Minimum-Delay Load-Balancing Through Non-Parametric Regression F. Larroca and J.-L. Rougier

1. Minimum-Delay Load-Balancing Through Non-Parametric RegressionF. Larroca and J.-L. Rougier IFIP/TC6 Networking 2009 Aachen, Germany, 11-15 May 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 ) is typically a convex increasing function that diverges as rl → cl; e.g. mean queuing delay • Why queuing delay? Simplicity and versatility IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

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

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

8. Agenda • Introduction • Attaining the optimum • Delay function approximation • Simulations • Conclusions IFIP/TC6 Networking 2009 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 • 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 fNϵ F where F is the set of continuous, non-decreasing and convex functions IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

10. 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. 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. An Example IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier

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

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

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

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

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

18. Thankyou! Questions? IFIP/TC6 Networking 2009 F. Larroca and J.-L. Rougier