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Theory and Applications of Point Processes / Théorie et
applications des processus ponctuels
(Org: Gail Ivanoff and/et David McDonald)
- FRANCOIS BACCELLI, École Normale Supérieure
Stochastic geometry and modelling of coverage and
capacity in CDMA networks
[PDF] -
Joint work with B. Blaszczyszyn and F. Tournois
(INRIA & Ecole Normale Supérieure)
The aim of the survey is to show that stochastic geometry provides an
efficient computational framework allowing one to predict
characteristics of large CDMA networks such as coverage or soft-handoff
level or capacity. The general idea consists in representing the
location of antennas and/or mobile stations as realizations of
stochastic point processes in the plane within a simple parametric
class, which takes into account the irregularities of antenna/mobile
patterns in a statistical way. This approach leads to new formulas and
simulation schemes allowing one to compute/estimate of the spatial
averages of these local characteristics in function of the model
parameters (density of antennas or mobiles, law of emission power,
fading law etc.) and to perform various parametric
optimizations.
- RALUCA BALAN, Universite de Sherbrooke, Sherbrooke, Quebec J1K 2R1
Markov point processes in Bayesian nonparametric statistics
[PDF] -
At the origin of this work, there is a Bayes property of a classical
Markov chain. We consider a class of random probability measures which
satisfy a Markov type property. This class includes the Dirichlet
process, the empirical process and a point process with Polya type
finite dimensional distributions. Our main result proves that this
class is ``closed'' in the Bayesian sense i.e., if the prior
distribution of the sample is Markov (in the specified sense), then its
posterior distribution will also be Markov. In particular, a neutral to
the right prior distribution leads to a neutral to the right posterior
distribution.
- JOHN BRAUN, University of Western Ontario
Point Processes and Applications
[PDF] -
This talk will discuss point processes in Rd, d=1, 2. We
will consider some applications to bio-systems. Some questions of
statistical inference will also be considered. In order to implement
some of the inference methods one needs to consider approximations to
the sampling distributions of certain statistics. These approximations
are typically limit theorems, but currently parametric and non
parametric bootstrap methods are of increasing interest.
- GAIL IVANOFF, University of Ottawa, Ottawa, Ontario K1N 6N5
Random clouds and censoring in survival analysis
[PDF] -
The theory of optional stopping is extended to general adapted random
sets called ``clouds''. In particular, a stopping theorem is proven for
martingales indexed by a class of sets. The theory may be applied to
survival analysis of spatial data censored by clouds. An analogue to
the classical Nelson-Aalen estimator of the integrated hazard is
defined, and its asymptotic behaviour is studied.
- DAVID MCDONALD, Department of Mathematics and Statistics, University of Ottawa,
Ontario K1N 6M5
Do birds of a feather flock together?
[PDF] -
One might wonder if the location of nests of red-wing blackbirds in the
Mer Bleue conservation area may be modelled as a Poisson process.
However it couldn't be a homogeneous Poisson process since some areas
are less desirable than others for nesting (part of the nesting area is
open water). The obvious alternative is that nest sites tend to clump
together perhaps for protection or alternatively these aggressive birds
tend to construct nests as far as possible from other nests.
- RICHARD SERFOZO, Georgia Institute of Technology
Reversible Markov processes on general spaces: spatial birth-death
and queueing
[PDF] -
This study describes the stationary distributions of spatial
birth-death and queueing processes that represent systems in which
discrete units (customers, particles) move about in an Euclidean or
partially ordered space where they are processed. These are reversible
Markov jump process with uncountable state spaces (sets of ``finite''
counting measures). Reversible Markov processes on countable state
spaces, introduced by Kolmogorov, have the exceptional property that
their stationary distributions have a canonical form: a simple ratio of
products of transition rates. We present an analogue of this result for
uncountable state spaces. This involves representing two-way
communication by certain Radon-Nikodym derivatives for measures on
product spaces. Included is a Kolmogorov criterion that establishes the
reversibility in the same spirit as one studies [y]-irreducible
Markov jump processes (Meyn and Tweedie (1993). Stationary Markov
Chains and Stochastic Stability). Related references for ``infinite''
birth-death processes are: N. Lopes Garcia, (1995). Birth and
death processes as projections of higher-dimensional Poisson
processes. Adv. in Appl. Probab., E. Glotzl, (1981). Time
reversible and Gibbsian point processes. I. Markovian spatial
birth and death processes on a general phase space. Math. Nachr.
- JIASHAN TANG, Carleton University
Balancing queues by mean field interaction
[PDF] -
Consider a queueing network with N nodes in which queue lengths are
balanced through mean-field interaction. When N is large, we study
the performance of such a network in terms of limiting results as N
goes to infinity.
(joint work with Don Dawson and Yiqiang Q. Zhao)
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