Modeling and analysis of BitTorrent-like P2P network

1
Modeling and analysis of BitTorrent-like P2P
network

• Fan Bin
• Oct,1st,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(?)
• The amount of time after which downloaders
  independently aborts its downloads Exp(?)
• The amount of time for which seeds stay Exp(?)
• 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
• ? arrival rate of new request
• µ uploading bandwidth
• c downloading bandwidth
• ? the rate at which downloaders abort downloads
• ? the rate at which seeds leave the system
• ? effectiveness of the file sharing

6
Further explanation

• The effectiveness of the file sharing ? is the
  fraction of pieces that a peer has on average.
• For example, a file is cut into 8 pieces
• ?(0/84/8)/225

Downloader20,2,5,6
Downloader1
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 Littles 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/clt1/ ?(1/ µ-1/ ?)
• The stability of equilibrium is determined by the
  eigenvalues of
• The system is stable
• When 1/cgt1/ ?(1/ µ-1/ ?)
• 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 kminx-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
Effectiveness of File Sharing

• In BitTorrent,
• A piece is typically 256kB
• Number of pieces is of the order of several
  hundreds
• K is typically 40
• So
• ? is very close to 1

20
Incentive Mechanism

• For peer i, di is its downloading rate, ui is
  its uploading rate.
• di is the gain and ui 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 ui
  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 doesnt 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
Opt Unchoking
Opt Unchoking
Free Rider
Opt Unchoking
25
Analysis

• The total average downloading rate of the
  free-rider
• In BitTorrent, nu4, thus the free-rider may get
  20 of the possible maximum of downloading rate

26
Comments

• ? 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