Content exchange in multiple
intergrated swarms
Ji Zhu, Laurent Massoulie, Ioannidis Stratis, Nidhi Hegde
Technicolor, Paris, 2012/01/07
Outline
 Combine swarms together to increase stability region
 Two file example
 Multiple file generalization
 Peer and piece selection generalization
 Inter-correlated swarms generalization
 Fluid limit analysis for meta-stability
 Random uniform
 Rarest first
 Design of the seed.
A two-file simple example
Parameters
Single seed, two files model
External arrival
Poi(λ1) for file 1, Poi(λ2) for file 2.
Peer departure
As soon as peers for file i (i=1,2) get file i.
Peer select
Random uniform, occurring as Poi(µ) for peers, Poi(Us) for the seed.
Piece select
Random useful
Peers uploading
Instantaneous, upload one piece at one time as Poi(µ).
Seed uploading
Instantaneous, upload one piece at one time as Poi(Us).
For file 1 λ1
Us
For file 2 λ2
µ
Depart
Stability region - one club analysis
Us
λ1
λ2
 If λi>Us, i=1 or 2. the one club grows to infinity.
 If λi<Us, the one club syndrome can get recovered.
 Any two large groups can not coexist long.
Generalize to K files with multiple
pieces.
 Theorem. Suppose there are K files, 1,2,…K. File i is
divided into F_i pieces. Peers without any pieces who want
to download file i arrive as Poi(λi). If random uniform peer
selection and random novel piece selection are applied, the
system is positive recurrent if
max i  U s
i
And the system is transient if
max i  U s
i
Generalize - peer and piece selection
 The stability region remains the same if peers apply weighted
uniform peer selection:
 Can rarest first piece selection increase the stability region? No.
Rarest first: when peer A contacts peer B, peer A downloads the
piece which is held by the least number of peers from the set of
pieces held by B but not held by A. Ties are broken by random
uniform selection.
Generalization - Swarm inter-correlation
 Theorem. Suppose peers without any pieces who want to
download pieces in a set C  F arrive as rate λ_C. If random
uniform peer selection and random novel piece selection (or
rarest first) are applied, the system is positive recurrent if for
all piece i,
max  C  U s
i
C:iC
and the system is transient if
max
i

C:iC
C
 Us
Questions after stability region




The time it takes for the system to go unstable.
Which is more stable? Random uniform? Rarest first?
Meta-stability?
Trade-off between sojourn time, seed rate and stability?
Fluid limit in a single swarm
Fluid limit in single swarm
 The original system converges to the fluid limit in high
arrival rate region.
 Single swarm, no seed, peers depart as soon as obtaining file.
 yC : number of peers holding the set of pieces C.
 Random uniform
 Rarest first. Piece order: 1<2<…<K.
Random Uniform - unstable
The trajectory of the total number of peers and the trajectory of the total piece distribution when K =4
80
The trajectory of the number peers in different types. Layer:1K=4
3
number of peers in different types
2.5
70
60
2
1.5
1
0
50
0
2
4
6
8
10
time
12
14
16
18
10
20
The trajectory of the number peers in different types. Layer:2K=4
number of peers in different types
9
40
30
8
7
6
5
4
3
2
1
0
20
0
2
4
6
8
10
12
time
The trajectory of the number peers in different types. Layer:3K=4
20
18
10
0
0
2
4
6
8
10
time
12
14
16
18
20
number of peers in different types
total piece distribution
0.5
 Parameter: K=4, λ = 5, µ = 1.
 Randomly chosen initial vector y_C(0) = rand(10).
16
14
12
10
8
6
4
2
0
0
2
4
6
8
10
time
12
14
16
18
20
14
16
18
20
Rarest first - Metastability
 Parameter: K=4, λ = 5, µ = 1.
 Randomly chosen initial vector y_C(0) = rand(10).
Rarest first - Metastability
 Parameter: K=4, λ = 5, µ = 1.
 Randomly chosen initial vector y_234=10, y_C = 0.3
for other|C|=K-1, y_C=0.1 for |C|<K-1.
Rarest first - Metastability
 Parameter: K=4, λ = 5, µ = 1.
 Randomly chosen initial vector y_234=10, y_C = 0.2
for other|C|=K-1, y_C=0.1 for |C|<K-1.
Meta-stability for the design of seed.
 Meta-stability indicates that under rarest first, the probability
for system to go unbalanced is quite small, though slightly
larger than zero.
 The seed does not need to help unless the system enters
unbalanced states (one club syndrome).
 In multiple swarm system, the seed can offer help to one
swarm only when pieces in that swarm become too
unbalanced.
Conclusion
 Combining swarms together help to decrease the load of the
seed.
 Meta-stability under rarest first indicates that the help from
the seed is not necessary when pieces are balanced in the
system.
 More work on monitoring of the piece distribution, when
and how the seed should offer help.
Questions?