Heterogeneous Congestion
Control Protocols
Steven Low
CS, EE
netlab.CALTECH.edu
with A. Tang, J. Wang, D. Wei, Caltech
M. Chiang, Princeton
Outline




Review: homogeneous case
Motivating experiments
Model
Equilibrium
 Existence, uniqueness, local stability
 Slow timescale control
Tang, Wang, Low, Chiang. Infocom March 2005
Tang, Wang, Hegde, Low. Telecommunications Systems, Dec 2005
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 ))
Reno, Vegas
p(t  1)  G ( p(t ), Rx (t ))
DT, RED, …
Network model: example
Reno:
Jacobson
1989
for
{
for
{
every RTT
W += 1
}
every loss
W := W/2
}
(AI)
(MD)
AI
2
i
x
1
xi (t  1)  2 
Ti
2
R
li
pl (t )
MD
l


pl (t  1)  Gl   Rli xi (t ), pl (t ) 
 i

TailDrop
Network model: example
FAST:
Jin, Wei, Low
2004
periodically
{
baseRTT
W :
W  
RTT
}
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

Duality model of TCP/AQM
 TCP/AQM
x*  F ( RT p * , x* )
p*  G ( p* , Rx * )
 Equilibrium (x*,p*) primal-dual optimal:
max U i ( xi )
subject to Rx  c
x 0
 F determines utility function U
 G guarantees complementary slackness
Kelly, Maloo, Tan 1998
 p* are Lagrange multipliers
Low, Lapsley 1999
Uniqueness of equilibrium
 x* is unique when U is strictly concave
 p* is unique when R has full row rank
Duality model of TCP/AQM
 TCP/AQM
x*  F ( RT p * , x* )
p*  G ( p* , Rx * )
 Equilibrium (x*,p*) primal-dual optimal:
max U i ( xi )
subject to Rx  c
x 0
 F determines utility function U
 G guarantees complementary slackness
Kelly, Maloo, Tan 1998
 p* are Lagrange multipliers
Low, Lapsley 1999
The underlying concave program also
leads to simple dynamic behavior
Duality model of TCP/AQM
 Equilibrium (x*,p*) primal-dual optimal:
max
x 0
U ( x )
i
i
subject to
Rx  c
Mo & Walrand 2000:
log xi
U i ( xi )  
(1   ) 1 xi1




1 :
  1.2:
2 :
 :
if   1
if   1
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 throughput
buffer size = 80 pkts




buffer size = 400 pkts
FAST and Reno share a single bottleneck router
NS2 simulation
Router: DropTail with variable buffer size
With 10% heavy-tailed noise traffic
Multiple equilibria:
throughput depends
on arrival
Dummynet experiment
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
Multiple equilibria:
throughput depends
on arrival
Dummynet experiment
eq 2
eq 1
eq 2
Path 1
52M
13M
path 2
61M
13M
path 3
27M
93M
eq 3 (unstable)
eq 1
Tang, Wang, Hegde, Low, Telecom Systems, 2005
Some implications
homogeneous
heterogeneous
equilibrium
unique
non-unique
bandwidth
allocation
on AQM
independent
dependent
bandwidth
allocation
on arrival
independent
dependent
 Duality model:
max U i ( xi ) s.t. Rx  c
x 0

* *
x  Fi   Rli pl , xi 
 l

*
i
 Why can’t use Fi’s of FAST and Reno in
duality model?
They use different prices!
i 

Fi  xi   i  xi  Rli pl 
Ti 
l

delay for FAST
xi2
1
Fi  2 
Ti
2
loss for Reno
R
li
l
pl
 Duality model:
max U i ( xi ) s.t. Rx  c
x 0

* *
x  Fi   Rli pl , xi 
 l

*
i
 Why can’t use Fi’s of FAST and Reno in
duality model?
They use different prices!
i 

Fi  xi   i  xi  Rli pl 
Ti 
l

1

p l    Rli xi (t )  cl 
cl  i

xi2
1
Fi  2 
Ti
2


p l  g l  pl (t ),  Rli xi (t ) 
i


R
li
l
pl
Homogeneous protocol
x
y
R
F1
Network
TCP
G1
FN
q
GL
R
T


xi (t  1)  Fi   Rli pl (t ), xi (t ) 
 l


pl (t  1)  Gl  pl (t ),

AQM

i Rli xi (t ) 
p
same price
for all sources
Heterogeneous protocol
x
y
R
F1
G1
Network
TCP
FN
GL
q
p
R
T


xi (t  1)  Fi   Rli pl (t ), xi (t ) 
 l

heterogeneous
prices for
 type j sources

xi (t  1)  Fi   Rli mlj  pl (t ) , xij (t ) 
 l

j
AQM
j
Heterogeneous protocols
 Equilibrium: p that satisfies


j
xi ( p)  f i   Rli ml ( pl ) 
 l

  cl
j j
yl ( p) :  Rli xi ( p) 
i,j
  cl if pl  0
j
j
Duality model no longer applies !
 pl can no longer serve as Lagrange multiplier
Heterogeneous protocols
 Equilibrium: p that satisfies


j
xi ( p)  f i   Rli ml ( pl ) 
 l

  cl
j j
yl ( p) :  Rli xi ( p) 
i,j
  cl if pl  0
j
j
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


j
xi ( p)  f i   Rli ml ( pl ) 
 l

  cl
j j
yl ( p) :  Rli xi ( p) 
i,j
  cl if pl  0
j
j
 Dynamic: dual algorithm


j
xi ( p (t ))  f i   Rli ml ( pl (t )) 
 l

p l   l  yl ( p (t ))  cl 
j
j
Notation
 Simpler notation: p is equilibrium if
y( p)  c
on bottleneck links
 Jacobian: J ( p) :
y
( p)
p
 Linearized dual algorithm:
p   J ( p* ) p(t)
See Simsek, Ozdaglar, Acemoglu 2005
for 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.,
y
det J ( p) : det
( p)  0
p
Theorem
 Almost all networks are regular
 A regular network has finitely many and
odd number of equilibria (e.g. 1)
Global uniqueness
m lj  [al ,21/ L al ] for any al  0
m lj  [a j ,21/ L a j ] for any a j  0
Theorem
 If price heterogeneity is small, then equilibrium is
globally unique
Corollary
 If price mapping functions mlj are linear and linkindependent, then equilibrium is globally unique
e.g. a network of RED routers almost always has
globally unique equilibrium
Local stability:`uniqueness’  stability
j
1/ L

ml  [al ,2 al ] for any al  0
m lj  [a j ,21/ L a j ] for any a j  0
Theorem
 If price heterogeneity is small, then the unique
equilibrium p is locally stable
Linearized dual algorithm:
*

p   J( p ) p(t)
Equilibrium p is locally stable if
Re  J( p)  0
Local stability:`converse’
Theorem
 If all equilibria p are locally stable, then it is
globally unique
Proof idea:
 For all equilibrium p:
I ( p)  (1) L
 Index theorem:
L
I
(
p
)

(

1
)

eq p