1. Interaction of Overlay Networks: Properties and Implications Joe W.J. Jiang Dah-Ming Chiu John C.S. Lui The Chinese University of Hong Kong

2. Outline Introduction to overlay routing. Mathematical modeling Properties of NEP Implications of overlay interactions Conclusion

3. Introduction to overlay routing Design philosophy of traditional IP-level routing: simple and scalable. Overlay and P2P networks: harnessing the benefits of a disruptive technology. Overlay creates a virtual topology, and assists user-oriented or application-oriented routing. Overlay nodes relay packets for each other. A large percent of traffic can find better routes by relaying packets with the assistance of overlay nodes.

4. Principles of overlay routing Traditional Overlay Routing select the best path route monitor / update / recovery. route oscillation / race condition. Selfish Overlay Routing (user-optimal) selfish routing, by T.Roughgarden split traffic at the source existence of Nash Equilibrium. probabilistic routing implementation. performance not optimized. Optimal Overlay Routing (overlay-optimal) to split traffic at the source. minimize the average end-to-end delay for the whole overlay a routing optimization at the overlay layer.

5. Motivation of our work There has been little focus on the “interaction” of “co-existence” of multiple overlays. Questions to be answered: What’s the form of interaction ? Is there routing instability ? Is the equilibrium efficient? Can the selfish behavior be led to an efficient equilibrium?

6. What is next? Introduction Mathematical modeling Properties of NEP Implications of overlay interactions Conclusion

7. Preliminary Physical underlying network link delay is a function of the aggregate traffic. delay function - d j (l j ) continuous non-decreasing convex end-to-end delay is the additive form of delays on each link Logical overlay network objective: minimize the average weighted delay for the whole overlay. the delay depends on routing decisions of all overlays average weighted delay (s,t) all source-sink pairs in the overlay P(s,t) set of overlay paths available for source- sink pair (s,t) y k traffic rate assigned to path k delay k end-to-end delay on overlay path k

8. Preliminary (continue) Basic assumptions multiple source-sinks fixed traffic demand constant underlying traffic routing info. obtained from a common routing underlay Form of interactions overlays transparent to each other routing decision dependent on other overlays Interaction occurs when overlays share common resources, e.g. physical links, bandwidths, nodes.

9. Routing Optimization y (s) routing decision for overlay s A (s) routing matrix for overlay s y (-s) routing decision for other overlays A routing matrix for all overlays H matrix indicating available paths for each source- sink pair delay (s) the average weighted delay of overlay s demand constraint capacity constraint non-negativity constraint Algorithmic Solution Apply any convex programming techniques Marginal cost network flow

10. Algorithmic Solutions Apply any convex programming techniques objective function is convex. feasible region is convex and compact. optimal value and optimizer can be found by the Lagrangian method. Marginal cost network flow for each physical link, replace the delay cost by the marginal cost of the weighted delay. marginal cost -- first derivative weighted delay -- rate of traffic traversing a link in its own overlay * delay on this link split traffic among all available paths, s.t. all paths with positive traffic flow have the same end-to-end cost, smaller than paths with zero-traffic.

11. What is next? Introduction Mathematical modeling Properties of NEP Implications of overlay interactions Conclusion

12. Overlay Routing Game Nash routing game player: all overlays strategy: feasible region of OVERLAY (s) preference: a smaller average delay Routing behaviors of overlays different routing update period calculation of the optimal routing strategy Is there routing instability?

13. Nash Equilibrium Point A feasible strategy profile y=(y (1),…, y (s),…, y (n) ) T is a Nash Equilibrium in the overlay routing game if for every overlay s ∈ N, delay (s) (y (1),…y (s),…y (n) ) ≤ delay (s) (y (1),…y *(s),…y (n) ) for any other feasible strategy profile y *(s). Existence of NEP Theorem In the overlay routing game, there exists a Nash Equilibrium if the delay function delay (s) (y (s) ; y (-s) ) is continuous, non-decreasing and convex.

14. six co-existing overlays one source-sink pair each overlay overlapping physical links, physical nodes different routing update period simulation topology Fluid Simulation average delay for six overlays v.s. simulation time traffic flow for six overlays v.s. simulation time transient period convergence number of curves equals number of available paths for each flow Interaction of routing decisions convergence of routing decisions different convergence rate

15. What is next? Introduction Mathematical modeling Properties of NEP Implications of overlay interactions Conclusion

16. Anomalies of routing equilibrium Interesting questions Is the equilibrium point efficient? Can the selfish behavior be led to an efficient equilibrium? Anomalies due to unregulated competition for common resources : sub-optimality slow convergence fairness paradox

17. Example of illustration 1 unit y1y1 1-y 1 y2y2 1-y 2

18. Sub-optimality d 15 (l) = 1+l y1y1 1-y 1 y2y2 1-y 2 d 34 (l) = l d 26 (l) = 2.5+l Other links zero delay Nash Equilibrium Point y 1 =0.5 y 2 =1.0 delay 1 =delay 2 =1.5 Not Pareto Optimal A Point on Pareto Curve y 1 =0.4 y 2 =0.9 delay 1 =1.48<1.5 delay 2 =1.43<1.5

19. Slow-convergence 5 unit 65 C 34 =? 10 C 34 =8 C 34 =6 slower convergence

20. Fairness Paradox 1 unit d 26 (l) = c+l d 15 (l) = a+l d 34 (l) = bl α 00 0 0 a, b, c, α are non-negative parameter of the delay functions Everything is symmetric except for the two private links common link: n3-n4 private links: n1-n5 overlay1 n2-n6 overlay2 Unfariness becomes unbounded!

21. War of Resource Competition 1 unit y1y1 1-y 1 y2y2 1-y 2 P g (y 1 +y 2 ) P c1 (1-y 1 ) P c2 (1-y 2 ) Min y 1 P g (y 1,y 2 )+(1-y 1 )P c1 (1-y 1 ) Min y 2 P g (y 1,y 2 )+(1-y 2 )P c2 (1-y 2 ) P c1 <P c2 ? >

22. What is next? Introduction Mathematical modeling Properties of NEP Implications of overlay interactions Conclusion

23. Study the interaction between multiple co- existing overlays. Formulate the game as a non-cooperative Nash routing game. Prove the existence of NEP. Show the anomalies and implications of the NEP.

24. Thank you for your attention! Q & A