Discrete One-Dimensional Coverage Process on a Renewal Process

Sandro Gallo and Nancy L. Garcia

Consider the following coverage model on \mathbb {N}, for each site i \in \mathbb {N}associate a pair (\xi _i, R_i) where (\xi _i)_{i \ge 0} is a 1-dimensional undelayed discrete renewal point process and (R_i)_{i \ge 0} is an i.i.d. sequence of \mathbb {N}-valued random variables. At each site where \xi _i=1 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.

The whole paper is available here.

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