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
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
min
 Function fl (rl )
 f (r )
l
l
l
measures the congestion on link l; e.g.
mean queuing delay
 Why queuing delay? Simplicity and versatility
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F. Larroca and J.-L. Rougier
Introduction
analytical expression of fl (rl ) 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 (rl ) (e.g. monotone
increasing)
• Learn it from measurements instead (reflect more
precisely congestion on the link)
• Optimize with this learnt function
 An
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F. Larroca and J.-L. Rougier
Agenda
Introduction
Attaining
Delay
the optimum
function approximation
Simulations
Conclusions
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F. Larroca and J.-L. Rougier
Problem Definition
delay on link l is given by Dl(rl)
 Our congestion measure: weighted mean end-to-end
queuing delay
 The problem:
 Queuing
ns
min
d
d
i 1
s
si
DP   r l Dl r l  :  f l r l 
l
ns
s.t.
d
i 1
 Since fl (rl ):=
si
 d s and d si  0 s, i
rl Dl (rl ) is proportional to the queue size,
we will use this value instead
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l
F. Larroca and J.-L. Rougier
Congestion Routing Game
 Path
P has an associated cost fP :
fP   fl ( rl )
l:lP
where fl(rl) 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:
rl
(d )    fl ( x)dx
l
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0
F. Larroca and J.-L. Rougier
Congestion Routing Game
'
happens if we use fl ( rl )  f l ( rl ) ?
 The equilibrium coincides with the minimum of:
 What
rl
(d )    f l ' ( x)dx   f l ( rl )  K
l
0
l
 To
solve our problem, we may play a Congestion
Routing Game with fl ( rl )  f l ' ( rl )
 To converge to the Equilibrium we will use REPLEX
 Important: fl(rl) should be continuous, positive and
non-decreasing
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F. Larroca and J.-L. Rougier
Agenda
Introduction
Attaining
Delay
the optimum
function approximation
Simulations
Conclusions
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F. Larroca and J.-L. Rougier
Cost Function Approximation
 What
should be used as fl (rl )?
1. That represents reality as much as possible
2. Whose derivative (fl(rl)) is:
a. continuous
b. positive => fl (rl ) non-decreasing
c. non-decreasing => fl (rl ) convex
address 1 we estimate fl (rl ) from measurements
 Weighted Convex Nonparametric Least-Squares
(WCNLS) is used to enforce 2.b and 2.c :
• Given a set of measurements {(ri,Yi)}i=1,..,N find fN ϵ F
 To
N
min
f N F
2



Y

f
(
r
)
 i i N i
i 1
where F is the set of continuous, non-decreasing and convex functions
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F. Larroca and J.-L. Rougier
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:
 N   j   j r  

f r   log   e


 j 1

*
N
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F. Larroca and J.-L. Rougier
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:
1
i 
f 0 ( ri )  Yi
where f0(ri) is the k-nearest neighbors estimation
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F. Larroca and J.-L. Rougier
An Example
 Measurements
obtained by injecting 72 hours worth of
traffic to a router simulator (C = 18750 kB/s)
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Agenda
Introduction
Attaining
Delay
the optimum
function approximation
Simulations
Conclusions
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F. Larroca and J.-L. Rougier
Performance Comparison
 Considered
scenario: Abilene along
with a week’s worth of traffic
 Performance if we used:
• the M/M/1 model instead
of
WCNLS
*
MaxU
*
W CNLS 



r
c

max
r
cl 
• Link
TotalUtilization
Mean DelayX Xmax




f
r
f
r
l
l l

l l lrl
l  l
l
fP l max
• A greedy algorithm
where
(MaxU)
l


lP


M/M/1
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cl
WCNLS
F. Larroca and J.-L. Rougier
Agenda
Introduction
Attaining
Delay
the optimum
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 (rl )
• 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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F. Larroca and J.-L. Rougier
Thank you!
Questions?
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