Robust Regression for Minimum-Delay Load-Balancing F. Larroca and J.-L. Rougier

1. Robust Regression for Minimum-Delay Load-BalancingF. Larroca and J.-L. Rougier 21st International Teletraffic Congress (ITC 21) Paris, France, 15-17 September 2009

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 (rl ) 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

3. Introduction • An 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 ITC 21 F. Larroca and J.-L. Rougier

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

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

6. Congestion Routing Game • Path P has an associated cost fP : 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: ITC 21 F. Larroca and J.-L. Rougier

7. 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: fl(rl) should be continuous, positive and non-decreasing ITC 21 F. Larroca and J.-L. Rougier

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

9. Cost Function Approximation • What should be used as fl (rl )? • That represents reality as much as possible • Whose derivative (fl(rl)) is: • continuous • positive => fl (rl ) non-decreasing • non-decreasing => fl (rl ) convex • To 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 where F is the set of continuous, non-decreasing and convex functions ITC 21 F. Larroca and J.-L. Rougier

10. 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

11. Cost Function Approximation • Why the weights? They address two problems: • Heteroscedasticity • Outliers • Weight wi indicates the importance of measurement i (e.g. outliers should have a small weight) • We have used: where f0(ri)is the k-nearest neighbors estimation ITC 21 F. Larroca and J.-L. Rougier

12. An Example • Measurementsobtained by injecting 72 hoursworth of traffic to a router simulator (C=18750kB/s) ITC 21 F. Larroca and J.-L. Rougier

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

14. Performance Comparison • Considered scenario: Abilene along with a week’s worth of traffic • Performance if weused: • the M/M/1 model instead of WCNLS • A greedyalgorithmwhere (MaxU) • Total Mean Delay • Link Utilization • M/M/1 WCNLS ITC 21 F. Larroca and J.-L. Rougier

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

16. 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? ITC 21 F. Larroca and J.-L. Rougier

17. Thankyou! Questions? ITC 21 F. Larroca and J.-L. Rougier