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A statistical Theory of Chord under Churn

A statistical Theory of Chord under Churn. Supriya Krishnamurthy, Sameh El-Ansary, Erik Aurell and Seif Haridi IPTPS 2005 Presented by Yookyung Jo. Contribution of the paper. Calculation of churn-related system properties from first principle Validity rate of a routing table entry

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A statistical Theory of Chord under Churn

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  1. 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 7 1 6 2 Node 3 : 2 keys 5 3 4

  7. Derivation of Chord system properties • Node interval distribution • Wrong successor pointer rate • Finger pointer failure rate • Lookup cost

  8. System parameters • K : # of keys • N : # of nodes • M : log2K (# of fingers) • λj : join rate per node • λf : failure rate per node (leave) • λj = λf N : stable • λs : stabilization rate per node • λs = r λf • α * λs : successor stabilization, • (1- α) * λs : finger stabilization • E.g. : (K,N,r,α) = (10^6, 1000, 50, 0.25)

  9. Simulation setup • N =1000 • K = 2^20 • S = 6 • 200 ≤ r ≤ 2000 • 0.25 ≤ α≤ 0.75

  10. Node interval distribution • Intx(r,α) : # of intervals of length x • P(x) : probability of intervals of length x

  11. Node interval distribution

  12. Wrong successor pointer prob. • W1(r,α) : # of nodes with 1st successor wrong • w1(r,α) : probability of such a node

  13. Wrong successor pointer prob.

  14. Wrong successor pointer prob.

  15. Finger failure probability • Fk(r,α) : # of nodes with kth finger failing • fk(r,α) : probability of a node having kth finger failing

  16. Finger failure probability

  17. Finger failure probability

  18. Finger failure probability Failure rate of longer finger is higher than shorter finger

  19. Lookup cost • Ct(r,α) : 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

  20. Lookup cost

  21. Lookup cost

  22. 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?)

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