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

c1 c7c6 c2 c5 c9c10c11c8 c4c3 2 2 Broker c2 Local colony 2 c2c7c6 Extended colony associated to 2

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)

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

c1 c7c6 c2 c5 c9c10c11c8 c4c3 2 T(2)={2,6,7} T(3)={3,8}T(4)={4,9,10} T(5)={5,11} T(1)={1,2, ….,11}

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

c1 c7c6 c2 c5 c9c10c11c8 c4c3 2 Success !

c1 c7c6 c2 c5 c9c10c11c8 c4c3 2

c1 c7c6 c2 c5 c9c10c11c8 c4c3 2 Success !

c1 c7c6 c2 c5 c9c10c11c8 c4c3 2 Success !

c1 c7c6 c2 c5 c9c10c11c8 c4c3 2 Failure!

N = # brokers/colonies (X 1, …, X N ) stationary version of membership process {X 1 (t), …,X N (t)} (X 1, …, X N ) iid rvs

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) X i ~ Poisson rv with mean  i = i.  i P(X i =k) = (  i ) k exp(-  i )/k!

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

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

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

Model extensions Compound requests R =(R 1, …, R M ) p i,m = Probability members in colony i grant service to sub-request R m Non-independent membership in different colonies Introduce workload, focus on execution time, network latency, … Introduce user mobility