Optimization Models for Heterogeneous Protocols€¦ · Application 1) Understand each layer in...
Transcript of Optimization Models for Heterogeneous Protocols€¦ · Application 1) Understand each layer in...
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??