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Simple stochastic models for Arigatoni overlay networks

Simple stochastic models for Arigatoni overlay networks. Philippe Nain INRIA ARIGATONI on WHEELS Kickoff meeting, Sophia Antipolis, February 26-27, 2007. c1. 1. c5. c2. c3. c4. 5. 2. 3. 4. 6. 7. 8. 9. 10. 11. c6. c7. c8. c9. c10. c11. Broker. Extended colony

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Simple stochastic models for Arigatoni overlay networks

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  1. Simple stochastic models for Arigatonioverlay networks Philippe Nain INRIA ARIGATONI on WHEELS Kickoff meeting, Sophia Antipolis, February 26-27, 2007

  2. c1 1 c5 c2 c3 c4 5 2 3 4 6 7 8 9 10 11 c6 c7 c8 c9 c10 c11 Broker Extended colony associated to 2 Local colony 2 c2 c2 c6 c7 2

  3. N Brockers always active : always able to handle a request (i.e. serve or forward a request to its predecessor) whether it is « local » or not • Members are dynamics : join a local colony, stay connected for a while and then leave (temporarily or permanently)

  4. Focus on single, atomic*, request R issued at brocker in at t=0 (brocker in ancestor of brockers in-1, …,io ) Xi(t) = membership of colony i at time t T(i) = set of nodes in tree rooted at i * Can be extented

  5. c1 T(1)={1,2, ….,11} 1 c5 c2 c3 c4 5 2 3 4 6 7 8 9 10 11 c6 c7 c8 c9 c10 c11 T(5)={5,11} T(2)={2,6,7} T(3)={3,8} T(4)={4,9,10}

  6. Focus on single, atomic*, request R issued at brocker in at t=0 (brocker in ancestor of brockers in-1, …,io ) Xi(t) = membership of colony i at time t T(i) = set of nodes in tree rooted at i With probability • pn (Xm(0), m T(in)), R served by extended colony in • 1- pn (Xm(0), m T(in)), R forwarded to brocker in-1 ; if so, with prob. pn (Xm(0), m  T(in-1)-T(in)), R served by colonies in T(in-1)-T(in)); otherwise, R forwarded to in-2, etc. * Can be extented

  7. c1 Success! 1 c5 c2 c3 c4 5 2 3 4 6 7 8 9 10 11 c6 c7 c8 c9 c10 c11

  8. c1 1 c5 c2 c3 c4 5 2 3 4 6 7 8 9 10 11 c6 c7 c8 c9 c10 c11

  9. c1 Success ! 1 c5 c2 c3 c4 5 2 3 4 6 7 8 9 10 11 c6 c7 c8 c9 c10 c11

  10. c1 Success ! 1 c5 c2 c3 c4 5 2 3 4 6 7 8 9 10 11 c6 c7 c8 c9 c10 c11

  11. c1 Failure! 1 c5 c2 c3 c4 5 2 3 4 6 7 8 9 10 11 c6 c7 c8 c9 c10 c11

  12. N = # brokers/colonies (X1, …, XN) stationary version of membership process {X1(t), …,XN(t)} (X1, …, XN) iid rvs

  13. Members join each colony according to independent Poisson processes (reasonnable assumption) • Intensity i for colony i Each member stays connected for a random time with an arbitrary distribution • i= Mean connection duration in colony i Proposition (membership distribution in colony i) Xi ~ Poisson rv with mean i= i . i P(Xi=k) = (i)k exp(-i)/k!

  14. Application 1 : probability of success/failure q(in,ij) = prob. R served at broker ij Q(in) = prob. R not served pi = probability member in colony i grants service (user availability) ; below p = pii

  15. No need to know maximal number of members in a colony; only need to know average membership Few input parameters

  16. Application 2 : same as #1 but with fixed membership i = membership in colony i Replace e-(1-p)f(l) by pf(l) in previous formulae:

  17. Model extensions • Compound requests R =(R1, …, RM) pi,m = Probability members in colony i grant service to sub-request Rm • Non-independent membership in different colonies • Introduce workload, focus on execution time, network latency, … • Introduce user mobility

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