PeertoPeer Systems

1
Peer-to-Peer Systems
Fundamentals and Design of Distributed Systems
D.H.J. Epema
Parallel and Distributed Systems Group
2
Peer-to-Peer (P2P) systems

• In most DSs, some nodes have more authority or
  functionality than others (e.g., in a
  client-server system)
• In peer-to-peer systems, all nodes have the same
  authority and functionality, i.e., these systems
  are decentralized
• Some of the first P2P were amateur systems for
  information storage and retrieval
• P2P systems are dynamic nodes can come and go
• Perhaps P2P systems arent that new Usenet since
  1979 (news propagation mechanism among UNIX
  machines)

3
A Peer

• Pronunciation pir
• Etymology Middle English, From Middle French
  per, from per, adjective, equal, from Latin par
• Date 13th century
• Meaning One that is of equal standing with
  another, especially, one belonging to the same
  societal group, especially based on age, grade,
  or status
• So in Dutch een gelijke

4
Applications of P2P systems

• Information storage/content distribution
• e.g., for music or video files
• file-oriented or search-oriented
• examples Napster, Freenet, Gnutella, Chord,
  Pastry, CAN, BitTorrent, KaZaA, eDonkey
• Computation
• tap unused processing capacity
• usually embarrassingly parallel applications
• example SETI_at_HOME (Search for Extra-Terrestrial
  Intelligence)
• Collaboration
• games, virtual meetings

5
Issues in P2P systems

• Searching (routing, locating) how to find (new)
  files/nodes, and how to route the replies back
• Downloading get the required contents
  efficiently
• Growth nodes can join the system
• Shrinking nodes can leave the system (fail or
  disconnect)
• Performance
• Scalability
• Freeriding deter users from only downloading
• Security e.g., anonymity
• The network of nodes participating in a P2P
  system as part of the Internet is an example of
  an overlay network

peers
6
Levels of P2P-ness

• True P2P
• a true lack of any central authority
• examples Freenet, Gnutella, Chord
• Central P2P
• central components in the system
• examples
• Napster (has a central location database)
• SETI_at_HOME (has a single central machine handing
  out the work)
• BitTorrent (search is centralized)
• Question Can a P2P system exist without a
  centrally managed component?

7
Topics

• Anonimity
• Graph structures
• graph properties
• three graph types
• Routing/searching
• Case studies
• Freenet
• Gnutella
• Chord/CFS
• CAN
• Pastry
• BitTorrent

8
Anonymity

• Some P2P systems have as one of their goals
  anonymity
• There are different forms of anonymity w.r.t. a
  document
• of the author
• of the publisher
• of the servers storing the document
• of the readers
• of the document servers do not know what
  documents they are storing
• of the query a server cannot tell what document
  it is using as a response

9
Small-world effect (1)

• In 1967, Stanley Milgram did the following
  experiment
• he gave the same letter to 160 random people in
  Omaha, Nebraska
• he asked them to get the letters to a stockbroker
  in Boston, Ma
• intermediaries had to know each other on a
  first-name basis
• 42 letters made it to the stockbroker
• with a median number of 5.5 intermediaries
• at the time, the US had a population of about 200
  million

10
Small-world effect (2)

• Apparently, path lengths in social networks tend
  to be short
• Many people only know people in a small social
  circle, but a few have connections in far-a-way
  places
• These highly connected nodes are very important
• Similar phenomenon in WWW
• number of clicks to get from any page to any
  other page
• is unidirectional
• portals play an important role here

11
Graph concepts

• Given a connected graph (directed or not)
• Degree of a node number of nodes it is connected
  to
• In a regular graph, all nodes have equal degree
• The average pathlength is average of the lengths
  of the (a) shortest path across all pairs of
  nodes
• The neighborhood of a node is the set of nodes it
  is connected to
• The clustering coefficient of a node is the
  fraction of potential links among the nodes in
  its neighborhood that is actually present
• The clustering coefficient of a graph is the
  average of the clustering coefficients of its
  nodes

degree3
neighborhood
6 potential links
12
Graph types

• We will consider three graph types
• Regular graphs
• high clustering coefficient, long paths
• Random graphs
• low clustering coefficient, short paths
• Power-law graphs
• varying clustering coefficients, short paths,
• found in practice
• (Small-world graphs high clustering coefficient,
  short paths)

13
Regular graphs lattices

• Characterized by three numbers
• dimension d
• number of nodes along one dimension n
• number of links in each of the 2d directions k/2
• Structure
• start with a torus of some dimension
• add (k/2)-1 connections in each direction to the
  nearest nodes

k4
d1, k2
14
Regular graphs 1-Lattices
neighborhood
k2
k4
Clustering coefficient
0
0.5
15
Regular graphs 2-Lattices
k2 k4
(4)

(8)
Clustering coefficient
0
6/28
number of potential connections
16
Properties of lattices

• Degree dk
• Average pathlength in a d-lattice is
  approximately
• d (1/2) (n/2) / (k/2) dn/2k
• Clustering coefficient of a 1-lattice
• Regular graphs have a
• high average pathlength
• high clustering coefficient
• So no small-world effect

1
n-1
maximum distance
dimension
step size
average
(3/4) ((k-2) / (k-1))
17
Random graphs

• Number of nodes n
• Probability of any potential link being present
  p
• Average degree kp(n-1)
• Total number of links pn(n-1)/2
• May not be connected
• Experimental result for large n and k5, the
  largest component of a random graph is almost
  equal to the complete graph

n-1 potential links
p
18
Properties of random graphs

• Clustering coefficient
• Average pathlength approximately
• Intuitive explanation
• start with a single node
• add its k neighbors
• add the k2 neighbors of those neighbors
• do this l times, with l the average pathlength
• then we have all nodes, so nkl
• Random graphs have a
• low average pathlength
• low clustering coefficient
• So no small-world effect

p
p
log(n)/log(k)
19
Degree distribution random graphs

• The degree in a random graph with n nodes has a
  binomial distribution
• This decreases very fast for large k

probability that degree is equal to k
k links out of n-1 potential ones
20
Comparison regular-random
Q has 1 of Ps neighbors as neighbor
Q
random graph




10,000
P has 100 neighbors
P
0.74
50
2.00
0.01
21
Power-law graphs (1) definition

• The fraction of nodes Pk with k links satisfies
• Pk Ck-?
• for some C, k, ? gt 0
• Pk decreases very slowly
• Usually the degree exponent ? is between 2 and 3
• Properties
• there exist a few nodes with many connections
• a low average path length (order log(n))
• random failures dont have a large impact
• Power-law graphs occur often in nature
• Other name scale-free networks

22
Power-law graphs (2) construction

• Deterministic method to construct power-law
  graphs
• start with a graph G consisting of a single node
• add to identical copies G and G at a lower
  level
• connect the root of G with all leaves of G and
  G
• repeat steps 2 and 3 as often as you like

9 nodes 4 leaves at lowest level
G1
GG2
27 nodes 8 leaves at lowest level
G
G
23
Power-law graphs (3) degree exp.

• Graph Gn after n steps
• number of nodes 3n
• number of leaves at lowest level 2n
• degree of root 2n1 2 (with induction, equal
  to 2n-2 2.2n-1)
• Hub root of the graph at some level
• Concentrate for computation of degree exponent on
  hubs
• Consider graph Gi after i steps
• After n steps (ngti), there are 3n-i copies of Gi
  (and of its root)
• Of these roots, a fraction of 2/3 never increase
  their degree

degree of root in Gn-1
links to new leaves
degree changed degree unchanged
Gi
3 x Gi
9 x Gi
ni1
ni2
ni
24
Power-law graphs (4) degree exp.
degree of Gi

• So there are (2/3)3n-i nodes (hubs) in Gn with
  degree 2i1-2 2i1
• So a fraction of (2/3)3-i23-(i1) has this
  degree (divide by 3n)
• Might as well say a relative fraction of 3-i has
  degree 2i
• As a consequence, the degree distribution
  satisfies
• Pk k(-ln 3/ln 2)
• So the degree exponent is ? ln 3 / ln 2
• Clustering coefficient is 0 (no triangles)

25
Power-law graphs (5)

• Stochastic methods for generating power-law
  graphs
• incremental growth add nodes with links or sets
  of links one by one
• preferential attachment connect new nodes to
  highly connected ones
• Node i has degree di
• Node i is chosen with probability Pidi/?di
• Start with some graph with m0 nodes and m0-1
  edges
• In each step
• with probability p add mm0 edges, with edges
  chosen according to probability distribution
  Pi
• with probability 1-p add a single node with m
  links, also according to the distribution Pi

26
Power-law graphs (6)

• From T. Bu and D. Towsley, On Distinguishing
  between Internet Power Law Topology Generators,
  IEEE Infocom 2002
• The topology of the network of Autonomous
  Systems in the Internet has a power-law structure
  with a degree exponent of -2.18
• The characteristic path length L of a graph is
  the median of the means of the shortest path
  lengths connecting every node to all other nodes
• Comparison of Internet AS graph with random
  graph with same characteristic pathlength

Internet
random
27
Routing/searching (1)

• In some P2P systems both nodes and files are
  identified by random numbers
• node ids for instance a hash of IP-addresses
• file keys for instance a hash on contents or
  keywords
• File search may be by
• file name
• key words
• file key
• Reply to a search can be
• unique (search for specific file)
• not unique (multiple replies satisfying key words)

28
Routing/searching (2)

• Distributed hash tables (DHTs)
• use of random numbers for nodes and files
• main problem mapping files to nodes
• mapping is fixed
• Unstructured P2P systems
• mapping of files to nodes is not fixed
• Guaranteed routing
• search for existing file is always successful
• examples Chord, DHTs in general
• Probabilistic routing
• search for existing file may fail
• examples Freenet, Gnutella

29
Probabilistic routing

• Messages contain a Time-To-Live (TTL, or
  hops-to-live) value which determines the maximum
  number of hops a request message can travel
• The TTL value is decremented in every node
  visited
• When the TTL value reaches zero, the message is
  not forwarded anymore
• Messages contain a pseudo-random pseudo-unique
  identifier, to determine whether a message has
  visited a node previously
• If so, the message may be discarded or sent one
  hop back

TTL
3
2
1
0
30
Freenet design goals

• Freenet is a cooperative distributed file system
  for anonymous information storage and retrieval
  with location independence and lazy replication
• Nodes volunteer disk space, no guarantees for
  permanent availability of files
• Design goals
• anonymity of both producers and consumers of
  information
• efficient dynamic storage and routing of
  information
• decentralization of network functions
• No
• broadcast searches for files
• centralized location indexes of files

31
Freenet file keys (1)

• Files with a keyword-signed key
• user chooses a descriptive text string for the
  file
• from this keyword, a public/private key pair is
  derived
• the public half is hashed to yield the file key
• the private half is used to sign the file
• the publisher of a file publishes descriptive
  text string
• user uses the file key to locate the file
• user uses the signature to check the file (do I
  get what I asked for)
• Problems
• two users may choose the same descriptive keyword
• users may insert junk files under popular
  keywords (key-squatting)

32
Freenet file keys (2)

• Files with a signed-subspace key
• enables users to have personal name spaces,
  identified by a randomly generated public/private
  key
• only owner of name space can add files (requires
  private key)

33
Freenet retrieving data (1)

• Every node maintains a routing cache with
  (file-key, node-id) pairs of files it expects to
  be at specific nodes

file key, node-id
F387
node 5
F783
F456
node 8
F124
node 1
34
Freenet retrieving data (2)

• A request for a file is sent to the node that
  stores the file with the closest file-key in the
  routing cache
• This is repeated along a chain of nodes

fid398
F398
1
8
5
35
Freenet retrieving data (3)

• A request travels a chain of length at most equal
  to the TTL
• In the message, also a depth is maintained
• increased at every hop
• used as TTL for the way back
• Nodes can play with these values to increase
  anonimity
• When the TTL reaches zero, the request is not
  forwarded, and a failure message is sent back

36
Freenet retrieving data (4)

• When a node holding the file is found, the file
  is sent back along the chain traveled by the
  request
• Each node along this chain
• caches the file
• records the pair (file key, id of originating
  node) in its routing cache
• To support anonymity, nodes along the way may
  pretend that they are the originator of the file

(398,5)
(398,5)
5
(398,5)
F398
requester for file F398
F398
F398
F398
37
Freenet storing data (1)

• A request for insert contains
• a file key
• a hops-to-live value (number of copies to
  create)
• The request for insert is routed as a request for
  a file
• Each node along the way checks whether a key
  collision occurs
• If so
• an error message is sent back
• the user has to choose a new file keyword and try
  again
• When no collision is detected, the file is stored
  along the whole chain traveled by the insert
  request

38
Freenet storing data (2)

• When no collision is detected
• a success message is sent back
• the file is stored along the whole chain traveled
  by the insert request
• the pair (file key, id of inserter) is entered in
  the routing caches of the nodes along the chain

(398,1)
(398,1)
(398,1)
1
5
F398
inserter of file F398
F398
F398
F398
39
Freenet effects of retrieving and storing

• The keys in a nodes routing table tend to get
  clustered
• a node attracts requests for similar keys
• a node attracts inserts for similar keys
• Popular data will get stored (logically) close to
  the requesters
• Requests and inserts increase connectivity,
  because nodes get to know about previously
  unknown nodes

40
Freenet remarks

• Both the routing tables and the file storage may
  implement LRU
• In general, it seems that Freenet in operation
  tends to a power-law network (connections are
  entries in the routing caches)

41
Gnutella (1)

• Gnutella is search-oriented a request may get
  multiple answers
• Requests are broadcast
• Requests carry a TTL (typical initial value 7)
• When a request revisits a node, it is discarded
• Nodes are usually connected to about 3-5 other
  nodes
• As a result, messages get to about 10,000 nodes
  (47 16,000)

1
2
3
42
Gnutella (2)

• Replies travel back along the route of the
  request
• Replies do not contain contents, but the IP
  address of the location found
• Downloads through HTTP
• Gnutella should be self-organizing fast nodes in
  the core of the network, slow ones at the fringes
• Was very popular in 1999-2002
• There are modifications to the protocol that make
  it more efficient

43
Distributed hash tables (DHTs)

• Nodes and file keys are random numbers of some
  number of bits
• Usually assumed to be uniformly distributed
• So nodes with ids that are close are with high
  probability diverse in geography, ownership, etc.
• Some mechanism is used to map (hash) keys to
  nodes in a distributed system
• Main question route requests for keys to the
  proper node
• DHTs usually
• have guaranteed routing
• do not support anonimity

44
Chord hashing (1)
no nodes in between

• Chord uses what is called consistent hashing
• Both files key and node-ids are m bits wide
  (e.g., m128)
• Assume the numbers 0 through 2m-1 to be arranged
  clock-wise along a ring
• Some of these represent actual nodes
• For a number k, successor(k) is the first node in
  the clock-wise direction equal to or larger than
  k
• Assume a network of N nodes

2
1
0
2m-1
actual nodes
successor
45
Chord hashing (2)

• A file with key k is mapped to the node with id
  successor(k)
• So every node is responsible for the files with
  keys that map to a segment of the ring

successor(k)
node id and file key coincide
k
nodes files mapped to
segment
responsible node
46
Chord locating (1)

• In principle easy
• every nodes keeps the id and the IP-address of
  its successor in the ring
• when locating the file with key k, a node sends
  the file key along the ring
• when the file key reaches node successor(k), it
  is clear if the file is present in the system
• reply sent to originator of request
• is not efficient (of order N/2)

k
successor(k)
47
Chord locating (2)
successor(n1) successor(n2) successor(n4) su
ccessor(n8)

• Additional data structure in every node
• the finger table
• contains entries of (node-id, IP-address) pairs
  at distances equal to powers of 2
• In node n,
• fingerisuccessor(n2i-1), i1,2,,m
• Many of the first elements of the finger table
  will be equal gaps between successive nodes in
  the ring are of expected size 2m/N
• With a million nodes (220), the gaps between
  nodes are of size 2108, and the first 108
  elements of the finger table are expected to
  coincide

n2i-1
n
n2i
48
Chord locating (3)

• When node n wants to locate key k
• if kfingeri for some i, check with node k
• if kltfinger1, check with node finger1
• else
• send a request to the largest node nfingeri
  with kgtn
• n repeats all of this

case 1
case 2
k
k
n
n
k
case 3a
49
Chord locating (4)

• Two possibilities
• replies back to originating node
• replies sent along to next nodes

node n
node n
k
finger1
file key k
n
fingeri
node n
n
k
k

finger1
fingeri1
node n
50
Chord locating (5)

• Complexity of look-up log(N)
• First complexity is of order at most m (
  log(ids) )
• Let node n look for key k
• Let node p be the last node preceding k (last
  step in search)
• Consider the interval n2i-1,n2i) that contains
  p
• In n, let fingerifsuccessor(n2i-1)
• Then fp
• Distance between n and f is at least 2i-1
• Distance between f and p is at most 2i-1
• So f is closer to p than to n distance is cut at
  least in half!

n
p

)
f
k
2i-1
2i-1
51
Chord locating (6)

• Assume nodes are uniformly distributed along the
  ring
• So a node covers a segment of the ring of
  length 2m/N
• Why complexity log(N) instead of m ?
• Reason many of the first elements in a finger
  table coincide because nodes are sparsely
  distributed
• So many of the last steps are covered by a single
  step

n
is fingerk for k not small
52
Chord locating (7)

• On average, on a segment of the ring of length L,
  there are N(L/2m) nodes (L/2m is the fraction of
  the ring considered)
• After A steps, the distance between the current
  query node and the key k will be reduced to at
  most 2m/2A (previous slide)
• Take A2log(N)log(N2) the distance between the
  current query node and the key k will be reduced
  to 2m/N2 (2AN2)
• Probability of having a node on a segment of
  length 2m/N2 is 1/N, which can be neglected
• So only A steps needed

after A steps
no nodes
53
Chord node joins (1)
relocate these file keys from n2 to n

• In principle easy
• a new node n has to know any existing node n in
  the ring
• and ask it to find the elements in its finger
  table by searching for successor(n2i), for
  i0,1,
• node n can then announce its presence to its
  successor (n2) to get the keys it should store
• However, the finger tables of other nodes may
  have to be adapted (in particular, of its
  predecessor (n1))
• In general, when data structures are suspected to
  be inconsistent, a node can execute steps 1 and 2
  

n
n2
n1
finger1
n
54
Chord node joins (2)
n

• Additional data structure predecessor
• Steps for n to join
• node n gets finger1n2 from n
• node n contacts n2
• n2 sends its predecessor (n1) back, which is now
  the predecessor of n
• n2 sets its predecessor to n
• node n contacts n1 so n1 can set its successor to
  n

n2
n1
n
successor predecessor
55
Content Addressable Network (CAN) (1)

• CAN is a DHT with a d-dimensional uniform hash
  function
• Keys are deterministically mapped to a point in a
  d-dimensional torus T
• T is split up into zones (dynamically as peers
  join and leave)
• Every peer is responsible for one zone

key
peer
hash function
zone
T, d2
56
CAN (2) Routing

• Every node maintains a routing table with
• the coordinates of the neighboring zones
• the IP addresses of the responsible peers
• Use greedy routing algorithm
• Complexity (assume n peers/zones) average path
  length is (d/4) n(1/d)

number of zones n(1/d)
5 neighbors
key requested
57
CAN (3) joining

• A peer A that wants to join
• randomly chooses a point in P in T
• contacts any CAN peer B
• and asks B to search for the zone/peer (say C)
  that holds P
• Then the zone of C is split up equally between A
  and C

C
A
P
B
58
CAN (4) leaving

• When a peer wants to leave, its zone is merged
  with a neighbor if that leads to a valid zone
• Otherwise, the neighbor with the smallest zone is
  temporarily responsible for two zone

59
PASTRY (1)

• Node ids and file keys are K bits e.g., K128
  bits
• Let N2K
• Node ids and file keys are assumed to be integers
  with digits in base 2b for some b
• If b4 and K128
• Pastry maps a file key to the node with id
  numerically closest to it
• Pastry uses hypercube-like routing
• Pastry uses a notion of (scalar) proximity
  (round-trip time, number of hops in the Internet)

node id
1001
1101
..
0101
a0
a1
a31
digits
60
Intermezzo Hypercubes
10
11

• n.2n-1 connections
• maximum distance n
• nodes that differ in 1 bit are connected

n 2
00
01
011

• Routing
• - scan bits from right to left
• - if different, send to neighbor
• with same bit different
• - repeat until end

000 -gt 111
111
010
110
n 3
001
101
000
100
61
Pastry (2) routing table

• Assume b2, K16
• Notation common prefix-next digit-rest of
  nodeid
• Routing table in node 10233102 (IP addresses
  omitted)

all other possible values
no node with suitable id known
No common prefix
Coincide in first bit
Coincide in first 2 bits
first 3 bits
Number of rows?
Number of columns?
62
Pastry (3) routing table

• Size of routing table (log2bN) x (2b-1)
• A node tries to fill all rows in its routing
  table
• The lower in the table, the more difficult this
  is (fewer and fewer nodes have the right prefix)
• Routing/searching for file key k in node n
• send the request to a node which has a longer
  common prefix with k than n
• if such a node does not appear in the routing
  table, send k to a node with a common prefix of
  equal length which is closer to k
• repeat this until found
• For most entries in the routing table, there are
  many choiceschoose one with good proximity

63
Pastry (4) routing table

• In node 10233102, search for key 10211011
• Routing complexity log2bN

64
Pastry (5) leaf set

• It is a rare coincidence that a routing table is
  completely filled
• The leaf set of a node is a set of nodes of size
  L, for some L
• The leaf set of node n contains the L/2
  numerically closest smaller nodeids and the L/2
  numerically closest larger nodeids
• Smallest (largest) of all these S (T)
• When routing for key k, always first check
  whether SkT
• If so, map k to closest of these
• Otherwise, use routing table

k
n
S
T
L/2
L/2
65
Pastry (6) node joins

• Assume a joining node n knows at least one node
  n1
• Problem how to fill the routing table of n
• Solution
• let n ask n1 to look for itself!!!!
• if n and n1 do not have a common prefix, n1 will
  send the request to node n2 which has the first
  digit in common with n
• so the second row of the routing table of n2 will
  have the first bit equal to the first bit of n
• so n uses the second row of the routing table of
  n2
• and n uses the third row of the routing table of
  n3
• etc.

use leaf set of z as n is numerically close to z
use first row
use second row
n1
n
n2
z
66
BitTorrent (1) searching

• BitTorrent is only a file downloading protocol
• It depends for file searching on other components
  (web sites with content lists and trackers)
• Trackers central components that keep track for
  each file which peers are currently downloading
  it
• A peer that wants to download a file gets a
  .torrent file from a web site with the address of
  a tracker

peers downloading same file
Suprnova
tracker
67
BitTorrent (2) downloading

• Files are split up in chunks (pieces of the file)
  of size, e.g., 256 KByte (on the order of 1000
  chunks per file)
• A swarm is the set of peers that have all or part
  of a file and are still online
• Peers in a swarm exchange the ids of the chunks
  they have
• Leechers have only a part of a file and barter
  for chunks of the file (tit-for-tat)
• Seeders have the complete file and upload for free

downloaders
seeder
68
BitTorrent (3) downloading

• Downloaders barter with a small number of peers
  (e.g., 4)
• Optimistic unchoking every 30 seconds, a
  downloader contacts a fifth peer to see whether
  is has better download performance
• If so, it replaces one of the four peers
• BitTorrent implements
• the rarest-first policy to ensure a uniform
  distribution of pieces among downloaders
• endgame mode prevent peers who have all but a
  few pieces from waiting too long to finish

69
BitTorrent analysis (1)

• A fluid model of the operation of BitTorrent for
  a single file
• Parameters of the model
• x(t) the number of downloaders
• y(t) the number of seeders
• ? the arrival rate of peers wanting to download
  the file
• T the rate with which downloaders abort the
  download
• ? the rate with which seeders leave the system
• µ the upload bandwidth of all peers
• c the download bandwidth all peers, cµ

downloaders
seeders
?
?
x(t)
y(t)
?
70
BitTorrent analysis (2)

• Maximum download capacity cx(t)
• Maximum upload capacity µ(?x(t) y(t))
• The parameter ? accounts for the reduced
  download effectiveness of downloaders because
  they do not have the complete file
• Arrival and departure distributions assumed to
  be exponential
• System evolution governed by two differential
  equations

become a seeder
arrivals
abort
number of downloaders
leave
number of seeders
71
BitTorrent analysis (3)

• In steady state dx(t)/dt dy(t)/dt 0
• Then
• Suppose ? is positive
• Then with ß-1maxc-1,?-1(µ-1-?-1),

72
BitTorrent analysis (4)

• Let T be the average download time of the
  downloaders that become seeders
• Arrival rate of downloaders that become seeders
  
• Fraction of downloaders that become seeders
• Littles formula
• average number of jobs in system N
• arrival rate ?
• average time in system W
• then N?W
• Apply here

?
N
W
73
BitTorrent analysis (5)

• So
• with
• Consequences
• BitTorrent scales well T does not depend on the
  arrival rate
• When download effectiveness ? increases, T
  decreases
• When departure rate of seeders ? increases, T
  increases
• Up to some point, increasing the download
  bandwidth c decreases T (until it is no
  bottleneck anymore)
• Similar statement for the upload bandwidth µ

74
BitTorrent analysis (6)

• Suppose downloader D is connected to a set O of k
  other downloaders
• ? 1 - P(D has no piece that any peers in O
  needs)
• Assume that the piece distributions at the peers
  are identical and independent
• Then ? 1 - P(D has no piece that peer j in O
  needs)k

O
D
75
BitTorrent analysis (7)

• Assume that
• the numbers of pieces in the downloaders are
  uniformly distributed on 0,1,,N-1, with N the
  number of pieces in the file
• the pieces at any downloader are taken at random
  and uniformly from the pieces of the file
  (rarest-piece first)
• Then a computation shows that
• So in practice, ? will be close to 1