1. P2P live streaming: optimality results and open problems Laurent Massoulié Thomson, Paris Research Lab Based on joint work with: Bruce Hajek, Sujay Sanghavi, Andy Twigg, Christos Gkantsidis, Pablo Rodriguez, Thomas Bonald, Fabien Mathieu and Diego Perino

2. 2 Context P2P systems for live streaming & Video-on-Demand – PPLive, Sopcast, TVUPlay, Joost, Verisign… Soon the main channel for multimedia diffusion?

3. 3 Epidemics for live streaming diffusion 1243 Data packets 12 2 Mechanism specification: selection rule for target node packet to transmit  Epidemics (one per packet) competing for resources

4. 4 Rough categories Structured vs Unstructured: – DHT’s vs everything else Trees vs Meshes: – Maintainance of trees along which to forward sub-streams, or not Push vs Pull: – Data selection: receiver-driven or sender-driven

5. 5 Which one is the winning design? Structured approaches: – Clear performance in static configurations – Structure to be maintained in the presence of user churn Epidemic approaches: – No explicit steps to take against churn – Comparable performance? YES!

6. 6 Outline Rate & Delay optimal schemes for symmetric networks [S. Sanghavi, B. Hajek, LM] [T. Bonald, LM, F. Mathieu, D. Perino] Rate-optimal schemes for asymmetric networks – Asymmetric access and multiple commodities [LM and A. Twigg] – Network constraints [LM, C. Gkantsidis, P. Rodriguez and A. Twigg] Open problems

7. 7 Symmetric network with access constraints … Scarce resource: access capacity Symmetry assumptions:  Complete communication graph  Uplink b/w ≡ 1 pkt / sec Bounds on optimal performance Throughput = N / (N-1)  1 (pkt / second) Delay = log 2 (N) where N: number of nodes

8. 8 Structured approaches Based on internal node disjoint trees e.g. odd pkts along blue tree. Even pkts along green tree How to reconstruct trees upon departures (and arrivals)?

9. 9 A naive epidemic scheme: random target / earliest useful pkt 124578 124 Sender’s packets Receiver’s packets 3 1 st useful packet Fraction of nodes reached Time 1 2 3 0.01 0.02 0 40 20 Privileges direct benefit to receiver

10. 10 A better scheme: random target / latest packet 124578 ?? Sender’s packets Receiver’s packets Latest packet ?????? Fraction of nodes reached Time Privileges system overall system benefit

11. 11  Diffusion at rate 63% of optimal and with optimal delay feasible (Do source coding at source over consecutive data windows) A better scheme: random target / latest packet Main result: For arbitrary  >0, each node receives each packet w.p. (1-  )(1-1/e) within delay (1+  ) log 2 (N), Independently for distinct packets

12. 12 A better scheme: random target / latest packet Main result: For arbitrary  >0, each node receives each packet w.p. 1-e -1/10 within delay log 2 (N), Independently for distinct packets

13. 13 Even better: random target / latest useful pkt ? Sender’s packets Receiver’s packets Latest useful pkt ??? 124578 1238

14. 14 I.e:Diffusion at rates arbitrarily close to optimal feasible under optimal delay ( plus constant) Even better: random target / latest useful pkt For arbitrary injection rates λ 0, Each peer receives fraction 1- 1/x of packets in time log 2 (N)+O(x).

15. 15 Asymmetric access constraints Network assumptions: – access capacities, c i – Everyone can send to everyone (complete communication graph) Injection rate: λ  Necessary condition for feasibility:

16. 16 Most deprived target / random useful packet 124578 Sender’s packets 157814 Potential receiver 1Potential receiver 2 5 Source policy: sends “fresh” packets if any (fresh = not sent yet to anyone)

17. 17 Most deprived target / random useful packet 124578 Sender’s packets 157814 Potential receiver 1Potential receiver 2 5 Neighborhood management: Periodically add random neighbor & suppress least deprived neighbor  Fixed neighborhood sizes

18. 18 Main result Provided λ < λ*, system state fluctuates around stable equilibrium point  Hence all packets are received at all nodes after time bounded in probability Many more schemes tested; best contenders so far:  Most Deprived Peer / Latest Useful packet  Latest Packet / Random Useful Peer

19. 19 Multiple commodities Several sources s, Dedicated receiver sets V(s) Can overlap Sources are not receivers Nodes cannot relay commodities they don’t consume …

20. 20 Multiple commodities Necessary conditions for feasibility: Bundled most deprived / random useful: do not distinguish between commodities when – measuring deprivation – Chosing random useful packet System is ergodic when Conditions hold with strict inequality

21. 21 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

22. 22  Based on local informations  No explicit construction of spanning trees Random useful packet selection and Edmonds’ theorem 14 5 124578 Main result: When injection rate λ strictly feasible, Markov process is ergodic ? ? ? ? ? ? ?? ?

23. 23 Proof highlights Fluid limits: renormalisation in time and space  Identify deterministic “fluid” dynamics  Prove their convergence to zero (with Lyapunov function) Corollary: An analytic proof of Edmonds’ combinatorial result

24. 24 Open problems: Performance under user churn Delay performance for asymmetric networks – Impact of topology Multiple commodities Performance with relay nodes – With or without network coding