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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