Minimum-Delay Load-Balancing
Through Non-Parametric
Regression
F. Larroca and J.-L. Rougier
IFIP/TC6 Networking 2009
Aachen, Germany, 11-15 May 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
is typically a convex increasing function
that diverges as rl → cl; e.g. mean queuing delay
 Why queuing delay? Simplicity and versatility
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IFIP/TC6 Networking 2009
F. Larroca and J.-L. Rougier
Introduction
A
simple model (M/M/1) is always 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
• Optimize with this learnt function
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IFIP/TC6 Networking 2009
F. Larroca and J.-L. Rougier
Agenda
Introduction
Attaining
Delay
the optimum
function approximation
Simulations
Conclusions
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IFIP/TC6 Networking 2009
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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IFIP/TC6 Networking 2009
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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IFIP/TC6 Networking 2009
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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IFIP/TC6 Networking 2009
F. Larroca and J.-L. Rougier
Agenda
Introduction
Attaining
Delay
the optimum
function approximation
Simulations
Conclusions
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IFIP/TC6 Networking 2009
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
 Convex Nonparametric Least-Squares (CNLS) 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
2


Y

f
(
r
)
 i N i
F
min
fN
i 1
where F is the set of continuous, non-decreasing and convex functions
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IFIP/TC6 Networking 2009
F. Larroca and J.-L. Rougier
Cost Function Approximation
 The
size of F complicates the problem
 Consider instead G (subset of F) a family of piecewiselinear 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
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IFIP/TC6 Networking 2009
F. Larroca and J.-L. Rougier
Cost Function Approximation
 This
regression function presents a problem: its
derivative is not continuous (cf. 2.b)
 A soft approximation of a piecewise linear function:
 N   j   j r  

f r   log   e


j

1


1
*
N

Our final approximation of the link-cost function:
f r  
*
N
1
N
e
N
  je
  j   j r 
  j   j r  j 1
j 1
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IFIP/TC6 Networking 2009
F. Larroca and J.-L. Rougier
An Example
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IFIP/TC6 Networking 2009
F. Larroca and J.-L. Rougier
Agenda
Introduction
Attaining
Delay
the optimum
function approximation
Simulations
Conclusions
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IFIP/TC6 Networking 2009
F. Larroca and J.-L. Rougier
NS-2 simulations
 The
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considered network:
IFIP/TC6 Networking 2009
F. Larroca and J.-L. Rougier
NS-2 simulations
 Alternative
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(“wrong”) training set:
IFIP/TC6 Networking 2009
F. Larroca and J.-L. Rougier
Agenda
Introduction
Attaining
Delay
the optimum
function approximation
Simulations
Conclusions
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IFIP/TC6 Networking 2009
F. Larroca and J.-L. Rougier
Conclusions and Future Work
We have presented a framework to converge to the actual
minimum total mean delay demand vector
 Two shortcomings of our framework:
• fl(rl) is constant outside the support of the observations
• Links with little or no queue size have a negligible cost
 Possible Solution: Add a “patch” function that is negligible
with respect to fl(rl) except at high loads
 How does fl(rl) behaves over time? Does it change? How
often?
 How does our framework performs when compared with
other mechanisms or simpler models?
 Faster and/or more robust alternative regression methods?
 Is REPLEX the best choice?

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Thank you!
Questions?
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IFIP/TC6 Networking 2009
F. Larroca and J.-L. Rougier