Stability of decentralised control mechanisms

1. Stability of decentralised control mechanisms Laurent Massoulié Thomson Research, Paris

2. Congestion control • Point-to-point data flows • Data rates regulated by TCP at end-points • Multipath versions: “Overlay” TCP

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

4. Outline • Proportional fairness for congestion control • New characterisation • Implications on stability and insensitivity • “Random useful” packet forwarding for p2p broadcasting • Optimality properties • Open questions

5. Network Bandwidth Allocation problem • Flows of distinct types, sS • Ns such flows • Which rate s to type s flows? • Vector (s )sS : must lie in set C • C: captures physical network constraints • Convex • Non-increasing

6. C2 C1 N0 N1 N2 Network capacity set C,single path flows • Fixed routes & link capacity constraints • (i)C  0+1 ≤ C1, 0+2 ≤ C2 • Polyhedral, convex non-increasing capacity set C

7. a 2c a c c b c b Network capacity set C, multi-path flows • Type s flows can use network paths from set P(s) • Bandwidth: s= pP(s)p • Network capacity (path var.): {p} C’ • path variables: ab + cba +bac ≤ c,… Capacity set C (class variables): b+ c ≤ 2c, a+ c ≤ 2c, b+ a ≤ 2c.

8. Utility-maximising allocations • Maximise sS Ns Us(s/Ns) over C • Distributed control mechanisms (single- and multi-path) known • A special case: Us(x)= ws [x1--1]/(1-) if  1, ws log(x) if  =1  (w,)-fair allocations In terms of Kuhn-Tucker multipliers: TCP square root formula: • “TCP-fairness” corresponds to =2, ws=1/RTT2s

9. The dynamic set-up • Type s file transfers: start at instants of Poisson process, rate s • File sizes: Exponential distribution (s) [or general i.i.d.] • Markov process: Ns++ at rate s, Ns-- at rate s s where s: result of congestion control (time scale separation assumption)

10. Objectives of congestion control • Maximise schedulable region, defined as R = Set of vectors of loadss=s/ssuch that Markov process ergodic • Make performance insensitive to assumption of exponential service times

11. Previous results • Optimal schedulable region R=int(C) • Exponential service times • Max-min fairness [Konstantopoulos et al. 99] • (w,a)-fairness [Bonald-M. 01] • General utility-maximisation schemes [Ye 03] • General i.i.d. service times • Balanced fairness [Bonald-Proutiere 02-04] exactly insensitive; no known distributed control to achieve it • Max-min fairness [Bramson 05]

12. Proportional fairness • Definition: • Alternative characterisation: where J: Fenchel-Legendre transform of (log of) capacity set C:

13. Main application Theorem Proportional Fairness achieves maximal schedulable region R=int(C) for arbitrary phase-type service time distributions (more generally, for original dynamics augmented by Markovian user routing)

14. Proof insights • PF “almost” reversible: • Suggests proof outline: • The “right” Lyapunov function is given by • Apply suitable Lyapunov function criteria for ergodicity (Foster, Rybko-Stolyar, Dai, Robert)

15. Reversible allocations • Markov process reversible iff for some F, in which case, stationary distribution:

16. Reversible allocations (ctd) • “Rate function” of equilibrium distribution decreases along “fluid dynamics” of system (by decrease of Kullback-Leibler divergence between current and stationary distributions)

17. Congestion control – summary • Characterisation of proportional fairness • Yields new stability result • Explains previously observed reversibility on particular topologies (hypergrids) • could yield finer results, e.g. • characterisation of rate function at equilibrium

18. Based on joint work with Andy Twigg, Christos Gkantsidis & Pablo Rodriguez P2P broadcasting

19. Broadcast problem Transmit data from source to all nodes Unstructured (overlay) network Nodes have no global knowledge Models many p2p applications Content distribution Video-on-Demand Live video streaming

20. 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, …

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

22. 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] s s s 1 1 + 1 b b b a a a 1 1 1 c c c

23. 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. a s 1 1 b 1 a b c 1 2 1 1

24. 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) λ New packets injected at rate λ s b a c

25. Assumptions: G: arbitrary edge-capacitated graph Min(mincut(G)): λ* Poisson packet arrivals at source at rate 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

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

27. Optimality proof λ Identify fluid dynamics: s s,a s,b s,a,c s,a,c s,a,b s,b,c 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: Random Block Choice s,a,b,c λ ??

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

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

30. 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 s 4 s s s b a b a b a b 2 a 2 c c c c 2

31. 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 • Can estimate |P(u) \ P(v)| efficiently

32. 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?

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

34. 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?

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

36. Grid networks

37. Delay/loss trade-off for RU policy Expected fraction of skipped packets is (1-1/k)D ~ e-D/k s network v 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.

38. Simulation Uplink capacity Fraction of nodes 128 0.2 384 0.4 1000 0.25 5000 0.15 • 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)

39. 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?

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