Epidemic Dissemination & Efficient Broadcasting in Peer-to-Peer Systems

1. Epidemic Dissemination & Efficient Broadcasting in Peer-to-Peer Systems Laurent Massoulié Thomson, Paris Research Lab Based on joint work with: Bruce Hajek, Sujay Sanghavi, Andy Twigg, Christos Gkantsidis and Pablo Rodriguez

2. Context • P2P systems for live streaming & Video-on-Demand • PPLive, Sopcast, TVUPlay, Joost, Kontiki… • Internet hosts form overlay network • Data exchanges between overlay neighbours • Aim: real time playback at all receivers • Soon the main channel for multimedia diffusion?

3. Diffusion of Code Red Virus

4. Diffusion of Code Red Virus Logistic curve (Verhulst 1838, Lotka 1925,…) Exponential growth Optimal global infection time:logarithmic in population size

5. Epidemics for live streaming diffusion Data packets 1 2 3 4 1 2 2 • Mechanism specification: selection rule for • target node • packet to transmit Epidemics (one per packet) competing for resources

6. Problem statement • Currently deployed systems rely on epidemic approach • Appeal of simple & decentralised schemes • Large user populations (103 – 106) • High churn (nodes join and leave) • “Cost of decentralisation? i.e., can epidemics make efficient use of communication resources? Metrics: rate and delay

7. Outline • Delay-optimal schemes [S. Sanghavi, B. Hajek, LM] • Rate-optimal schemes [LM, C. Gkantsidis, P. Rodriguez and A. Twigg] • Outlook

8. The access constraint scenario Scarce resource: access capacity • Models DSL / Cable uplink bandwidth limitations • Normalised: 1 packet / second … • Bounds on optimal performance • Throughput = N / (N-1)  1 (pkt / second) • Delay = log2(N) where N: number of nodes

9. 0.02 0.01 0 20 40 Challenge Fraction of nodes reached Naïve approach • Random target • First useful packet 1 2 3 Sender’s packets 1 2 4 5 7 8 1st useful packet 1 2 3 4 Time Receiver’s packets Tension between timeliness of delivery and diversity

10. The “random target / latest packet” policy Fraction of nodes reached Sender’s packets 1 2 4 5 7 8 Latest packet ? ? ? ? ? ? ? ? Receiver’s packets Time

11. The “random target / latest packet” policy • Diffusion at rate 63% of optimal and with optimal delay feasible (Do source coding at source over consecutive data windows) Main result: Each node receives each packet w.p. 1-1/e  63% with optimal delay ( less than log2(N) ), Independently for distinct packets.

12. Proof idea Nodes that have pkt with label  t Fraction of nodes 1 Same dynamics as single epidemic diffusion translated logistic curve Nodes that have pkt with label  t+1 time t t+1 Number of transmission attempts for packet t: N  area between curves = N  Number of nodes receiving t:

13. Outline • Delay-optimal schemes [S. Sanghavi, B. Hajek, LM] • Rate-optimal schemes [LM, C. Gkantsidis, P. Rodriguez and A. Twigg] • Outlook

14. Access constraints scenario • Network assumptions: • access capacities, ci • Everyone can send to everyone (complete communication graph) • Statistical assumptions: • source creates fresh packets at instants of Poisson process with rate λ • Packet transmission time from node i: Exponential r.v. with mean 1/ci  Optimal broadcast rate:

15. The “Most deprived neighbour / random useful packet” policy Sender’s packets 1 2 4 5 7 8 5 1 5 7 8 1 4 Potential receiver 1 Potential receiver 2 Source policy: sends “fresh” packets if any (fresh = not sent yet to anyone)

16. Main result • Provided λ < λ*, Markov process describing system state is ergodic. • Hence all packets are received at all nodes after time bounded in probability Proof: identifies “workload” as Lyapunov function for fluid dynamics of Markov process Open questions: • Magnitude of delays (simulations suggest logarithmic) • Extension to general, not complete graphs

17. Extension to limited neighborhoods • Each node maintains shortlist of neighbours • Sends to most deprived from neighbour set • Periodically adds randomly chosen neighour, and dumps least deprived Neighbourhood size stays fixed Ergodicity result still holds: fluid dynamics unchanged Q: impact of neighborhood size?

18. Network constraints • Graph connecting nodes • Capacities assigned to edges • Achievable broadcast rate [Edmonds, 73]: • Equals maximal number of edge-disjoint spanning trees that can be packed in graph • Coincides with minimal max-flow ( = min-cut) between source and arbitrary receiver

19. Random useful packet selection and Edmonds’ theorem 1 2 4 5 7 8 Based on local informations No explicit construction of spanning trees 5 1 4 Main result: When injection rate λ strictly feasible, Markov process is ergodic ? ? ? ? ? ? ? ? ?

20. Proof idea λ s s s,1 s,2 1 2 3 s,1,3 s,1,2 s,2,3 Original network Induced network s,1,2,3 λ ? Variables xA: Number of packets present exactly at nodes in set A • Fluid Renormalisation: • The xA obey deterministic dynamics • Convergence to zero of fluid trajectories: • shown by using Lyapunov function

21. Comments • Provides “analytical” proof of Edmond’s theorem • Delays?

22. Conclusions • Epidemic diffusion • Straightforward implementation • Efficient use of bandwidth resources • Random & local decisions lead to global optimum

23. Outlook • Open problems • Schemes both delay- and rate- optimal? • Concurrent stream diffusions? • Stability proofs without the Lyapunov function?