minimum-delay load-balancing through non-parametric regression f. larroca and j.-l. rougier ifip/tc6...
TRANSCRIPT
Minimum-Delay Load-Balancing Through Non-Parametric
Regression
F. Larroca and J.-L. Rougier
IFIP/TC6 Networking 2009
Aachen, Germany, 11-15 May 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 ) is typically a convex increasing function that diverges as l → cl; e.g. mean queuing delay
Why queuing delay? Simplicity and versatility
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
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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 (l ) (e.g. monotone increasing)
• Learn it from measurements instead• Optimize with this learnt function
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
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Agenda
Introduction
Attaining the optimum
Delay function approximation
Simulations
Conclusions
IFIP/TC6 Networking 2009 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
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
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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:
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
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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 Important: l(l) should be continuous, positive and
non-decreasing
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
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Agenda
Introduction
Attaining the optimum
Delay function approximation
Simulations
Conclusions
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
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Cost Function Approximation What should be used as fl (l )?
1. That represents reality as much as possible
2. 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 Convex Nonparametric Least-Squares (CNLS) 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
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
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Cost Function Approximation The size of F complicates the problem Consider instead G (subset of F) a 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
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
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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:
Our final approximation of the link-cost function:
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
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An Example
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
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Agenda
Introduction
Attaining the optimum
Delay function approximation
Simulations
Conclusions
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
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NS-2 simulations The considered network:
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
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NS-2 simulations Alternative (“wrong”) training set:
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
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Agenda
Introduction
Attaining the optimum
Delay function approximation
Simulations
Conclusions
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
page 17
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:
• l(l) 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 l(l) except at high loads
How does l(l) 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?
IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
page 18 IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier
Thank you!
Questions?