1. Optimal peer-to-peer broadcasting schemes Laurent Massoulié Thomson Research, Paris Joint work with A. Twigg, C. Gkantsidis and P. Rodriguez

2. Peer-to-peer broadcasting Pplive, Sopcast, TVUPlay,Joost,… Hosts exchange data with “overlay” neighbors Aim: real-time playback at all hosts

3. Broadcast problem Goal: Efficient decentralized schemes  Metrics: broadcast rate & playback delay Constraints:  Edge capacities (well studied, centralized) [distributed]  Node capacities (less explored) Models different nodes in P2P networks: ADSL, cable, …

4. Outline Rate-optimal scheme for edge-capacitated networks Node-capacitated networks Application: video streaming Summary

5. Edge-capacitated case: background λ* = min number of edges to disconnect some node from s Can be achieved by packing edge-disjoint spanning trees [Edmonds,Lovasz, Gabow,…] centralized algorithms broadcast rate, λ* = min [ mincut(s,i): i  V ] [Edmonds, 1972] 1 1 1 a s b c 1 1 1 a s b c a s b c +

6. Challenges Aim for decentralised schemes No explicit tree construction  simplifies management with node churn Manage tension between timeliness and diversity  in-order delivery from s to a & b reduces potential rate from 2 to 1. 1 1 1 1 a s b 1 2 1 a b c

7. Random Useful packet forwarding Let P(u) = packets received by u for each edge (u,v) send a random packet from P(u) \ P(v) New packets injected at rate λ λ a s b c

8. Assumptions: G: arbitrary edge-capacitated graph Min(mincut(G)): λ * Poisson packet arrivals at source at rate λ < λ * Pkt transfer time along edge (u,v): Exponential random variable with mean 1/c(u,v) Theorem With RU packet forwarding, Nb of pkts present at source not yet broadcast: A stable, ergodic process. RU packet forwarding: Main result

9. a s b s,a s s,b s,a,b c s,a,cs,b,c s,a,b,c s,a,c Correct description of state space: Number of packets X A present exactly at nodes u  A, for any set of nodes A (plus state of packets in flight on edges) Optimality of RU – proof

10. Optimality proof s,a s s,b s,a,bs,a,cs,b,c s,a,b,c s,a,c Identify fluid dynamics: λ ?? λ Random Block Choice These capture the original system’s dynamics after some space/time rescaling; Prove that solution of fluid dynamics converges to zero when λ < λ* by exhibiting suitable Lyapunov function:

11. Outline Rate-optimal scheme for edge-capacitated networks Node-capacitated networks Application: video streaming Summary

12. Node-capacitated case P2P networks constrained by node upload capacity:  Cable, ADSL

13. Node-capacitated case P2P networks constrained by node upload capacity:  Cable, ADSL How to allocate upload capacity to neighbours?  By Edmonds thm, optimum can be achieved by assigning node capacities to edges and packing spanning trees a s b c 4 2 2 a s b c 2 a s b c a s b c

14. Most-deprived neighbour selection for each node u  choose a neighbour v maximizing |P(u)\P(v)|  If u=source, and has fresh pkt, send random fresh pkt to v  Otherwise send random pkt from P(u)\P(v) to v Distributed: uses only local information

15. Optimality properties Let λ* be the optimal rate that can be achieved by a feasible allocation of edge capacities {c* ij }. Theorem: For the complete graph and injection rate λ < λ*, system ergodic under fresh/RU pkt forwarding to most deprived neighbour. More general networks?

16. Outline Optimal & decentralized packet forwarding in edge-capacitated networks Node-capacitated networks Application: video streaming Summary

17. Video streaming Model  Assume feasible injection rate λ  Source begins sending at time 0  At time D, users start playing back at rate λ Packets not yet received are skipped  p = fraction of skipped packets How much delay to achieve target p?

18. Grid networks 40x40 grid Add shortcut edges with Pr=0.01 Place source in centre of grid

19. Grid networks

20. Delay/loss trade-off for RU policy Expected fraction of skipped packets is (1-1/k) D ~ e -D/k s v network A toy model: Let k=expected Nb of packets s has and v doesn’t Approximate the network by the following: Source begins with k packets 1..k Source receives new packets at rate λ Source gives randomly useful packets to v at rate λ k reflects connectivity between s and v Fraction of skipped packets decreases exponentially with delay D Can be used to determine suitable playback delay at receiver v.

21. Simulation 0.155000 0.251000 0.4384 0.2128 Fraction of nodes Uplink capacity Random graph (n=500,p=0.05) Distribution of node capacities as observed in Gnutella [Bharambe et al] Optimal rate, λ* ≤ 1180 Delay < 1000 inter-pkt send times (<1min)

22. Conclusions Edge-capacitated networks  Random Useful pkt forwarding achieves optimal broadcast rate  Future: Understand topology impact on delays Extend to dynamic networks Node-capacitated networks  “Most deprived” neighbour selection appears to perform well Proven rate-optimal for complete graphs Future: optimal for other networks?