterogeneous protocols
13M61Mpath 293M27Mpath 3
13M52MPath 1
eq 2eq 1eq 1
eq 2
Tang, Wang, Hegde, Low, Telecom Systems, 2005
eq 3 (unstable)
>=≤
=
∑
∑
0 if
)(
)(
, ll
lji
ji
jli
ll
jl
jli
ji
ji
pcc
pxR
pmRF(p)x
Multi-protocol: J>1
Duality model no longer applies !n pl can no longer serve as Lagrange
multiplier
n TCP-AQM equilibrium p:
n TCP-AQM equilibrium p:
>=≤
=
∑
∑
0 if
)(
)(
, ll
lji
ji
jli
ll
jl
jli
ji
ji
pcc
pxR
pmRF(p)x
Multi-protocol: J>1
Need to re-examine all issuesn Equilibrium: exists? unique? efficient? fair?n Dynamics: stable? limit cycle? chaotic?n Practical networks: typical behavior? design guidelines?
Results: existence of equilibrium
n Equilibrium p always exists despite lack of underlying utility maximization
n Generally non-uniquen Network with unique bottleneck set but
uncountably many equilibrian Network with non-unique bottleneck sets
each having unique equilibrium
Results: regular networks
nRegular networks: all equilibria p are locally unique
Theorem (Tang, Wang, Low, Chiang, Infocom 2005)
nAlmost all networks are regularnRegular networks have finitely many
and odd number of equilibria (e.g. 1)
Proof: Sard’s Theorem and Index Theorem
Results: global uniqueness
Theorem (Tang, Wang, Low, Chiang, Infocom 2005)
nIf all equilibria p have I(p) = (-1)L
then p is globally unique nIf all equilibria p all locally stable, then
it is globally unique
( )( )
><−
=
∂∂== ∑
0det if 1 0det if 1
: )(index
)(: )(:)(,
(p)(p)
pI
ppy
(p)pxRpy ji
ji
jli
JJ
J
Results: global uniqueness
Theorem (Tang, Wang, Low, Chiang, Infocom 2005)
nFor J=1, equilibrium p is globally unique if R is full rank (Mo & Walrand ToN 2000)
nFor J>1, equilibrium p is globally unique if J(p) is negative definite over a certain set
)(: )(:)(,
ppy
(p)pxRpy ji
ji
jli ∂
∂== ∑ J
Results: global uniqueness
Theorem (Tang, Wang, Low, Chiang, Infocom 2005)
nIf price mapping functions mlj are `similar’,
then equilibrium p is globally unique
nIf price mapping functions mlj are linear and
link-independent, then equilibrium p is globally unique
0any for ]2,[
0any for ]2,[/1
/1
>∈
>∈jjLjj
l
llL
lj
l
aaam
aaam
&
&
Summary: equilibrium structure
Uni-protocolnUnique bottleneck
setn Unique rates &
prices
Multi-protocoln Non-unique bottleneck
sets n Non-unique rates &
prices for each B.S.
n always oddn not all stablen uniqueness
conditions
Experiments: non-uniqueness
Discovered examples guided by theoryn Numerical examples n NS2 simulations (Reno + Vegas)n DummyNet experiments (Reno + FAST)n Practical network??