On Self Adaptive Routing in Dynamic Environments

1. On Self Adaptive Routing in Dynamic Environments -- A probabilistic routing scheme Haiyong Xie, Lili Qiu, Yang Richard Yang and Yin Zhang@ Yale, MR and AT&T Presented by Joe, W.J.Jiang 28-08-2004

2. Outline • Overview of Adaptive Routing • Related Work • Probabilistic Routing Scheme • Convergence Analysis • Simulation Results • Conclusion

3. Where are you? • Overview of Adaptive Routing • Related Work • Probabilistic Routing Scheme • Convergence Analysis • Simulation Results • Conclusion

4. Introduction to adaptive routing • Routing in the Internet:interior gateway routing – OSPFexterior gateway routing – BGP • Static routing, based on hop counts • There is an inherent inefficiency in IP routing from user’s perspective: latency, bandwidth, loss rate, etc • Adaptive routing, allowing end hosts to select routes by themselves.

5. Selfish Routing (user-optimal routing) • Each end host selects a route with minimum latency. • Shortest path routing, metric -- latency, additive • Two approaches:source routing -- Nimrodoverlay routing -- Detour, RON • Selfish by nature -- selfish routing

6. Illustration of source routing n1-n2-n3-n4-n5 n1 n2 n3 n4 n5

7. Illustration of overlay routing

8. Problems I -- Oscillation • Ring Network (Data Networks) • Simultaneous Overlay Network 1+ Mbps (L2) Primary Paths Alternate Paths 2 Mbps L1 Sources Bottleneck Phy. Link Destinations 1 Mbps (L3) Ov.Nw. Nodes (2 Ovns)

9. Problem II -- Performance Degradation • Nash Equilibrium • Well known that Nash Equilibrium do not in general optimize social welfare. • Braess’s Paradox 1 x 1 Performance degradation:selfish routing : global optimal= 2/(0.5+1) = 4/3 1/2 0 s t 1/2 1 x

10. Where are you? • Overview of Adaptive Routing • Related Work • Probabilistic Routing Scheme • Convergence Analysis • Simulation Results • Conclusion

11. Related Work • Wardrop equilibrium: a research aspect in economics of transportation. • The proof of existence of unique equilibrium and some extensions. • Network optimal routing - Data Networks- Frank-Wolfe Method- Projection MethodThese are centralized algorithms. • Distributed version for optimal routing - Parallel and Distributed Computation

12. Related Work (Cont) • “How bad is selfish routing?”- There exists unique Nash Equilibrium for selfish routing under network flow model.- The performance (average delay) ratio between selfish routing and global routing could be unbounded for arbitrary network.- The upper bound for network with linear delay function is 4/3. • “On selfish routing in Internet-like environment”- Based on simulation, selfish routing and global optimal routing exhibit similar performance, under different network topology and traffic models.

13. Related Work (cont) • If individual users are allowed to select routes selfishly without coordination, how to ensure these behaviors will converge to an equilibrium? • “Dynamic Cesaro-Wardrop equilibration in Networks”- a model to ensure the convergence of probabilistic routing scheme • “On self adaptive routing in dynamic environments”

14. Where are you? • Overview of Adaptive Routing • Related Work • Probabilistic Routing Scheme • Convergence Analysis • Simulation Results • Conclusion

15. Routing Scheme - Data Path Component • Data path component- similar to distance vector routing- destination could be all overlay nodes.- - a generalization of normal Internet routing.

16. Routing Scheme - Control Path Component • Control path component - how routing probabilities are updated. • Selfish routing, Wardrop routing, user-optimal routing • property - Given a source-destination pair with a given amount of traffic, the routes with positive traffic should have equal latency, no larger than those unused routes for this source-destination pair.

17. Routing Scheme: Notation • ljithe latency of link from node i to its neighbor j • Ljikthe estimated delay from i to destination k through node j • qjikthe internal probability from node i to destination k through neighbor j • pjikthe routing probability from node i to destination k through neighbor j • {qjik} will converge to the Wardrop equilibrium • {pjik} are ε- approximate of {qjik}

18. Update of routing probabilities (1) • Node i first computes the new delay Δjik = lji + Ljk • Ljkis the estimated latency from node j to node k • node i update the new latency estimationLjik = (1-α(n)) Ljik + α(n) Δjik • α(n) is the delay learning factor. • then node i computes its overall delay estimation Lik to destination k

19. Update of routing probabilities (2) • Node i reports Lik to its neighbors after some delay, and the delay is a random value between T/2 to T, to avoid synchronization. • node i updates its internal routing probabilities: • β(n’) is routing learning factor • ξjik is i.i.d uniform random routing vectors to add disturbance to avoid non-Wardrop solutions

20. Update of routing probabilities (3) • Projection: node i projects the internal routing probabilities to the subspace of [0,1]N(i), which is equivalent to solving the following problem:

21. Update of routing probabilities (4) • Node i compute the routing probabilities:

22. Protocol to implement user-optimal routing

23. Comments on measuring • About measuring lji , two approaches:- measured by node i- measured by node j • The advantage of the second method:- unnecessary for clock synchronization- Δjik = lji + Ljk, there is an offset which is just the clock difference between i and the destination, independent of j.-- overhead is to stamp packets

24. Probabilistic Scheme for network optimal routing • Overview of network optimal routingto solve the convex programming:

25. Probabilistic Scheme for network optimal routing (cont) Proved in “How bad is selfish routing”.

26. Probabilistic Scheme for network optimal routing (cont) • For network optimal routing, replace lji with marginal cost function: mcji = lji + fjisji • sji is the rate of change in the latency from node i to node j at traffic amount fji • Without knowing the analytical expression of latency functions. • However, the paper does not mention the scheme to measure the rate of change in the latency.

27. Where are you? • Overview of Adaptive Routing • Related Work • Probabilistic Routing Scheme • Convergence Analysis • Simulation Results • Conclusion

28. Convergence analysis - Intuition • Consider a network with only two links • p1, p2 >=0, p1+p2=1 • Five cases.- (a) link 1 has higher latency- (b) link 1 has lower latency- (c) link 1 and 2 has the same latency- (d) link 1 has all of the traffic- (e) link 2 has all of the traffic

29. Convergence Analysis - Assumption • A1 - latency function is continuous, non-decreasing and bounded. • A2 - the updates are frequent enough compared with the change rate in the underlying network states. • A3 -

30. Convergence Analysis - Assumption • A4 -

31. Where are you? • Overview of Adaptive Routing • Related Work • Probabilistic Routing Scheme • Convergence Analysis • Simulation Results • Conclusion

32. Evaluation Methodologies • Network topologies: ATT, Sprint, Tiscali • Traffic demands • Traffic stimuli:- Traffic spike- Step function- Linear function • Performance metrics:- average latency- average convergence time- link utilization

33. Dynamics of user-optimal routing and network-optimal routing

34. Conclusion • Overview of Adaptive Routing • Related Work • Probabilistic Routing Scheme • Convergence Analysis • Simulation Results • Conclusion

35. Conclusion • This paper introduces a probabilistic routing scheme to achieve both user-optimal (selfish) routing and network optimal routing. • An application of enforcement learning. • Not consider the issue of fairness between users (or overlays).

36. Thank you!