We consider a point process i + ξi, where i ∈ Z and the ξi’s are i.i.d. random variables with variance σ. This process, with a suitable rescaling of the distribution of ξi’s, converges to the Poisson process in total variation for large σ. We then study a simple queueing system with our process as arrival process, and we provide a complete analytical description of the system. Although the arri...

Modelling structured multivariate point process data has wide ranging applications like understanding neural activity, developing faster file access systems and learning dependencies among servers in large networks. In this paper, we develop the Poisson network model for representing multivariate structured Poisson processes. In our model each node of the network represents a Poisson process. T...

This paper investigates an alternative way of using the Stein-Chen method in Poisson approximations. There are three principal bounds stated in terms of reduced Palm probabilities for general point processes. The rst two are for the accuracy of Poisson random variable approximation to the distribution of the number of points in a point process with respect to the total variation metric and Wass...

We consider an extension of the Poisson Hail model where the service speed is either zero or infinity at each point of the Euclidean space. We use and develop tools pertaining to sub-additive ergodic theory in order to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hails...

Let the nodes of a Poisson point process move independently in R according to Brownian motions. We study the isolation time for a target particle that is placed at the origin, namely how long it takes until there is no node of the Poisson point process within distance r of it. In the case when the target particle does not move, we obtain asymptotics for the tail probability which are tight up t...

This article describes the batch Markovian arrival process (BMAP), a point process that is characterized by Markov-modulated batch arrivals of random size. The BMAP is a generalization of many well-known processes including the Markovian arrival process (MAP), the Poisson process, and the Markov-modulated Poisson process. It provides a common framework for modeling arrival processes in a variet...

By dealing with interarrival times as exponentially distributed fuzzy random variables, a fuzzy random homogeneous Poisson process and a fuzzy random compound Poisson process are respectively defined. Several theorems on the two processes are provided, respectively.

In this paper a nerve cell (or neuron) is modeled as a system that transforms a semimartingale input into a point process output. This extends a result of van der Boogaard et al. (1986) in which neurons transform point process inputs into point process outputs. The semimartingale input model proposed here encompasses Poisson, Gauss-Poisson, diffusion and Ito-Markov processes which have been con...

The characteristic independence property of Poisson point processes gives an intuitive way to explain why a sequence of point processes becoming less and less repulsive can converge to a Poisson point process. The aim of this paper is to show this convergence for sequences built by superposing repulsive point processes. We use Papangelou intensities and Stein’s method to prove this result with ...