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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
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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
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l
llf )(min
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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
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Agenda
Introduction
Attaining the optimum
Delay function approximation
Simulations
Conclusions
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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
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s
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, 0 and s.t.
: min
1
1
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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
)()(
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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
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Kfdxxfdl
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)()()(0
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Agenda
Introduction
Attaining the optimum
Delay function approximation
Simulations
Conclusions
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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
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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
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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
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iii Yf
)(1
0
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An Example
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Measurements obtained by injecting 72 hours worth of traffic to a router simulator (C = 18750 kB/s)
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Agenda
Introduction
Attaining the optimum
Delay function approximation
Simulations
Conclusions
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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
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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
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Agenda
Introduction
Attaining the optimum
Delay function approximation
Simulations
Conclusions
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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?
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Thank you!Questions?