Discrete One-Dimensional Coverage Process on a Renewal Process

Sandro Gallo and Nancy L. Garcia

Consider the following coverage model on N, for each site i ∈ N associate a pair (ξi, Ri) where (ξi)i≥0 is a 1-dimensional undelayed discrete renewal point process and (Ri)i≥0 is an i.i.d. sequence of N-valued random variables. At each site where ξi = 1 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.

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