IEG5270 Advanced Topics in P2P Networking Modeling and analysis of p2p content distribution algorith

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IEG5270 Advanced Topics in P2P
NetworkingModeling and analysis of p2p content
distribution algorithmsPart 1

• Dah Ming Chiu
• Chinese University of Hong Kong

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Recap

• The secrets of P2P content distribution
• Use multiple trees!
• Tree-based (push)
• Data-driven (pull)
• Utilize peers resources to satisfy goal
• File downing (as fast as possible)
• Streaming (as continuous as possible at given
  playback rate)
• BitTorrent
• A working, data-driven, p2p file-downing
  algorithm
• Bootstrapping metainfo, tracker
• Peer selection algorithm
• Chunk selection algorithm

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Models

• Why do we need models?
• To understand the limit of p2p system
• To understand the key factors, for designing
  better algorithms
• To help operate a p2p system
• The complexity of modeling a p2p system
• It is a large scale distributed system
• A large number of peers, and each with its own
  state
• The complexity of modeling the network conditions
• Links, routers, ISPs, bottlenecks
• The complexity of modeling the peer dynamics
• How do they arrive and leave the system
• Accuracy versus simplicity/usefulness

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Two types of models

• Throughput models
• Try to understand the capacity (maximum
  throughput) of BT-like systems
• A large number of models in this category
• Different levels of abstraction
• Different peer dynamics
• Different variations of the BT algorithm
• Exploring different metrics
• P2P streaming models
• YP Zhou et al is the first (?)
• Model the playback buffer to study playback
  performance

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The uplink sharing model

• Assume away the network complexity
• Only the uplink can be the bottleneck in peer
  communications
• Practically all models make this assumption
• Its like the Poisson arrival assumption in
  studying a queue
• The term is coined in Mundingers thesis

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Mundiners thesis

• Summarized in
• Optimal Scheduling of P2P File Dissemination
• To appear in Journal of Scheduling
• Study the problem using a discrete model
• Closer to real situation
• Linkage to previous work on scheduling
• The results can be easily appreciated using fluid
  assumption
• Most other models make fluid assumption there
  are many chunks, each very small
• Key assumptions
• Uplink sharing model
• Static peer population

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Related work

• The broadcast problem
• Makespan the number of rounds for all N nodes
  to receive M message from a sender
• Unidirectional communication 2M log2(N)
• Bidirectional communication M log2(N)
• In general, if peer is uplink capacity is Ci,
  (server is C0), what is the minimum makespan?
• In this case, peers serve each other
  asynchronously

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Convert to a synchronous problem

• First show minimum makespan can be achieved using
  a synchronous system
• Upload to one peer at a time
• Upload chunks in discrete time slots

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Prove equal capacity case by construction
The M2, N3 case
The first phase of the Mgt2, Ngt3 case
In the subsequent phase, peers start
back-filling Construct a strategy to finish all
pieces See paper for details
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Discussion

• The meaning of the formula
• For sufficiently large M, the minimum makespan
  changes very slowly with increasing peer
  population

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The unequal capacity case

• What is the minimum makespan for peers with
  different uplink capacities?
• It is a MILP (Mixed integer linear programming
  problem)
• See paper for details
• The following lemma is used to discretize the
  problem

But there is a simple closed-form solution,
asymptotically when M is very large!
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Consider simple examples(small N and M)

• C1C2CN, but CS can be different
• Example 1 N2, M1, two cases
• Both peers download from server
• Peer 1 download from server, peer 2 download from
  peer 1

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Example 2

• N2, M2, four cases
• Everything downloaded from server
• One peer downloads from server, 2nd peer download
  from 1st
• One peer downloads from server, 2nd peer downs
  partly from 1st and partly from server
• Each peer downloads exactly one chunk from
  server, and the other chunk from each other

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Example 2 analysis

• N2, M2, four cases
• Everything downloaded from server
• One peer downloads from server, 2nd peer download
  from 1st
• One peer downloads from server, 2nd peer downs
  partly from 1st and partly from server
• Each peer downloads exactly one chunk from
  server, and the other chunk from each other

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Minimum makespan for NM2
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The M? case fluid assumption

• Instead of makespan, think in terms of throughput
• Given the uplink bandwidth from the server and
  peers, how do we allocate it so content can reach
  each peer at the (same) maximum rate?
• Mundinger proved something more general

For us, it is more intuitive, and more
immediately useful to look at the single server
case
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The simpler/more useful result

• Given the uplink sharing model (right figure) and
  very large M, the maximum throughput is

This is a special case of Theorem 4.
DM Chiu et al, Can network coding help in P2P
networks, invited paper at 2nd NETCOD workshop,
2006
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Proof step 1 R is upper bound

• C0 is uplink bandwidth from server
• Server must be able to send content out at least
  once it is a clear upper bound
• C0 sumj(Cj) is the total uplink bandwidth from
  server and all peers
• Each peer must receive servers content from some
  where.
• The total demand is therefore nR, which must
  equal to the total supply

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Proof step 2 Realizing R by construction

• One 1-hop tree (from sender)
• N 2-hop trees (from each peer)
• Assign rates optimally, satisfying the uplink
  constraints
• Two cases
• C0 gt C/(n-1) where C sum Ci
• C0 lt C/(n-1)

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Maximum rate achieved

• Case 1 C0 gt C/(n-1) where C sum Ci
• Assign rate Ci/(n-1) to the ith 2-hop spanning
  tree
• The ith spanning tree can deliver Ci/(n-1) to
  each other peer
• Assign rate C0 C/(n-1) to the 1-hop spanning
  tree
• The server can deliver (C0-C/(n-1))/n to each
  peer
• Each peer receives (C0 C)/n
• Case 2 C0 lt C/(n-1)
• Assign rate C0Ci/C to the ith 2-hop spanning
  tree, which can then deliver to all other peers
  at that rate
• Each peer receives sum(C0Ci/C) C0

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Summary

• Simple static model
• Uplink sharing model
• Large M fluid assumption
• Static population (1 server n peers)
• Peers always able to help each other
• One simple extension more seeders
• m seeders and n downloaders
• Reason some peers stay around after becoming
  seeders
• Try work this out yourself

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Dynamic peers model

• Qiu and Srikant, Modeling and performance
  analysis of BitTorrent-like peer-to-peer
  networks, Sigcomm 2004 (Google Scholar 200)

In Mundingers model
a constant n a constant 1 0 Constant
Ci Unbounded 0
this is assumed to be 1
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A fluid model of population

• System equations
• In steady state
• where

?gt0 assumed
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Downloading time

• Apply Littles Law

Average number of peers who will complete
downloads
Average rate downloads are completed
Average downloading time
where
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Observations

• T is not dependent on arrival rate ?!
• When sharing efficiency ? increases, T decreases
• When seeders departure rate ? increases, T
  increases
• Initially, when downloading rate c increases, T
  decreases, but c is large enough (uplinks become
  bottleneck), c no longer affects T
• Same observation about peer uplink rate ?
• Normally c gt ? for a given peer but if ? lt ?,
  then there will be abundant uplink capacity (due
  to helpers), and downlink will be the bottleneck

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Effect of sharing efficiency ?

• In the capacity equation (Mundinger), ?1 is
  assumed
• The analysis so far assumes 0lt?lt1
• When ?0, (no sharing by downloaders), two cases
• If rate of seeds leaving (?) is less than the
  rate a seed can help a peer download a file (?),
  then the system is limited by the downloading
  capacity T 1/c
• If otherwise
• This equation means y will decrease to 0, and the
  system dies.
• Note, in this model, there is no server, only
  seeders.

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Stability

• It is not sufficient to derive a steady state
  population by setting dx/dt dy/dt 0
• Must also show that the system will converge to
  the steady state
• See detailed discussion of local stability in
  paper

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Simulation results
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Comparison to a queuing system

• A regular queue models a client-server system,
  when the arrival rate exceeds the service rate,
  the system blows up (queue goes to infinity)
• The analysis of a p2p system is similar to that
  for a queue, except the customers are also
  servers
• The arrival of a customer also brings some
  additional service capacity (?gt0 case)
• This makes T stay finite the p2p queue is
  infinitely scalable!
• A queue may different service models FCFS, LCFS,
  priority, PS
• What is the effect of different service models in
  a p2p system?

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Tradeoff in rate allocationin uplink sharing
model

• B Fan, DM Chiu and J Lui, The delicate tradeoffs
  in BitTorrent-like file sharing protocol design,
  ICNP 2006
• Similar to Qiu Srikants model
• But multiple classes of peers
• Arrival rate
• Uplink capacity
• Downlink capacity
• fat and thin peers if 2 classes
• No seeders

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Feasible rate allocation space

• Any u and d satisfying

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Performance metric 1average downloading time

• In steady state
• Number of type i peers
• Total upload capacity
• C??
• ciui/di ?
• ci is the share ratio of peer i
• Download time of type i peers

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Performance metric 2 fairness

• Fairness index
• It can be applied to any quantity x. In this
  case, let xi ci
• Therefore, fairness index

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Tradeoff

• Each rate allocation (u,d) yields a different T
  and F
• The space of different strategies and achievable
  (T,F)
• Consider three specific allocations
• Optimal downloading time
• Optimal fairness
• in terms of share ratio
• Max-min rate allocation
• Equalize the downloading rate of all peers,
  unless it is limited by the downlink capacity

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Optimizing downloading time

• Problem definition
• Note
• assuming uploading rate uplink capacity
• Smaller type number means higher uplink
  capacity
• Applying standard Lagrange multiplier methods and
  KKT conditions
• Type 1 peer gets less downloading rate
  proportionally speaking
• Type 1 peer may even get less downloading rate
  than thin peers

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Optimizing fairness

• This means equalizing the share ratios
• Therefore
• By same Lagrange multiplier methods

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Max-min rate allocation

• Problem definition
• Can use the water filling algorithm to find the
  solution.
• If none of the downlinks are bottlenecks

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Compare the results

• Since we have the closed form results for these
  cases
• The concept of Pareto optimal, or non-inferior
  allocations

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Apply it to distributed algorithms

• Rate allocation analysis so far assumes knowledge
  only available in centralized algorithms
• Examples of distributed algorithms found in BT
• Tit-for-tat
• This yields the most fair rate allocation, in
  equilibrium.
• Peers form clusters
• Random
• This gives a solution similar to max-min
• Each peer gets average uploading rate in steady
  state
• Possible to adopt a weighted average of the two
• Select ns neighbors using tit-for-tat, and na
  neighbors randomly

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Different mix of TFT and random
For small na, the mixed strategy is not able to
find best matched neighbors, (total number of
peers 100)
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Recap

• So far, we considered three models
• Mundingers model
• Gives throughput capacity for small N,M, or very
  large M
• But static peer population
• Qiu-Srikant model
• Gives average delay (hence also capacity) for
  dynamic peer population, and various other
  parameters
• But sharing efficiency modeled by one parameter ?
• Single class of peers
• Fan-Chiu-Lui model
• Multiple classes of peers, and the tradeoff of
  throughput and fairness
• Assumes perfect sharing efficiency
• How to more accurately model sharing efficiency?