A statistical Theory of Chord under Churn

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A statistical Theory of Chord under Churn

• Supriya Krishnamurthy, Sameh El-Ansary, Erik
  Aurell and Seif Haridi
• IPTPS 2005
• Presented by Yookyung Jo

2
Contribution of the paper

• Calculation of churn-related system properties
  from first principle
• Validity rate of a routing table entry
• Lookup performance
• Previously,
• Simulation
• Upper, lower bounds

3
Contribution of the paper

• New analysis methodology for distributed systems,
  based on master equation

4
Master equation

• Time evolution of the probability of a system
  being in a particular state
• Gain/loss terms
• Steady state 0
• Technique used in statistical physics

5
Chord review

• Peer to peer key-lookup service with O(log(N))
  hop
• Key managed by its successor
• Invariants
• Hard successor list, key responsibility
• Soft finger table (log N lookup)
• Lookup process
• Churn join, leave process
• Aggressive join
• Stabilization successor, finger

6
Chord review
Node 0 5 keys
Node 1 1 keys
0
1
7
Node 3 2 keys
2
6
3
5
4
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Derivation of Chord system properties

• Node interval distribution
• Wrong successor pointer rate
• Finger pointer failure rate
• Lookup cost

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System parameters

• K of keys
• N of nodes
• M log2K ( of fingers)
• ?j join rate per node
• ?f failure rate per node (leave)
• ?j ?f ? N stable
• ?s stabilization rate per node
• ?s r ?f
• a ?s successor stabilization,
• (1- a) ?s finger stabilization
• E.g. (K,N,r,a) (106, 1000, 50, 0.25)

9
Simulation setup

• N 1000
• K 220
• S 6
• 200 r 2000
• 0.25 a 0.75

10
Node interval distribution

• Intx(r,a) of intervals of length x
• P(x) probability of intervals of length x

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Node interval distribution
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Wrong successor pointer prob.

• W1(r,a) of nodes with 1st successor wrong
• w1(r,a) probability of such a node

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Wrong successor pointer prob.
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Wrong successor pointer prob.
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Finger failure probability

• Fk(r,a) of nodes with kth finger failing
• fk(r,a) probability of a node having kth finger
  failing

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Finger failure probability
17
Finger failure probability
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Finger failure probability
Failure rate of longer finger is higher than
shorter finger
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Lookup cost

• Ct(r,a) lookup cost for a key that is t keys
  away
• one hop cost time out cost
• a(m) prob. Of at least 1 node in interval m

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Lookup cost
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Lookup cost
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Conclusion Comments

• Derived churn-related system properties of Chord
• Master equation based approach
• Novel, Makes analysis tractable
• Useful for the analysis of large-scale
  distributed systems
• Critique
• Simplification for analysis (Hop cost time out
  cost?)