Core Persistence in Peer-to-Peer Systems Relating Size to Lifetime

1. Core Persistence in Peer-to-Peer Systems Relating Size to Lifetime V. Gramoli, A-M. Kermarrec, A. Mostefaoui, M. Raynal, B. Sericola

2. Context • Large-Scale Dynamic Systems • Nodes join and leave the System • Rejoining nodes might not hold the data • Nodes maintain no global information • Data Persistence Problem • For a data, if all its owners leave, it becomes lost • Observations on Peer-to-Peer (P2P) Systems • Highly dynamic • Never empty Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

3. Goal • Guaranteeing Persistence despite Dynamics • Major Challenge • Providing • Required probability, p, and • The system churn, c, • …data must be replicated • Adjusting replication period, δ, • Adjusting replication size, q. Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

4. System Churn • Large-Scale Distributed System • n interconnected nodes • each w/ unique ID • w/o global knowledge • Dynamic System • Nodes join/leave the system • A joining node is new • Data • Data is initially replicated at a subset of nodes, called a core. Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

5. Churn Model, c • Churn: • System dynamism intensity. • It represents: • Rate of arrival and departure by node by unit of time. • We observe the system at two instants • Let Q be the initial core, and q its size, • Let A be the set of replaced nodes, α its size, • Let Q’ be the resulting core (after replacement). Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

6. Churn Model Nodes w/ data. Nodes w/o data t time Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

7. Churn Model Nodes w/ data. Nodes w/o data t time Core Q at time t, |Q| = q Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

8. Churn Model Nodes w/ data. Nodes w/o data t t + δ time After period δ = 2 and with churn c= 0,2 Core Q at time t, |Q| = q Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

9. Churn Model Nodes w/ data. Nodes w/o data t t + δ time After period δ = 2 and with churn c= 0,2 Replaced nodes A, |A| = α Core Q at time t, |Q| = q Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

10. Churn Model Nodes w/ data. Nodes w/o data t t + δ time After period δ = 2 and with churn c= 0,2 Replaced nodes A, |A| = α Core Q’ at time t+δ, |Q’| = q Core Q at time t, |Q| = q Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

11. Churn Model • Evolution of the amount of initial nodes • t0n initial nodes • t1n-nv = n(1-v) initial nodes ... • tin(1-v)i initial nodes • ti+1n(1-v)i - n(1-v)iv = n(1-v)i+1 initial nodes • We choose α = ┌n-n(1-v)δ┐the number of nodes replaced after δtime units Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

12. Data Availability • Initially, q nodes own the data (replicas) • α nodes are replaced uniformly at random • How many data replicas remain after δ time units in a system w/ churn c? Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

13. Data Availability • Preliminary Observation • Number β = |Q’ ∩ A| of nodesthat owned the data and leave the system is bounded: max(0, α + q - n) ≤ β ≤ min(α, q) a b Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

14. Data Availability • Probability of β = k replicas have been replaced? Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

15. Looking for the data • Initially, qreplicas. • δ time units later, q system nodes are uniformly drawn at random. • What is the probability of finding the data after this δ time units in a system w/ churn c? Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

16. Looking for the data • Probability of missing the data • Random drawing, at uniform, and w/o replacement of q nodes. LetE = Q’ \ A. (disjoint events) Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

17. Looking for the data • Probability of missing the data Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

18. Core size for n = 104 the core size α/n = probability of missing the data Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

19. Probability, Dynamism, and Core Lifetime • Varying churn, size, and probability Proba of finding data α/n Core size for Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

20. Conclusion • Retrieving a data is paradoxally easy! • Storage Applications • Modifying the data at q nodes • Accessing the up-to-date data by contacting q nodes • Cores are probabilistic quorums • Future Research • Modeling the churn using a more realistic model (Markovian continu). • Specifying a protocol for probabilistic data consistency/persistence in dynamic system. Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola

21. Some References • A Quorum based protocol for searching objects in P2P ntwks. K. Miura, T. Tagawa, and H. Kakugawa. IEEE Trans. on Parallel and Distributed Systems, 17(1):25–37, 2006. • Probabilistic quorums for dynamic systems. I. Abraham and D. Malkhi. Distributed Computing, 18(2):113–124, 2005. • Reconfigurable distributed storage for dynamic ntwks. G. Chockler, S. Gilbert, V. Gramoli, P. M. Musial, and A. A. Shvartsman. In Proc. of 9th Int’l Conf. on Principles of Distributed Systems, 2005. Gramoli, Kermarrec, Mostefaoui, Raynal, Sericola