A one-dimensional Poisson growth model with non-overlapping intervals.
type: Item
instance of: scholarly article
title: A one-dimensional Poisson growth model with non-overlapping intervals.
zbMATH Open document ID: 1047.60006
DOI: 10.1016/S0304-4149(00)00049-1
published in: Stochastic Processes and their Applications
publication date: 2004-09-22T00:00:00Z
review text: Dynamic lilypond model [\textit{O. Häggström} and \textit{R. Meester}, Random Struct. Algorithms 9, 295--315 (1996; Zbl 0866.60088)] and Poisson germ model with maximally non-overlapping spherical grains (PMS) [\textit{D. J. Daley}, \textit{D. Stoyan} and \textit{H. Stoyan}, Adv. Appl. Probab. 31, 610--624 (1999; Zbl 0944.60023)] are two names of the germ-grain model \(\Xi \), the one-dimensional version of which is examined in the present paper. In the model, spherical grains grow at an equal rate from Poisson distributed germs and stop whenever two neighbour grains touch. Whereas computer simulations have been used in the above cited paper by \textit{D. J. Daley} et al. concerned with \(\Xi \subset \mathbb R^d,\;d=1,2,3,\) several analytic results based mainly on the properties of the uncovered complement \(\mathfrak A=\Xi ^c\) are established here. The main results are as follows (for the unit intensity of germs). First, the grain lengths \(V\) have distribution Pr\({(V>y)={\text e}^{-y}\exp ({\text e}^{-y}-1)}\), hence the line fraction of \(\Xi \) is \(\overline {w_1}=E(V)=\text{Pr}(x\in \Xi \mid x\in \mathbb R^1)=1-{\text e}^{-1}\) and Pr\((0\in \Xi )=1-{\text e}^{-1/2}\). Further, it is shown that the uncovered components of \(\mathfrak A\) (more precisely: of its realizations) constitute the regeneration sets of a standard regenerative process; the Kingman \(p\)-function [\textit{J. F. C. Kingman}, ``Regenerative phenomena'' (1972; Zbl 0236.60040)] is then derived and the distribution of the number \(N\) of contiguous grains in a covered interval is found: Pr\((N=k)=(k-1)/k!,\;k=2,3,\dots ,\) and \(E(N)=\) e. The length of a randomly chosen set of contiguous grains has the distribution function \(G(x)=1-{\text e}^{-y}I_1(2\sqrt x)/\sqrt x\), where \(I_1(.)\) is the modified first order Bessel function; hence the mean length of covered intervals composing \(\Xi \) is \(\alpha ={\text e}-1\), that one of uncovered intervals forming \(\mathfrak A\) is \(\beta =…
reviewed by: Ivan Saxl
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 62M30
zbMATH DE Number: 2102350
zbMATH Keywords: alternating renewal process
zbMATH Keywords: germ-grain model
zbMATH Keywords: grain-length distribution
zbMATH Keywords: lilypond model
zbMATH Keywords: Stienen model
author: D. J. Daley
author: Colin L. Mallows
author: Larry Shepp
MaRDI profile type: Publication
cites work: Q3331506
cites work: The volume fraction of a Poisson germ model with maximally non-overlapping spherical grains
cites work: Nearest neighbor and hard sphere models in continuum percolation
cites work: The stochastic theory of regenerative events
cites work: Q5646191
cites work: Q4237322
cites work: Q4842952
Wikidata QID: Q126777902
full work available at URL: s0304-4149(00)00049-1
OpenAlex ID: W2131800074
Recommended article: One-dimensional Poisson growth models with random and asymmetric growth
Recommended article: A model for random instantaneous growth on an interval
Recommended article: A one dimensional model of population growth
Recommended article: A linear random growth model
Recommended article: Q3778580
Recommended article: A nonlinear stochastic growth model on discrete time domains
Recommended article: Q3483132
description: scientific article; zbMATH DE number 2102350