1. Modeling and analysis of BitTorrent-like P2P network Fan Bin Oct,1 st,2004

2. Review of BitTorrent Downloaders –To have a part (none) of the file Seeds –To have the complete file Piece selection –Rarest First Peer selection –Choking algorithm –Optimistic unchoking

3. Brief Description of this paper Deterministic Fluid model –Steady-state performance –Local Stability –Characterizing Variability File Sharing Model –Effectiveness of File Sharing Incentive Mechanism –Nash Equilibrium in BitTorrent network Optimistic Unchoking –Free-riding effect

4. Basic assumptions Number of Peers that arrive ~ Poisson( λ ) The amount of time after which downloaders independently aborts its downloads ~ Exp(θ) The amount of time for which seeds stay ~ Exp( γ ) All peers have the same uploading bandwidth μ All peers have the same downloading bandwidth c c≥ μ

5. Notations of the model x(t) number of downloaders at time t y(t) number of seeds at time t λ arrival rate of new request μ uploading bandwidth c downloading bandwidth θ the rate at which downloaders abort downloads γ the rate at which seeds leave the system η effectiveness of the file sharing

6. Further explanation The effectiveness of the file sharing η is the fraction of pieces that a peer has on average. For example, a file is cut into 8 pieces: η=(0/8+4/8)/2=25% Downloader1:{} Downloader2:{0,2,5,6}

7. Modeling If no constraint on downloading bandwidth, the total uploading rate will be: Considering the downloading bandwidth Total uploading rate of system:

8. Deterministic fluid model For the evolution of the number of peers( Downloaders and seeds)

9. Equilibrium Equilibrium is a state of a system which does not change In a system described by differential equations, then equilibria can be estimated by setting a derivative (all derivatives) to zero.

10. Steady-State Performance To study the system in steady-state: We obtain the equilibrium: Here

11. Average downloading time According to Little’s Law is the fraction of downloaders that will be seeds. Is the average rate to complete So, the average downloading time:

12. Stability An equilibrium is considered stable if the system always returns to it after small disturbances. If the system moves away from the equilibrium after small disturbances, then the equilibrium is unstable.

13. Stability For the differential equation like: Assume is the equilibrium, The stability of the system is determined by the eigenvalue of the Jacobian Matrix

14. Stability and eigenvalue if the eigenvalues are negative or complex with negative real part, then the equilibrium point is a sink If the eigenvalues are positive or complex with positive real part, then the equilibrium point is a source If the eigenvalues are real number with different sign (one positive and one negative), then the the equilibrium point is a saddle.

15. Local Stability When 1/c<1/ η(1/ μ-1/ γ) –The stability of equilibrium is determined by the eigenvalues of –The system is stable When 1/c>1/ η(1/ μ-1/ γ) –The stability of equilibrium is determined by the eigenvalues of –The system is stable So in general cases, the system will reach the equilibrium, x and y will keep same.

16. Variability Preset a simple characterization of the variance of x and y around the equilibrium point using Gaussian approximation.

17. File Sharing Model Two Assumptions: –Peer i, connected to k=min{x-1,K} other peers. Here K is the maximum number of downloaders to connect. –The file has been cut into N pieces, uniformly distributed in the system

18. Effectiveness of File Sharing

19. In BitTorrent, –A piece is typically 256kB –Number of pieces is of the order of several hundreds –K is typically 40 So –η is very close to 1

20. Incentive Mechanism For peer i, d i is its downloading rate, u i is its uploading rate. d i is the gain and u i is the cost. A peer wants to maximize the gain, also wants to minimize the cost.

21. Incentive Mechanism Study the BitTorrent system as a non- cooperative game Nash equilibrium is a set of uploading rate {u i *} s.t

22. Incentive Mechanism If the network consists of a finite number of groups of peers, in which, all peers have the same physical uploading bandwidth, the Nash equilibrium exists.

23. Optimistic unchoking Optimistic unchoking is used to explore better peers to connect Free-riding means a peer doesn’t contribute anything to the system while it attempts to obtain service from other peers Optimistic unchoking has effect on making free-riding

24. Simple Example Free Rider Opt Unchoking

25. Analysis The total average downloading rate of the free-rider: In BitTorrent, n u =4, thus the free-rider may get 20% of the possible maximum of downloading rate

26. Comments η may be the function of time t, not the constant fraction. Uploading rate and downloading rate of one peer may interfere due to the physical bandwidth constraint. The Nash Equilibrium is in different groups of peers respectively