Discrete one-dimensional coverage process on a renewal process

TL;DR

This paper studies a discrete coverage process on the natural numbers driven by a renewal process and random interval lengths, providing sharp conditions for complete coverage and extending related results in rumor spreading and percolation.

Contribution

It introduces a new coverage model on the natural numbers using renewal processes and random intervals, with precise criteria for coverage probability.

Findings

Derived sharp conditions for coverage probability being positive or null.

Extended existing results in rumor processes to this renewal-based coverage model.

Connected coverage conditions to classical percolation and Boolean models.

Abstract

We consider the {following} coverage model on $\mathbb{N}$. For each site $i\in \mathbb{N}$ we associate a pair $(\xi_i, R_i)$ where $\{\xi_0, \xi_1, \ldots \}$ is a 1-dimensional {undelayed} discrete renewal point process and $\{R_0,R_1,\ldots\}$ is an i.i.d. sequence of $\mathbb{N}$-valued random variables. At each site where $\xi_i=1$ we start an interval of length $R_i$. Coverage occurs if every site of $\mathbb{N}$ is covered by some interval. We obtain sharp conditions for both, positive and null probability of coverage. As corollaries, we extend results of the literature of rumor processes and discrete one-dimensional Boolean percolation.