Content exchange in multiple intergrated swarms

1. Content exchange in multiple intergrated swarms Ji Zhu, Laurent Massoulie, Ioannidis Stratis, NidhiHegde Technicolor, Paris, 2012/01/07

2. 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.

3. A two-file simple example For file 1 λ1 Us µ Depart For file 2 λ2

4. Stability region - one club analysis • Ifλi>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. Us λ1 λ2

5. 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 And the system is transient if

6. 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.

7. Generalization - Swarm inter-correlation • Theorem. Suppose peers without any pieces who want to download pieces in a set 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, and the system is transient if

8. 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?

9. 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. • : number of peers holding the set of pieces C. • Random uniform • Rarest first. Piece order: 1<2<…<K.

10. Random Uniform - unstable • Parameter: K=4, λ = 5, µ = 1. • Randomly chosen initial vector y_C(0) = rand(10).

11. Rarest first - Metastability • Parameter: K=4, λ = 5, µ = 1. • Randomly chosen initial vector y_C(0) = rand(10).

12. 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.

13. 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.

14. 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.

15. 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?