1. Minimum-Delay Load-Balancing Through Non-Parametric Regression F. Larroca and J.-L. Rougier IFIP/TC6 Networking 2009 Aachen, Germany, 11-15 May 2009

2. 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 ) is typically a convex increasing function that diverges as  l → c l ; e.g. mean queuing delay Why queuing delay? Simplicity and versatility IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

3. page 2 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 f l (  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

4. page 3 Agenda Introduction Attaining the optimum Delay function approximation Simulations Conclusions IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

5. 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 IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

6. 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: IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

7. 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:  l (  l ) should be continuous, positive and non-decreasing IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

8. page 7 Agenda Introduction Attaining the optimum Delay function approximation Simulations Conclusions IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

9. 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 Convex Nonparametric Least-Squares (CNLS) 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 IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

10. page 9 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

11. page 10 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

12. page 11 An Example IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

13. page 12 Agenda Introduction Attaining the optimum Delay function approximation Simulations Conclusions IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

14. page 13 NS-2 simulations The considered network: IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

15. page 14 NS-2 simulations Alternative (“wrong”) training set: IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

16. page 15 Agenda Introduction Attaining the optimum Delay function approximation Simulations Conclusions IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

17. page 16 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

18. page 17IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier Thank you! Questions?