on self adaptive routing in dynamic environments

On Self Adaptive Routing in Dynamic Environments

-- A probabilistic routing scheme

Haiyong Xie, Lili Qiu, Yang Richard Yang and Yin Zhang@ Yale, MR and AT&T

Presented by Joe, W.J.Jiang28-08-2004

Outline

• Overview of Adaptive Routing• Related Work• Probabilistic Routing Scheme• Convergence Analysis• Simulation Results• Conclusion

Where are you?


Introduction to adaptive routing

• Routing in the Internet:interior gateway routing – OSPFexterior gateway routing – BGP

• Static routing, based on hop counts• There is an inherent inefficiency in IP routing

from user’s perspective: latency, bandwidth, loss rate, etc

• Adaptive routing, allowing end hosts to select routes by themselves.

Selfish Routing (user-optimal routing)

• Each end host selects a route with minimum latency.

• Shortest path routing, metric -- latency, additive

• Two approaches:source routing -- Nimrodoverlay routing -- Detour, RON

• Selfish by nature -- selfish routing

Illustration of source routing

n1 n2 n3 n4 n5

n1-n2-n3-n4-n5

Illustration of overlay routing

Problems I -- Oscillation

• Ring Network (Data Networks)• Simultaneous Overlay Network

Primary Paths

BottleneckPhy. Link

1+ Mbps(L2)

2 MbpsL1

1 Mbps(L3)

Sources

Destinations

Alternate Paths

Ov.Nw. Nodes(2 Ovns)

Problem II -- Performance Degradation

• Nash Equilibrium• Well known that Nash Equilibrium do not in general o

ptimize social welfare.• Braess’s Paradox

x 1

s t

x1

1/2

1/2

0

1

Performance degradation:selfish routing : global optimal= 2/(0.5+1) = 4/3

Where are you?


Related Work

• Wardrop equilibrium: a research aspect in economics of transportation.

• The proof of existence of unique equilibrium and some extensions.

• Network optimal routing - Data Networks- Frank-Wolfe Method- Projection MethodThese are centralized algorithms.

• Distributed version for optimal routing - Parallel and Distributed Computation

Related Work (Cont)

• “How bad is selfish routing?”- There exists unique Nash Equilibrium for selfish routing under network flow model.- The performance (average delay) ratio between selfish routing and global routing could be unbounded for arbitrary network.- The upper bound for network with linear delay function is 4/3.

• “On selfish routing in Internet-like environment”- Based on simulation, selfish routing and global optimal routing exhibit similar performance, under different network topology and traffic models.

Related Work (cont)

• If individual users are allowed to select routes selfishly without coordination, how to ensure these behaviors will converge to an equilibrium?

• “Dynamic Cesaro-Wardrop equilibration in Networks”- a model to ensure the convergence of probabilistic routing scheme

• “On self adaptive routing in dynamic environments”

Where are you?


Routing Scheme - Data Path Component• Data path component

- similar to distance vector routing- destination could be all overlay nodes.-

- a generalization of normal Internet routing.

0 1 ikj

j

ikj pp ，

Routing Scheme - Control Path Component

• Control path component - how routing probabilities are updated.

• Selfish routing, Wardrop routing, user-optimal routing

• property - Given a source-destination pair with a given amount of traffic, the routes with positive traffic should have equal latency, no larger than those unused routes for this source-destination pair.

Routing Scheme: Notation

• lji the latency of link from node i to its neighbo

r j• Lj

ik the estimated delay from i to destination k through node j

• qjik the internal probability from node i to desti

nation k through neighbor j• pj

ik the routing probability from node i to destination k through neighbor j

• {qjik} will converge to the Wardrop equilibrium

• {pjik} are ε- approximate of {qj

ik}

Update of routing probabilities (1)• Node i first computes the new delay

Δjik = lj

i + Ljk

• Ljk is the estimated latency from node j to node k• node i update the new latency estimation

Ljik = (1-α(n)) Lj

ik + α(n) Δjik

• α(n) is the delay learning factor.• then node i computes its overall delay estimation Lik t

o destination k

)('

''iNj

jkj

ikj

ik LpL

Update of routing probabilities (2)• Node i reports Lik to its neighbors after some delay, a

nd the delay is a random value between T/2 to T, to avoid synchronization.

• node i updates its internal routing probabilities:

• β(n’) is routing learning factor• ξj

ik is i.i.d uniform random routing vectors to add disturbance to avoid non-Wardrop solutions

ikj

ikj

ikikj

ikj

ikj LLqnqq '

Update of routing probabilities (3)• Projection: node i projects the internal routing

probabilities to the subspace of [0,1]N(i), which is equivalent to solving the following problem:

jx

x

qx

j

iNjj

iNj

ikjj

allfor 10over

1 subject to

minimize

)(

)(

2

Update of routing probabilities (4)• Node i compute the routing probabilities:

)(

1iN

qp ikj

ikj

Protocol to implement user-optimal routing

Comments on measuring

• About measuring lji , two approaches:

- measured by node i- measured by node j

• The advantage of the second method:- unnecessary for clock synchronization- Δj

ik = lji + Ljk, there is an offset which is just the clock

difference between i and the destination, independent of j.- - overhead is to stamp packets

ikj

ikj

ikikj

ikj

ikj LLqnqq '

Probabilistic Scheme for network optimal routing

• Overview of network optimal routingto solve the convex programming:

)()( ,)()( minimize eeeeeEe

ee fflfcfcfC

PP

P f

Eeff

,...,k}{irf

P

PePpe

iPP

P

i

0

1

:

Probabilistic Scheme for network optimal routing (cont)

eeeeeeeeeeee fflflfflfcfl )()())(()()(*

. instancefor

mequilibriuNash at isit ifonly and if optimal is

for feasible flow aThen above. as defined function

cost marginal with , edgeeach for function convex a is

in which instancean be Let :

)(G,r,l

(G,r,l)

fl

e

(x)lx(G,r,l)Theorem

*

*

e

Proved in “How bad is selfish routing”.

Probabilistic Scheme for network optimal routing (cont)

• For network optimal routing, replace lji with m

arginal cost function: mcj

i = lji + fj

isji

• sji is the rate of change in the latency from no

de i to node j at traffic amount fji

• Without knowing the analytical expression of latency functions.

• However, the paper does not mention the scheme to measure the rate of change in the latency.

Where are you?


Convergence analysis - Intuition

• Consider a network with only two links

• p1, p2 >=0, p1+p2=1• Five cases.

- (a) link 1 has higher latency- (b) link 1 has lower latency- (c) link 1 and 2 has the same latency- (d) link 1 has all of the traffic- (e) link 2 has all of the traffic

Convergence Analysis - Assumption

• A1 - latency function is continuous, non-decreasing and bounded.

• A2 - the updates are frequent enough compared with the change rate in the underlying network states.

• A3 -

Convergence Analysis - Assumption

• A4 -

Where are you?


Evaluation Methodologies

• Network topologies: ATT, Sprint, Tiscali• Traffic demands• Traffic stimuli:

- Traffic spike- Step function- Linear function

• Performance metrics:- average latency- average convergence time- link utilization

Dynamics of user-optimal routing and network-optimal routing

Conclusion


Conclusion

• This paper introduces a probabilistic routing scheme to achieve both user-optimal (selfish) routing and network optimal routing.

• An application of enforcement learning.• Not consider the issue of fairness

between users (or overlays).

Thank you!

on self adaptive routing in dynamic environments

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