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
Context

P2P systems for live streaming & Video-on-Demand
– PPLive, Sopcast, TVUPlay, Joost, Verisign…

2
Soon the main channel for multimedia diffusion?
Epidemics for live streaming diffusion
Data packets
1 2 3 4
1 2
2
Mechanism specification: selection rule for
• target node
• packet to transmit
Epidemics (one per packet) competing for resources
3
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
4
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!
5
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]

6
Open problems
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)
7
•Delay = log2(N) where N: number of nodes
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)?
8
A naive epidemic scheme: random target / earliest useful pkt
Fraction of nodes reached
Sender’s packets
0.02
1 2
4 5
1
7 8
1st
2
useful packet
1 2 3 4
3
0.01
Receiver’s packets
Privileges direct benefit to receiver
0
20
40
Time
9
A better scheme: random target / latest packet
Fraction of nodes reached
Sender’s packets
1 2
4 5
7 8
Latest packet
? ? ? ? ? ? ? ?
Receiver’s packets
Privileges system overall system benefit
10
Time
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
 Diffusion
at rate 63% of optimal and with optimal delay
feasible
(Do source coding at source over consecutive data
windows)
11
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
12
Even better: random target / latest useful pkt
Sender’s packets
1 2
4 5
7 8
Latest useful pkt
1 ?
2 ?
3
?
8
?
Receiver’s packets
13
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).
I.e:
Diffusion at rates arbitrarily close to optimal
feasible under optimal delay ( plus constant)
14
Asymmetric access constraints

Network assumptions:
– access capacities, ci
– Everyone can send to everyone (complete communication graph)

Injection rate: λ
 Necessary condition for feasibility:


1


  *  min cs ,
ci 
N 1


i



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Most deprived target / random useful packet
Sender’s packets
1 2
1
5
4 5
5
7 8
Potential receiver 1
7 8
1
4
Potential receiver 2
Source policy: sends “fresh” packets if any
(fresh = not sent yet to anyone)
16
Most deprived target / random useful packet
Sender’s packets
1 2
1
5
4 5
5
7 8
Potential receiver 1
7 8
1
4
Potential receiver 2
Neighborhood management:
Periodically add random neighbor & suppress least deprived
neighbor

17 Fixed neighborhood sizes
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
18
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:
s  c s , s  S
 V
 s 
s 1
sK
c ,
u
u sK Vs
KS
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
20
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
21
Random useful packet selection
and Edmonds’ theorem
1 2
1
4 5
5
7 8
4
Main result:
When injection rate λ strictly feasible,
Markov process is ergodic
?
?
?
?
Based on local informations
No explicit construction of spanning trees
22
?
?
?
?
?
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
23
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
24