equilibrium of heterogeneous protocols steven low cs, ee netlab.caltech.edu with a. tang, j. wang,...
TRANSCRIPT
Equilibrium of Heterogeneous Protocols
Steven Low
CS, EEnetlab.CALTECH.edu
with A. Tang, J. Wang, ClatechM. Chiang, Princeton
Network model
F1
FN
G1
GL
R
RT
TCP Network AQM
x y
q p
))( ),(( )1(
))( ),(( )1(
tRxtpGtp
txtpRFtx T
Reno, Vegas
DT, RED, …
liRli link uses source if 1 IP routing
TCP-AQM:
Duality model
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
TCP-AQM:
Duality model
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:
Vegas, FAST, STCP HSTCP (homogeneous
sources) Reno (homogeneous
sources) infinity XCP (single link
only)
(Mo & Walrand 00)
Congestion control
F1
FN
G1
GL
R
RT
TCP Network AQM
x y
q p
iililll
il
lliii
txRtpGtp
txtpRFtx
)( ),( )1(
)( ,)( )1(
same pricefor 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
Multiple equilibria: multiple constraint sets
eq 1 eq 2
Path 1 52M 13M
path 2 61M 13M
path 3 27M 93M
eq 1
eq 2
Tang, Wang, Hegde, Low, Telecom Systems, 2005
eq 1 eq 2
Path 1 52M 13M
path 2 61M 13M
path 3 27M 93M
eq 1
eq 2
Tang, Wang, Hegde, Low, Telecom Systems, 2005
eq 3 (unstable)
Multiple equilibria: multiple constraint sets
Multiple equilibria: single constraint sets
1 1
11x
21x
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
Duality model no longer applies ! pl can no longer serve as
Lagrange multiplier
TCP-AQM equilibrium p:
TCP-AQM equilibrium p:
Multi-protocol: J>1
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-protocolUnique
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
TCP-AQM equilibrium p:
Multi-protocol: J>1
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 regularRegular 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
Theorem (Tang, Wang, Low, Chiang, Infocom 2005)
If all equilibria p all locally stable, then it is globally unique
Linearized gradient projection algorithm:
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-protocolUnique
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