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

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

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

Challenge

Naïve approach Random target First useful packet

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

Tension between timeliness of delivery and diversity

10

The “random target / latest packet” policy

1 2 4 5 7 8

? ?

Sender’s packets

Receiver’s packets

Latest packet

??????

Fraction of nodes reached

Time

11

Diffusion at rate 63% of optimal and with optimal delay feasible

(Do source coding at source over consecutive data windows)

The “random target / latest packet” policy

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

t

Proof idea

1 1

11 1

eNN

NN

time

Fraction of nodes

t+1

Nodes that have pkt with label t

Nodes that have pkt with label t+1

Number of transmission attempts for packet t: N area between curves = N

1

Number of nodes receiving t:

Same dynamics as single epidemic diffusiontranslated logistic curve

sfesfsfsf

Nf

1 11

,1

0

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:

i

is cN

c1

1 , min*

15

The “Most deprived neighbour / random useful packet” policy

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)

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 neighboursSends to most deprived from neighbour setPeriodically adds randomly chosen neighour, and

dumps least deprivedNeighbourhood 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 graphCoincides with minimal max-flow ( = min-cut) between source and arbitrary receiver

19

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

?

??

?

?

?

??

?

20

Proof idea

s

1 2

3Original network

Variables xA: Number of packets present exactly at nodes in set A

•Fluid Renormalisation: The xA obey deterministic dynamics

s,1

s

s,1,2,3

s,2

s,1,3 s,2,3Induced network

s,1,2

λ

λ ?

•Convergence to zero of fluid trajectories:

shown by using Lyapunov function VAxxL AA : sup)(

21

Comments

Provides “analytical” proof of Edmond’s theoremDelays?

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?