equilibrium of heterogeneous protocols steven low cs, ee netlab.caltech.edu with a. tang, j. wang,...

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