Interaction of Overlay Networks:

Properties and Implications

Joe W.J. Jiang Dah-Ming Chiu John C.S. LuiThe Chinese University of Hong Kong

Outline

Introduction to overlay routing.

Mathematical modeling

Properties of NEP

Implications of overlay interactions

Conclusion

Introduction to overlay routing

Design philosophy of traditional IP-level routing: simple and scalable.

Overlay and P2P networks: harnessing the benefits of a disruptive technology.

Source

Destination

Overlay Node

Physical Node

Physical Link

Logical Link

Source

Destination

Overlay Node

Physical Node

Physical Link

Logical Link

Source

Destination

Overlay Node

Physical Node

Physical Link

Logical Link

Overlay creates a virtual topology, and assists user-oriented or application-oriented routing. Overlay nodes relay packets for each other. A large percent of traffic can find better routes by

relaying packets with the assistance of overlay nodes.

Principles of overlay routing

Traditional Overlay Routingselect the best pathroute monitor / update / recovery.route oscillation / race condition.

Selfish Overlay Routing (user-optimal)selfish routing, by T.Roughgarden

split traffic at the source

existence of Nash Equilibrium.

probabilistic routing implementation.

performance not optimized.

Optimal Overlay Routing (overlay-optimal)to split traffic at the source.

minimize the average end-to-end delay for the whole overlay

a routing optimization at the overlay layer.

Motivation of our work

There has been little focus on the “interaction” of “co-existence” of multiple overlays.

Questions to be answered:What’s the form of interaction ?

Is there routing instability ?

Is the equilibrium efficient?

Can the selfish behavior be led to an efficient equilibrium?

What is next?

Introduction

Mathematical modeling

Properties of NEP

Implications of overlay interactions

Conclusion

Preliminary

Physical underlying network

link delay is a function of the aggregate traffic.

delay function - dj(lj)

continuous

non-decreasing

convex

end-to-end delay is the additive form of delays on each link

Logical overlay network

objective: minimize the average weighted delay for the whole overlay.

the delay depends on routing decisions of all overlays

average weighted delay

),( ),(ts tsPk

kk delayy

(s,t)all source-sink pairs

in the overlay

P(s,t)set of overlay paths available for source-si

nk pair (s,t)

yk

traffic rate assigned to path

k

delayk

end-to-end delay on overlay pat

h k

Preliminary (continue)

Basic assumptionsmultiple source-sinks

fixed traffic demand

constant underlying traffic

routing info. obtained from a common routing underlay

Form of interactionsoverlays transparent to each other

routing decision dependent on other overlays

Interaction occurs when overlays share common resources, e.g. physical links, bandwidths, nodes.

Routing Optimization

0 ,

, s.t.

Minimize

,,,,;

)(

||

1

),(

)()()()()(

)()()(

s

f

R

k

fsks

i

iiTsTss

sss

yCAy

xyFf

yADAydelay

yxCHAyOVERLAY

f

y(s)

routing decision for overlay s

A(s)

routing matrix for overlay s

y(-s)

routing decision for other overlays A

routing matrix for all overlays

Hmatrix indicatingavailable paths

for each source-sink pair

delay(s)

the average weighted delay

of overlay s

demand constraint

capacity constraint

non-negativity constraint

Algorithmic Solution Apply any convex programming techniques Marginal cost network flow

Algorithmic Solutions

Apply any convex programming techniquesobjective function is convex.feasible region is convex and compact.optimal value and optimizer can be found by the Lagrangian method.

Marginal cost network flowfor each physical link, replace the delay cost by the marginal cost of the weighted delay.marginal cost -- first derivativeweighted delay -- rate of traffic traversing a link in its own overlay * delay on this linksplit traffic among all available paths, s.t. all paths with positive traffic flow have the same end-to-end cost, smaller than paths with zero-traffic.

What is next?

Introduction

Mathematical modeling

Properties of NEP

Implications of overlay interactions

Conclusion

Overlay Routing Game

Nash routing gameplayer: all overlays

strategy: feasible region of OVERLAY(s)

preference: a smaller average delay

Routing behaviors of overlaysdifferent routing update period

calculation of the optimal routing strategy

Is there routing instability?

Nash Equilibrium Point

A feasible strategy profile y=(y(1),…, y(s),…, y(n))T is a Nash Equilibrium in the overlay routing game if for every overlay s∈N,

delay(s)(y(1),…y(s),…y(n)) ≤ delay(s)(y(1),…y*(s),…y(n))

for any other feasible strategy profile y*(s) .

Existence of NEP

Theorem In the overlay routing game, there exists a Nash Equilibrium if the delay function delay(s)(y(s) ; y(-s)) is continuous, non-decreasing and convex.

six co-existing overlays

one source-sink pair each overlay

overlapping physical links, physical nodes

different routing update period

31

40

Overaly 1Overlay 2Overlay 3Overlay 4Overlay 5Overlay 6

S1

S2

S3

S4

S5

S6

D1D2

D3

D4

D5

D6

32

3334

35

38

3736

29

2827

26

30

3944

4543

6

4

1

3

2

5

9

7

10

8

1213

41 42

25

24

23

22

21

20

18

17

19

16 15

14

simulation topology

Fluid Simulation

average delay for six overlays v.s. simulation time

traffic flow for six overlays v.s. simulation time

transient period

convergence

number of curves equals number of available paths for each flow

Interaction of routing decisions

convergence of routing decisions

different convergence rate

What is next?

Introduction

Mathematical modeling

Properties of NEP

Implications of overlay interactions

Conclusion

Anomalies of routing equilibrium

Interesting questionsIs the equilibrium point efficient?

Can the selfish behavior be led to an efficient equilibrium?

Anomalies due to unregulated competition

for common resources:sub-optimality

slow convergence

fairness paradox

Example of illustration

3

21

5

src1 src2

sink1 sink2

4

6Overlay 2

Overlay 1

1 unit 1 unit

3

21

5

src1 src2

sink1 sink2

4

6

y1

1-y1

y2

1-y2

Sub-optimality

3

21

5

src1 src2

sink1 sink2

4

6Overlay 2

Overlay 1

d15(l) = 1+l

y1

1-y1

y2

1-y2

d34(l) = ld26(l) = 2.5+lOther links zero delay

Nash Equilibrium Point

y1=0.5 y2=1.0

delay1=delay2=1.5

Not Pareto Optimal

A Point on Pareto Curve

y1=0.4 y2=0.9

delay1=1.48<1.5delay2=1.43<1.5

Slow-convergence

3

21

5

src1 src2

sink1 sink2

4

6Overlay 2

Overlay 1

5 unit 5 unit

6 5C34=?

10 10

10 10

C34=8

C34=6

slower convergence

Fairness Paradox

3

21

5

src1 src2

sink1 sink2

4

6Overlay 2

Overlay 1

1 unit 1 unit

d26(l) = c+l

d15(l) = a+l

d34(l) = blα

0 0

0 0

a, b, c,αare non-negative parameter of the delay functions

Everything is symmetric except for the two private links

common link: n3-n4private links: n1-n5 overlay1

n2-n6 overlay2

2

1

2

1

1

11

2

3 ,

4

1

2

3

42

3

2

3

2

3 ,1

2

3

22

3 ,1

delay

delayca

delaydelay

cab

Unfariness becomes unbounded!

War of Resource Competition

1 unit 1 unit

y1

1-y1

y2

1-y2

Pg(y1+y2

)

Pc1(1-y1) Pc2(1-y2)

Min y1Pg(y1,y2)+(1-y1)Pc1(1-y1)Min y2Pg(y1,y2)+(1-y2)Pc2(1-y2)

Pc1<Pc2

?>

What is next?

Introduction

Mathematical modeling

Properties of NEP

Implications of overlay interactions

Conclusion

Conclusion

Study the interaction between multiple co-existing overlays.

Formulate the game as a non-cooperative Nash routing game.

Prove the existence of NEP.

Show the anomalies and implications of the NEP.

Thank you for your attention!

Q & A