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?

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Epidemics for live streaming diffusion

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Data packets

1 2

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Mechanism specification: selection rule for• target node• packet to transmit

Epidemics (one per packet) competing for resources

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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

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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

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1st useful packet

Fraction of nodes reached

Time

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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

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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

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Sender’s packets

1 5 7 8 1 4

Potential receiver 1 Potential receiver 2

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Source policy: sends “fresh” packets if any(fresh = not sent yet to anyone)

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Most deprived target / random useful packet

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Sender’s packets

1 5 7 8 1 4

Potential receiver 1 Potential receiver 2

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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

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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