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Optimization Models for
Heterogeneous Protocols
Steven Low
CS, EE
netlab.CALTECH.edu
with J. Doyle, S. Hegde, L. Li, A. Tang, J. Wang, Clatech
M. Chiang, Princeton
Outline
Internet protocols
Horizontal decomposition
 TCP-AQM
 Some implications
Vertical decomposition
 TCP/IP, HTTP/TCP, TCP/wireless, …
Heterogeneous protocols
Application
TCP/AQM
IP
Link
max
x 0
U ( x )
i
i
subj to Rx  c
i
Internet Protocols
pl(t)
xi(t)
Application
TCP/AQM
IP
Link
 Protocols determines network behavior
 Critical, yet difficult, to understand and
optimize
 Local algorithms, distributed spatially
and vertically  global behavior
 Designed separately, deployed
asynchronously, evolves independently
Internet Protocols
pl(t)
xi(t)
Application
TCP/AQM
IP
Link
 Protocols determines network behavior
Need

Critical,to
yetreverse
difficult, to engineer
understand and
optimize
 Local algorithms, distributed spatially
…and
tovertically
understand
stack and
 global behavior

Designed separately,
deployed
network
as whole
asynchronously, evolves independently
Internet Protocols
pl(t)
xi(t)
Application
TCP/AQM
IP
Link
 Protocols determines network behavior
Need

Critical,to
yetreverse
difficult, to engineer
understand and
optimize
 Local algorithms, distributed spatially
…and
tovertically
forward
engineer
 global
behavior

Designed
separately,
deployed
new
large
networks
asynchronously, evolves independently
Internet Protocols
pl(t)
xi(t)
Application
Minimize response time (web layout)
TCP/AQM
Maximize utility (TCP/AQM)
IP
Minimize path costs (IP)
Link
Minimize SIR, max capacities, …
Internet Protocols
 Each layer is abstracted as an optimization
problem
 Operation of a layer is a distributed solution
 Results of one problem (layer) are parameters of
others
 Operate at different timescales
Application
TCP/AQM
IP
Link
max
x 0
U ( x )
i
i
subj to Rx  c
i
Protocol Decomposition
 Each layer is abstracted as an optimization
problem
 Operation of a layer is a distributed solution
 Results of one problem (layer) are parameters of
others
 Operate at different timescales
Application
TCP/AQM
IP
Link
1) Understand each layer in isolation, assuming
other layers are designed nearly optimally
2) Understand interactions across layers
3) Incorporate additional layers
4) Ultimate goal: entire protocol stack as solving
one giant optimization problem, where individual
layers are solving parts of it
Layering as Optimization Decomposition
 Layering as optimization decomposition




Network
Layers
Layering
Interface
NUM problem
Subproblems
Decomposition methods
Primal or dual variables
 Enables a systematic study of:



Network protocols as distributed solutions to global
optimization problems
Inherent tradeoffs of layering
Vertical vs. horizontal decomposition
(M. Chiang, Princeton)
Outline
Internet protocols
Horizontal decomposition
 TCP-AQM
 Some implications
Vertical decomposition
 TCP/IP, HTTP/TCP, TCP/wireless, …
Heterogeneous protocols
Application
TCP/AQM
IP
Link
max
x 0
U ( x )
i
i
subj to Rx  c
i
Network model
x
y
R
F1
Network
TCP
G1
FN
q
AQM
GL
R
T
p
Rli  1 if source i uses link l
IP routing
x(t  1)  F ( RT p(t ), x(t ))
p(t  1)  G ( p(t ), Rx (t ))
Reno, Vegas
DT, RED, …
Network model: example
TCP Reno: currently deployed TCP
AI
xi2
1
xi (t  1)  2 
Ti
2
R
li
pl (t )
MD
l


pl (t  1)  Gl   Rli xi (t ), pl (t ) 
 i

TailDrop
Rli  1 if source i uses link l
IP routing
x(t  1)  F ( RT p(t ), x(t ))
p(t  1)  G ( p(t ), Rx (t ))
Reno, Vegas
DT, RED, …
Network model: example
TCP FAST: high speed version of Vegas
i 

xi (t  1)  xi (t )    i  xi (t ) Rli pl (t ) 
Ti 
l

1

pl (t  1)  p l (t )    Rli xi (t )  cl 
cl  i

Rli  1 if source i uses link l
IP routing
x(t  1)  F ( RT p(t ), x(t ))
p(t  1)  G ( p(t ), Rx (t ))
Reno, Vegas
DT, RED, …
Duality model
 TCP-AQM:
 Equilibrium (x*,p*) primal-dual optimal:
 F determines utility function U
 G determines complementary slackness condition
 p* are Lagrange multipliers
Uniqueness of equilibrium
 x* is unique when U is strictly concave
 p* is unique when R has full row rank
Duality model
 TCP-AQM:
 Equilibrium (x*,p*) primal-dual optimal:
 F determines utility function U
 G determines complementary slackness condition
 p* are Lagrange multipliers
The underlying concave program also
leads to simple dynamic behavior
Duality model
 Global stability in absence of feedback delay
 Lyapunov function
 Kelly, Maulloo & Tan (1988)
 Gradient projection
 Low & Lapsley (1999)
 Singular perturbations
 Kunniyur & Srikant (2002)
 Passivity approach
 Wen & Arcat (2004)
 Linear stability in presence of feedback delay
 Nyquist criteria
 Paganini, Doyle, Low (2001), Vinnicombe (2002), Kunniyur
& Srikant (2003)
 Global stability in presence of feedback delay
 Lyapunov-Krasovskii, SoSTool
 Papachristodoulou (2005)
 Global nonlinear invariance theory
 Ranjan, La & Abed (2004, delay-independent)
Duality model
 Equilibrium (x*,p*) primal-dual optimal:
(Mo & Walrand 00)




  1 : Vegas, FAST, STCP
  1.2: HSTCP (homogeneous sources)
  2 : Reno (homogeneous sources)
  infinity: XCP (single link only)
Outline
Internet protocols
Horizontal decomposition
 TCP-AQM
 Some implications
Vertical decomposition
 TCP/IP, HTTP/TCP, TCP/wireless, …
Heterogeneous protocols
Application
TCP/AQM
IP
Link
max
x 0
U ( x )
i
i
subj to Rx  c
i
FAST Architecture
Each component
 designed independently
 upgraded asynchronously
Is large queue necessary for
high throughput?
Efficient and
Friendly
One-way delay
Requirement for VoIP


DSL upload (max upload capacity 512kbps)
Latency: 10ms
 Fully utilizing
bandwidth
 Does not disrupt
interactive
applications
Resilient to
Packet Loss
Current TCP collapses
at packet loss rate
bigger than a few %!


Lossy link, 10Mbps
Latency: 50ms
Implications
 Is fair allocation always inefficient ?
 Does raising capacity always
increase throughput ?
Intricate and surprising interactions in large-scale
networks … unlike at single link
Implications
 Is fair allocation always inefficient ?
 Does raising capacity always
increase throughput ?
Intricate and surprising interactions in large-scale
networks … unlike at single link
Daulity model
(Mo, Walrand 00)
   1 : Vegas, FAST, STCP
   1.2: HSTCP (homogeneous sources)
   2 : Reno (homogeneous sources)
   infinity: XCP (single link only)
Fairness
(Mo, Walrand 00)
   0: maximum throughput
   1: proportional fairness
   2: min delay fairness
   infinity: maxmin fairness
Fairness
(Mo, Walrand 00)
 Identify allocation with 
 An allocation is fairer if its  is larger
Efficiency: aggregate throughput
 Unique optimal rate x()
 An allocation is efficient if T() is large
Conjecture
Conjecture
T() is nonincreasing
i.e. a fair allocation is always inefficient
Example 1
max
throughput
proportional
fairness
maxmin
fairness
0
1/(L+1)
1/2
1
L/L(L+1)
1/2
Conjecture
T() is nonincreasing
i.e. a fair allocation is always inefficient
Example 1
Conjecture
T() is nonincreasing
i.e. a fair allocation is always inefficient
Results
Theorem: Necessary & sufficient
condition for general networks (R, c)
provided every link has a 1-link flow
Corollary 1: true if N(R)=1
Results
Theorem: Necessary & sufficient
condition for general networks (R, c)
provided every link has a 1-link flow
Corollary 1: true if N(R)=1
Results
Theorem: Necessary & sufficient
condition for general networks (R, c)
provided every link has a 1-link flow
Corollary 2: true if
 N(R)=2
 2 long flows pass through same# links
Counter-example
 Theorem: Given any
network where
0>0, there exists
Compact example
0
Counter-example
There exists a network such that
dT/d > 0 for almost all >0
Intuition
 Large  favors expensive flows
 Long flows may not be expensive
Max-min may be more efficient than
proportional fairness
expensive
long
Outline
Internet protocols
Horizontal decomposition
 TCP-AQM
 Some implications
Vertical decomposition
 TCP/IP, HTTP/TCP, TCP/wireless, …
Heterogeneous protocols
Application
TCP/AQM
IP
Link
max
x 0
U ( x )
i
i
subj to Rx  c
i
Protocol decomposition
Applications
TCP-AQM
TCP/
AQM
IP
Link
max max max
c
R
x 0
subject to
U ( x )
i
i
i
Rx  c,  c  B
T
TCP-AQM:
 TCP algorithms maximize utility with different
utility functions
Congestion prices coordinate across protocol layers
Protocol decomposition
Applications
IP
TCP/
AQM
IP
Link
TCP-AQM
max max max
c
R
x 0
subject to
U ( x )
i
i
i
Rx  c,  c  B
T
TCP/IP:
 TCP algorithms maximize utility with different
utility functions
 Shortest-path routing is optimal using congestion
prices as link costs ……
Congestion prices coordinate across protocol layers
Two timescales
 Instant convergence of TCP/IP
 Link cost = a pl(t) + b dl
price
 Shortest path routing R(t)
TCP/AQM
IP
a p(0)
a p(1)
R(0)
R(1)
static
…
…
R(t), R(t+1) ,
TCP-AQM/IP Model
TCP
x(t )  arg max
x0
U ( x )
i
i
i
subject to
p(t )  arg min
p 0
AQM
R(t ) x  c


U i ( xi )  xi  Rli (t ) pl 
i  max
xi  0
l

  cl pl
l
Link cost
IP
Ri (t  1) arg min  Rli (apl (t )  bdl )
Rli
l
Questions
Does equilibrium routing Ra exist ?
What is utility at Ra?
Is Ra stable ?
Can it be stabilized?
TCP/AQM
IP
a p(0)
a p(1)
R(0)
R(1)
…
…
R(t), R(t+1) ,
Equilibrium routing
Theorem
1. If b=0, Ra exists iff zero duality gap
 Shortest-path routing is optimal with
congestion prices
 No penalty for not splitting
Primal : max max
R
x 0
Dual :

min 
p 0

U ( x )
i
i
subject to Rx  c
i



U i ( xi )  xi max  Rli pl    pl cl 
i max
Ri
xi  0 
l
 l

Protocol decomposition
Applications
IP
TCP/
AQM
IP
Link
TCP-AQM
max max max
c
R
x 0
subject to
U ( x )
i
i
i
Rx  c,  c  B
T
TCP/IP (with fixed c):
 Equilibrium of TCP/IP exists iff zero duality gap
 NP-hard, but subclass with zero duality gap is P
 Equilibrium, if exists, can be unstable
 Can stabilize, but with reduced utility
Inevitable tradeoff bw utility max & routing stability
Protocol decomposition
Applications
Link
TCP/
AQM
IP
Link
IP
TCP-AQM
max max max
c
R
x 0
subject to
U ( x )
i
i
i
Rx  c,  c  B
T
TCP/IP with optimal c:
 With optimal provisioning, static routing is optimal using
provisioning cost  as link costs
TCP/IP with static routing in well-designed network
Summary
Link
IP
TCP-AQM
max max max
c
R
x 0
subject to
U ( x )
i
i
i
Rx  c,  c  B
T
 TCP algorithms maximize utility with different
utility functions
 IP shortest path routing is optimal using congestion prices
as link costs, with given link capacities c
 With optimal provisioning, static routing is optimal using
provisioning cost  as link costs
Congestion prices coordinate across protocol layers
Outline
Internet protocols
Horizontal decomposition
 TCP-AQM
 Some implications
Vertical decomposition
 TCP/IP, HTTP/TCP, TCP/wireless, …
Heterogeneous protocols
Application
TCP/AQM
IP
Link
max
x 0
U ( x )
i
i
subj to Rx  c
i
Congestion control
x
y
R
F1
Network
TCP
G1
FN
q
AQM
GL
R
T


xi (t  1)  Fi   Rli pl (t ), xi (t ) 
 l



pl (t  1)  Gl  pl (t ),  Rli xi (t ) 
i


p
same price
for all sources
Heterogeneous protocols
x
y
R
F1
Network
TCP
G1
FN
q
AQM
GL
R
T
p


Heterogeneous
xi (t  1)  Fi   Rli pl (t ), xi (t ) 
 l

prices for
 type j sources
j
j
j
j
xi (t  1)  Fi   Rli ml  pl (t ) , xi (t ) 
 l

Heterogeneous protocols
eq 2
eq 1
eq 2
Path 1
52M
13M
path 2
61M
13M
path 3
27M
93M
eq 1
Tang, Wang, Hegde, Low, Telecom Systems, 2005
Heterogeneous protocols
eq 2
eq 1
eq 3 (unstable)
eq 2
Path 1
52M
13M
path 2
61M
13M
path 3
27M
93M
eq 1
Tang, Wang, Hegde, Low, Telecom Systems, 2005
Multi-protocol: J>1
 TCP-AQM equilibrium p:
Duality model no longer applies !
 pl can no longer serve as Lagrange
multiplier
Multi-protocol: J>1
 TCP-AQM equilibrium p:
Need to re-examine all issues
 Equilibrium: exists? unique? efficient? fair?
 Dynamics: stable? limit cycle? chaotic?
 Practical networks: typical behavior? design guidelines?
Results: existence of equilibrium
 Equilibrium p always exists despite
lack of underlying utility maximization
 Generally non-unique
 Network with unique bottleneck set but
uncountably many equilibria
 Network with non-unique bottleneck sets
each having unique equilibrium
Results: regular networks
Regular networks: all equilibria p are
locally unique
Theorem (Tang, Wang, Low, Chiang, Infocom 2005)
Almost all networks are regular
Regular 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)
For J=1, equilibrium p is globally
unique if R is full rank (Mo & Walrand ToN 2000)
For J>1, equilibrium p is globally
unique if J(p) is negative definite over
a certain set
Results: global uniqueness
Theorem (Tang, Wang, Low, Chiang, Infocom 2005)
If price mapping functions mlj are `similar’,
then equilibrium p is globally unique
If price mapping functions mlj are linear and
link-independent, then equilibrium p is
globally unique
Summary: equilibrium structure
Uni-protocol
Unique bottleneck
set
 Unique rates &
prices
Multi-protocol
 Non-unique bottleneck
sets
 Non-unique rates &
prices for each B.S.
 always odd
 not all stable
 uniqueness
conditions