Download FAST v3

Equilibrium of Heterogeneous
Protocols
Steven Low
CS, EE
netlab.CALTECH.edu
with A. Tang, J. Wang, Clatech
M. Chiang, Princeton
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, …
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
 Equilibrium (x*,p*) primal-dual optimal:
(Mo & Walrand 00)




a  1 : Vegas, FAST, STCP
a  1.2: HSTCP (homogeneous sources)
a  2 : Reno (homogeneous sources)
a  infinity: XCP (single link only)
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

Multiple equilibria:
multiple constraint sets
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:
multiple constraint sets
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
Multiple equilibria:
single constraint sets
x11
1
1
x12




Smallest example for multiple equilibria
Single constraint set but infinitely many equilibria
J=1: prices are non-unique but rates are unique
J>1: prices and rates are both non-unique
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?
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
Multi-protocol: J>1
 TCP-AQM equilibrium p:
 Simpler notation: equilibrium p iff
Multi-protocol: J>1

 Linearized gradient projection algorithm:

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, i.e.
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: regular networks
Proof idea:
Sard’s Theorem: critical value of cont
diff functions over open set has
measure zero
Apply to y(p) = c on each bottleneck
set  regularity
Compact equilibrium set  finiteness
Results: regular networks
Proof idea:
 Poincare-Hopf Index Theorem: if there exists a
vector field s.t. dv/dp non-singular, then
 Gradient projection algorithm defines such a vector
field
 Index theorem implies odd #equilibria
Results: global uniqueness
 Linearized gradient projection algorithm:
Theorem (Tang, Wang, Low, Chiang, Infocom 2005)
If all equilibria p all locally stable, then it is
globally unique
Proof idea:
 For all equilibrium p:

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