Simple stochastic models for Arigatoni
overlay networks
Philippe Nain
INRIA
ARIGATONI on WHEELS
Kickoff meeting, Sophia Antipolis, February 26-27, 2007
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2 Broker
c2 Local colony 2
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c7 Extended colony
associated to 2

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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
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
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T(1)={1,2, ….,11}
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T(4)={4,9,10}
T(5)={5,11}
T(2)={2,6,7}
T(3)={3,8}
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
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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
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Success !
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Success !
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Success !
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Failure!
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N = # brokers/colonies
(X1, …, XN) stationary version of membership
process {X1(t), …,XN(t)}
(X1, …, XN) 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)
Xi ~ Poisson rv with mean i = i . i
P(Xi=k) = (i)k exp(-i)/k!
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
q in , i j   e (1 p ) f ( j ) .  (1  e (1 p ) f ( l ) )
n
l  j 1
n
Q(in )   (1 e (1 p ) f ( l ) )
l 0
with f (l ) :

m
mT ( il ) T ( il 1 )
f (l )  mean membership in nodes in set T (il )  T (il 1 )
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 pf(l) in previous formulae:
q in , i j   p
n
f ( j)
.  (1  p f ( l ) )
l  j 1
n
Q(in )   (1  p f ( l ) )
l 0
with f (l ) :

m
mT ( il ) T ( il 1 )
f (l )  mean membership in nodes in set T (il )  T (il 1 )
Model extensions
Compound requests R =(R1, …, RM)
pi,m = Probability members in colony i grant service to
sub-request Rm
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Non-independent membership in different colonies
Introduce workload, focus on execution time,
network latency, …
Introduce user mobility