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Simple Stochastic Models for Arigatoni Overlay Networks

This study focuses on a simple stochastic model for Arigatoni overlay networks. The network structure is designed based on an extended colony system, where brokers are always active and members join and leave dynamically. The model considers atomic requests issued at brokers and the probability of service within the network. With an emphasis on single requests and membership dynamics, the research provides insights into the network's operational efficiency.

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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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