Robust Regression for Minimum-Delay Load-Balancing F. Larroca and J.-L. Rougier 21st International Teletraffic Congress (ITC 21) Paris, France, September 2009

page 1 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 f l (  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

page 2 Introduction An analytical expression of f l (  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 f l (  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 3 Agenda Introduction Attaining the optimum Delay function approximation Simulations Conclusions ITC 21 F. Larroca and J.-L. Rougier

page 4 Problem Definition Queuing delay on link l is given by D l (  l ) Our congestion measure: weighted mean end-to-end queuing delay The problem: Since f l (  l ):=  l D l (  l ) is proportional to the queue size, we will use this value instead ITC 21 F. Larroca and J.-L. Rougier

page 5 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

page 6 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 Important:  l (  l ) should be continuous, positive and non-decreasing ITC 21 F. Larroca and J.-L. Rougier

page 7 Agenda Introduction Attaining the optimum Delay function approximation Simulations Conclusions ITC 21 F. Larroca and J.-L. Rougier

page 8 Cost Function Approximation What should be used as f l (  l )? 1.That represents reality as much as possible 2.Whose derivative (  l (  l )) is: a.continuous b.positive => f l (  l ) non-decreasing c.non-decreasing => f l (  l ) convex To address 1 we estimate f l (  l ) from measurements Weighted Convex Nonparametric Least-Squares (WCNLS) is used to enforce 2.b and 2.c : Given a set of measurements {(  i,Y i )} i=1,..,N find f N F where F is the set of continuous, non-decreasing and convex functions ITC 21 F. Larroca and J.-L. Rougier

page 9 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

page 10 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 f 0 (  i ) is the k-nearest neighbors estimation ITC 21 F. Larroca and J.-L. Rougier

page 11 An Example ITC 21 F. Larroca and J.-L. Rougier Measurements obtained by injecting 72 hours worth of traffic to a router simulator (C = kB/s)

page 12 Agenda Introduction Attaining the optimum Delay function approximation Simulations Conclusions ITC 21 F. Larroca and J.-L. Rougier

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 13 Performance Comparison ITC 21 F. Larroca and J.-L. Rougier Total Mean DelayLink Utilization M/M/1 WCNLS

page 14 Agenda Introduction Attaining the optimum Delay function approximation Simulations Conclusions ITC 21 F. Larroca and J.-L. Rougier

page 15 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 f l (  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 16ITC 21 F. Larroca and J.-L. Rougier Thank you! Questions?