A Statistical Physics approach for Modeling P2P Systems

1
A Statistical Physics approach for Modeling P2P
Systems

• Giovanna Carofiglio1, R.Gaeta2, M.Garetto1,
• P.Giaccone1, E.Leonardi1, M.Sereno2

1 Politecnico di Torino, 2 Università di Torino
Italy
MAMA Workshop joint with ACM SIGMETRICS
2005 Banff, June 6-10, 2005
2
Outline

• Motivation
• Basic Model
• Extended Model
• Content Search
• Download effects

3
P2P System Architecture
server
clients
peers

• A possible definition

Decentralized, self-organizing distributed
systems, in which all or most communication is
symmetric.
4
Peer-to-Peer traffic

• P2P is the single largest generator of traffic
• P2P traffic significantly outweights web traffic
• P2P traffic is continuing to grow

5
P2P Applications

• File Sharing
• BitTorrent, KaZaA, Gnutella, eDonkey, Napster,
  etc.
• DHTs
• Chord, CAN, Pastry, Tapestry
• Wireless Ad hoc Networking

• Communication
• Voice Over IP Skype
• Instant Messaging
• Distributed Computation
• Seti_at_home, UnitedDevices, Distributed Science

6
Motivation

• Most of the Internet traffic is generated by p2p
  applications.
• Performance studies of p2p systems may be useful
  to drive the design of future applications.
• Analytical models help analyzing
• large and complex p2p networks.

7
Modeling techniques

• Traditional Markov Models
• A detailed microscopic description is provided
  but with a huge space-state.
• It is computationally expensive to analyze large
  systems like p2p systems (with million of users
  and contents shared).
• Fluid models
• Network dynamics are described with an increased
  level of abstraction, neglecting stochastic
  information.
• Scalability the model is based on a set of
  differential equations invariant w.r.t. the size
  of the network (n.users, link cap)

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Model description

• We model a generic p2p system without focusing
  on a particular implementation.

• Based on a fluid approach like in 1 and 2,
  our model evolves in a second-order
  diffusion approximation where stochasticity in
  networks dynamics plays a relevant role.

• The model provide a description of
  users/contents dynamics both in transient and in
  steady state.

1F. Clevenot, P. Nain, A Simple Model for the
Analysis of SQUIRREL, Infocom 2004, Hong Kong,
Mar 2004. 2D. Qiu, R. Srikant, Modeling and
Performance Analysis of BitTorrent like
Peer-to-Peer Networks, Sigcomm 2004, U.S.A.
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Model structure
Users dynamics
Contents dynamics
Search phase
Download phase
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Outline
2

• Motivation
• Basic Model
• Extended Model
• Content Search
• Download effects

11
Users dynamics (1)

• The number of users joining the p2p network
  dynamically changes according to
• Enter-leave dynamics
• ? u new users arrival rate 1/µu
  average subscription time
• 
  
• Active-Sleeping mode
• 1/µas average active time 1/µsa
  average sleeping time

• Users in sleeping mode do not interact at all
  with the other users of the community.

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Users dynamics (2)
The evolution of the number of users in active or
sleeping mode, Ua and Us respectively, can be
described by two fluid differential equations
sleeping users who become active
new users
active users who become sleeping
active users who leave the system
active users who become sleeping
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Content Dynamics

• The evolution of the number of available copies
  of a content is driven by 2 phenomena
• the generation of new copies (downloads or
  off-on transitions)
• the cancellation of existing copies

? average request rate 1/µh , 1/µh average
content holding time for active/sleeping users
Note psps(µh ) is the probability that
sleeping users have the considered content when
they become active.
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Brownian Motion

• Content dynamics are modelled through
  a Second-Order Diffusion
  Approximation

Each content is a particle with instantaneous
position x(t) moving accordingly to a Brownian
motion.
Langevin equation
The evolution of the pdf f(x,t) over
follows
Fokker Planck equation
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Content diffusion equation

• The pdf F(x,t) of the number of copies follows
  the F.P. equation with boundary conditions
  for

Introduction of new contents in the system

• A content can disappear when are no more copies
  available. The rate at which a content disappear
  is

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Diffusion Parameters

• m(x,t) expresses the average speed at which
  the content-particle moves along the x
  axis.

hh variation coefficient of holding time hr
variation coefficient of inter request
time

• The variance s2(x,t) expresses the burstiness
  of the processes.

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Case Content disappearance (1)

• In a single-content scenario we study the
  probability that the content disappears as a
  function of the users dynamics.

Network parameters
Initial condition

• Active Users 10
• Sleeping Users 10
• Copies Availables 1

• ? u users arrival rate 0.1 ut/s
• 1/µu avg subscription time 4000 s
• 1/µas avg active period 400 s
• 1/µsa avg sleeping period 400 s
• ? average request rate
• 1/µh ,1/µh avg content holding time for
  a/s users 100 s

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Case Content disappearance (2)
Che grafico facciamo vedere? Modello e simulatore
michele a confronto? Solo Modello?
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Outline
2

• Motivation
• Basic Model
• Extended Model
• Content Search
• Download effects

20
Dual distribution

• Relations between users and contents dynamics

• The number of active and sleeping users at time
  t

• The number of copies available at time t

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Dual equations

• Ga(x,t) and Gs(x,t) are the pdf of the number
  of active and sleeping users having x contents

active users who become sleeping or leave the
system
sleeping users who become active
new users
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Diffusion parameters

• As for the contents diffusion equation m(x,t)
  expresses the average speed at which the
  copy-particle moves along the x axis, while
  s2(x,t) expresses the variance of the associated
  process.

ra rate of generation of new copies da/s rate
of cancellation of existing copies
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Multi-contents case (1)

• In a multi-content scenario, still assuming ideal
  search and download we study the steady state
  distribution of the contents among users.

Network parameters
Initial condition

• ? u users arrival rate 0 ut/s
• 1/µu avg subscription time inf
• 1/µas avg active period 6 h
• 1/µsa avg sleeping period 18 h
• ? average request rate 2 c/h
• ? c contents introduction 1/600 c/s
• 1/µh ,1/µh avg content holding time for
  a/s users 10 h, 8 h

• Active Users 2500
• Sleeping Users7500
• Copies Availables 1

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Multi-contents case (2)
Che grafici facciamo vedere? Modello e simulatore
michele a confronto? Solo Modello?
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Outline
2

• Motivation
• Basic Model
• Extended model
• Content Search
• Download effects

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The contents trasfer rate

• In a non-ideal p2p system the transfer rate of
  the contents dynamically changes according to

• the probability of a successful search phit(x,t)
  (related to content diffusion, search algorithm)

• the probability of a successful download
  pdown(x,t) (related to network congestion, user
  impatience, on-off dynamics)

The effective retrieval rate becomes

• Both search and download require to know F(x,t)
  and provide it as a function of time.

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Search Phase

• Search algorithm flooding in an unstructured
  p2p network

For each content request a query message is
forwarded to all the neighbors up to the distance
max_ttl

• Graph Model
  The P2P network
  topology is modeled as a random finite graph.

Active peer
Application-level connection
We consider Generalized Random Graph (GRG) to
allow an arbitrary vertex degree distribution.
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GRG Model

• Given the probability distribution pk that a
  vertex has k edges departing from it, we can
  define the generating function
• It can be shown that the generating function of
  the number of
• the first neighbors with a copy of the
  content is

a x/Ua X copies Uaactive users

• The composition of these generating functions
  gives the generating function of the number of
  neighbors at distance h

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GRG Topology

• Now we can define the generating function for
  the number of neighbors at distance up to max_ttl
  that have a copy of the content
• Hence it derives the hit probability

• To compute the pdf of the GRG nodes degree
  we adopt a M/M/8 queue

Assuming that an external observer
joins the network customers
connections established
in queue by the observer
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Outline
2

• Motivation
• Basic Model
• Extended Model
• Content Search
• Download effects

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Download Phase

• Assumptions
• The transport network is ideal
• Infinite bandwidth on the client side
• The peer from which downloading the desired
  content is rqndomly chosen between those storing
  that content.

The dynamics of dowload at each peer are modelled
by a M/G/1-PS queue.
Problem
The
download request rate incoming at peers is not
known a priori!

• It depends on
• The contents distribution at peers
• The policy used by the system to distribute
  the load among peers

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Probability of successful download (1)
Single Content Case

• Let ? is the popularity of a content, present in
  x copies in the network where there are Ua active
  peers
• Download request rate
• Assuming that the requests form a Poisson
  process, the queue becomes a M/G/1-PS with
  average delay
• Given a download rate y ?sphit the probability
  of successful download is

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Probability of successful download (2)

• The overall probability of successful download is

( F(x) is the pdf of the number of copies
available for the content )
Multiple Content Case
From F(x) we derive the probability that a peer
has k contents, present in x copies
The overall download request rate seen by a peer
is
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Probability of successful download (3)

• Since all Z(x) are independent we can approximate
  the distribution of Y around its average with
  a normal distribution
• The probability of successful download becomes

Notes

• my and sy are the first two moments of Y
• The integral is restricted to the interval
  for numerical reasons.

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Conclusions

• We defined a stochastic fluid model of a p2p
  system able to describe users and contents
  dynamics both in transient and stationary regime.
  
• A support model permits to consider the effects
  of the search and the download on the system
  performance.
• Analytical solution of the equations in steady
  state
• Model Extension to classes of different users
• Model Extension to classes of different contents
• Comparison beetween model and simulations in
  realistic scenarios.

Work in progress
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• Thank you!