Discrete One-Dimensional Coverage Process on a Renewal Process

Abstract

We consider the following coverage model on N. For each site i ∈ N we associate a pair (ξi, Ri) where {ξ0, ξ1, . . .} is a 1-dimensional undelayed discrete renewal point process and {R0, R1, . . .} is an i.i.d. sequence of N-valued random variables. At each site where ξi = 1 we start an interval of length Ri. Coverage occurs if every site of 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.