Robust Regression for Minimum-Delay Load-Balancing

F. Larroca and J.-L. Rougier

21st International Teletraffic Congress (ITC 21)

Paris, France, 15-17 September 2009

page 2

Introduction Current traffic is highly dynamic and unpredictable How may we define a routing scheme that performs well

under these demanding conditions? Possible Answer: Dynamic Load-Balancing

• We connect each Origin-Destination (OD) pair with several pre-established paths

• Traffic is distributed in order to optimize a certain function

Function fl (l ) measures the congestion on link l; e.g. mean queuing delay

Why queuing delay? Simplicity and versatility

ITC 21 F. Larroca and J.-L. Rougier

l

llf )(min

page 3

Introduction An analytical expression of fl (l ) is required: simple

models (e.g. M/M/1) are generally assumed What happens when we are interested in actually

minimizing the total delay? Simple models are inadequate We propose:

• Make the minimum assumptions on fl (l ) (e.g. monotone increasing)

• Learn it from measurements instead (reflect more precisely congestion on the link)

• Optimize with this learnt function

ITC 21 F. Larroca and J.-L. Rougier

page 4

Agenda

Introduction

Attaining the optimum

Delay function approximation

Simulations

Conclusions

ITC 21 F. Larroca and J.-L. Rougier

page 5

Problem Definition Queuing delay on link l is given by Dl(l) Our congestion measure: weighted mean end-to-end

queuing delay The problem:

Since fl (l ):=l Dl (l ) is proportional to the queue size, we will use this value instead

ITC 21 F. Larroca and J.-L. Rougier

isddd

fDDd

sis

n

isi

s lll

llll

n

iPsid

s

s

, 0 and s.t.

: min

1

1

page 6

Congestion Routing Game Path P has an associated cost P :

where l(l) is continuous, positive and non-decreasing Each OD pair greedily adjusts its traffic distribution to

minimize its total cost Equilibrium: no OD pair may decrease its total cost by

unilaterally changing its traffic distribution It coincides with the minimum of:

ITC 21 F. Larroca and J.-L. Rougier

Pll

llP:

)(

l

l

l

dxxd

0

)()(

page 7

Congestion Routing Game

What happens if we use ? The equilibrium coincides with the minimum of:

To solve our problem, we may play a Congestion Routing Game with

To converge to the Equilibrium we will use REPLEX ImportantImportant: l(l) should be continuous, positive and

non-decreasing

ITC 21 F. Larroca and J.-L. Rougier

)()( 'llll f

Kfdxxfdl

lll

l

l

)()()(0

'

)()( 'llll f

page 8

Agenda

Introduction

Attaining the optimum

Delay function approximation

Simulations

Conclusions

ITC 21 F. Larroca and J.-L. Rougier

page 9

Cost Function Approximation What should be used as fl (l )?

1. That represents reality as much as possible2. Whose derivative (l(l)) is:

a. continuousb. positive => fl (l ) non-decreasingc. non-decreasing => fl (l ) convex

To address 1 we estimate fl (l ) from measurements Weighted Convex Nonparametric Least-Squares (WCNLS) is

used to enforce 2.b and 2.c : • Given a set of measurements {(i,Yi)}i=1,..,N find fN ϵ F

where F is the set of continuous, non-decreasing and convex functions

ITC 21 F. Larroca and J.-L. Rougier

N

iiNiiFf

fYN 1

2)(min

page 10

Cost Function Approximation The size of F complicates the problem Consider G (subset of F) the family of piecewise-linear

convex non-decreasing functions The same optimum is obtained if we change F by G We may now rewrite the problem as a standard QP one Problem: its derivative is not continuous (cf. 2.a) Soft approximation of a piecewise linear function:

ITC 21 F. Larroca and J.-L. Rougier

N

jN

jjef1

* log1

page 11

Cost Function Approximation Why the weights? They address two problems:

• Heteroscedasticity• Outliers

Weight i indicates the importance of measurement i (e.g. outliers should have a small weight)

We have used:

where f0(i) is the k-nearest neighbors estimation

ITC 21 F. Larroca and J.-L. Rougier

iii Yf

)(1

0

page 12

An Example

ITC 21 F. Larroca and J.-L. Rougier

Measurements obtained by injecting 72 hours worth of traffic to a router simulator (C = 18750 kB/s)

page 13

Agenda

Introduction

Attaining the optimum

Delay function approximation

Simulations

Conclusions

ITC 21 F. Larroca and J.-L. Rougier

l

l

PlP c

max

Performance if we used: • the M/M/1 model instead of WCNLS• A greedy algorithm where (MaxU)

Considered scenario: Abilene along with a week’s worth of traffic

page 14

Performance Comparison

ITC 21 F. Larroca and J.-L. Rougier

• Total Mean Delay

l

WCNLSll

lll ffX * lMaxU

lllllccX maxmax * • Link Utilization

M/M/1 WCNLS

page 15

Agenda

Introduction

Attaining the optimum

Delay function approximation

Simulations

Conclusions

ITC 21 F. Larroca and J.-L. Rougier

page 16

Conclusions and Future Work

We have presented a framework to converge to the actual minimum total mean delay demand vector

Impact of the choice of fl (l )• Link Utilization: not significant (although higher maximum than

the optimum, the rest of the links are less loaded)• Mean Total Delay: very important (using M/M/1 model

increased10% in half of the cases and may easily exceed 100%)

Faster alternative regression methods? Ideally that result in a continuously differentiable function

Is REPLEX the best choice?

ITC 21 F. Larroca and J.-L. Rougier

page 17 ITC 21 F. Larroca and J.-L. Rougier

Thank you!Questions?