Modelling and Performance Analysis of BitTorrent-Like Peer-to-Peer Networks

Contents Simple deterministic fluid model – to calculate average file transfer delay in BitTorrent Stochstic fluid model – characterizes number of peers around equilibrium Model to study efficiency of BitTorrent Incentive mechanism in BitTorrent and its effect on n/w performance

Brief Introduction to BitTorrent Single large file is divided into pieces of 256 KB each Peer downloads different pieces from different peers Each peer uploads to fixed number of users (Default 4) Uploads to peers having best downloading rates Optimistic unchoking

Deterministic fluid model x(t) number of downloaders (also known as leechers) in the system at time t. y(t) number of seeds in the system at time t. λ the arrival rate of new requests( Poisson process) μ the uploading bandwidth of a given peer. We assume that all peers have the same uploading bandwidth. c the downloading bandwidth of a given peer. We assume that all peers have the same downloading bandwidth. θ the rate at which downloaders abort the download. γ the rate at which seeds leave the system. η indicates the effectiveness of the file sharing

dx/dt= λ − θ x(t) − min{cx(t), μ ( η x(t) + y(t))}, dy/dt= min{cx(t), μ ( η x(t) + y(t))} − γ y(t), To study steady state performance dx(t)/dt=dy(t)/dt= 0 and obtain equilibrium values x and y By using Little’s law (( λ − θ x)/ λ )x = ( λ − θ x)T, T =1/( θ + β ) where 1/ β = max{1/c, 1/ η ( 1/ μ − 1/ γ )}.

Remarks The average downloading time T is not related to λ, hence p2p system scales very well When η increases, T decreases When γ increases, T increases because a larger γ means that there are fewer seeds in the system Initially, when c increases, T decreases. However, once c is large enough, increasing c further will not decrease T, because the downloading bandwidth is no longer the bottleneck. A similar observation can be made regarding the uploading bandwidth μ.

Importance of η When η = 0 and γ > μ dy(t)/dt ≤ ( μ − γ )y(t) number of seeds will exponentially decrease to zero and the system dies When η > 0, the system reaches a steady state no matter what γ is. η = 1− P{downloader j needs no piece from downloader i}^ k η is close to 1 in BitTorrent

Stability around equilibrium Eigenvalues of matrix obtained from differential equations have strict negative real parts in all cases hence system is stable around equilibrium Stochastic fluid model : From the solution of stochastic differential equation it can be seen that in steady-state, the number of seeds and downloaders is distributed as Gaussian random variables it involves showing that the original stochastic process converges to the deterministic and stochastic differential equation limits when the arrival rate goes to ∞

Incentive Mechanism in BitTorrent In BitTorrent a peer decides to upload to peers from which it can get highest downloading rates Here optimistic unchoking is ignored and assumed that each peer has global information of uploading rates Stepwise peer selection algorithm Each peer tries to maximize downloading rate(gain) and minimize uploading rate(cost),this can be seen as non-cooperative game given game rules (peer selection algorithm) In a general n/w setting, nash equilibrium point exists with equilibrium uploading b/w = max physical uploading b/w of peer

Optimistic unchoking and free riding In Bittorrent n/w peer has the rate information about peers from which it is downloading. Hence optimistic unchoking is used to explore the network and obtain information about other peers and this gives opportunity to free-ride. In a network with group of peers having same b/w μ,free rider gets average downloading rate ≈ μ /(n u + 1) In Bittorrent, n u = 4 (default),hence free rider gets 20% of the max downloading rate

Fairness, incentives and performance in peer-to-peer networks

Contents To study sensitivity of service capacity of P2P networks, following models are proposed 1.Branching process model in transient regime 2.Markov chain model in stationary regime Fairness and incentives in P2P system

Throughput in BitTorrent N/w

Transient regime Deterministic model to understand file sharing mechanism Assumptions : Initially only one copy of file available n= 2^k demands, limited upload capacity = b,unlimited download bandwidth, file size = s A peer can only serve a document once it has been fully downloaded. Then average delay per peer = t log 2 n where t= s/b

Average delay seen by peers scales as log2 n which is favorable relative to the linear scaling of n in case of fixed set of servers Delay is further reduced by a factor of 1/m in case of multipart download

Branching process model Following model takes care of randomness of service times and abnormal departures of peers

Observations : 1.Variability in generation times improves the growth exponent 2.Increased parallelism typically may decrease the growth exponent 3.Uncooperative peers and parallel downloads : if seeds leave the system with probability p, here vp>1 and to keep growth rates higher keep higher v

Stationary regime : Markov chain model x- number of downloaders y- number of seeds (x,y) – two dimensional markov chain with following generator matrix Can find mean number of seeds,downloaders and mean delay for this system