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
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P2P live streaming: optimality results and open problems Laurent Massoulié Thomson, Paris Research Lab Based on joint work with: Bruce Hajek, Sujay Sanghavi,
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P2P live streaming:optimality results and open problems
Laurent MassouliéThomson, 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
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Context
P2P systems for live streaming & Video-on-Demand– PPLive, Sopcast, TVUPlay, Joost, Verisign…
Soon the main channel for multimedia diffusion?
3
Epidemics for live streaming diffusion
1 2 43
Data packets
1 2
2
Mechanism specification: selection rule for• target node• packet to transmit
Epidemics (one per packet) competing for resources
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
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!
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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
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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 = log2(N) where N: number of nodes
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Structured approaches
Based on internal node disjoint treese.g. odd pkts along blue tree.Even pkts along green tree
How to reconstruct trees upon departures (and arrivals)?
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A naive epidemic scheme: random target / earliest useful pkt
1 2 4 5 7 8
1 2 4
Sender’s packets
Receiver’s packets
3
1st useful packet
Fraction of nodes reached
Time
12
3
0.01
0.02
04020
Privileges direct benefit to receiver
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A better scheme: random target / latest packet
1 2 4 5 7 8
? ?
Sender’s packets
Receiver’s packets
Latest packet
??????
Fraction of nodes reached
Time
Privileges system overall system benefit
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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+) log2(N), Independently for distinct packets
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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 log2(N), Independently for distinct packets
13
Even better: random target / latest useful pkt
?
Sender’s packets
Receiver’s packets
Latest useful pkt
???
1 2 4 5 7 8
1 2 3 8
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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<1, and x>0,Each peer receives fraction 1- 1/x of packets in time log2(N)+O(x).
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Asymmetric access constraints
Network assumptions:
– access capacities, ci
– Everyone can send to everyone (complete communication graph)
Injection rate: λ
Necessary condition for feasibility:
i
is cN
c1
1 , min*
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Most deprived target / random useful packet
1 2 4 5 7 8
Sender’s packets
1 5 7 8 1 4
Potential receiver 1 Potential receiver 2
5
Source policy: sends “fresh” packets if any(fresh = not sent yet to anyone)
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Most deprived target / random useful packet
1 2 4 5 7 8
Sender’s packets
1 5 7 8 1 4
Potential receiver 1 Potential receiver 2
5
Neighborhood management:Periodically add random neighbor & suppress least deprived neighbor Fixed neighborhood sizes
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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 packetLatest Packet / Random Useful Peer
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Multiple commodities
Several sources s, Dedicated receiver sets V(s) Can overlap
Sources are not receivers Nodes cannot relay commodities they don’t consume
…
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Multiple commodities
Necessary conditions for feasibility:
Bundled most deprived / random useful: do not distinguish between commodities when
– measuring deprivation– Chosing random useful packet
SKcV
Ssc
sKs Vu
us
Ks
s
ss
, 1
,
System is ergodic when Conditions hold with strict inequality
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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 graphCoincides with minimal max-flow ( = min-cut) between source and arbitrary receiver
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Based on local informations
No explicit construction of spanning trees
Random useful packet selection and Edmonds’ theorem
1 4
51 2 4 5 7 8
Main result:
When injection rate λ strictly feasible,
Markov process is ergodic
?
??
?
?
?
??
?
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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
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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