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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

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

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

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

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?

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

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

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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