Let $\xi_1, \xi_2, \ldots$ be a Poisson point process of density
$\lambda$ on $(0,\infty)^d, \; d \geq 1$ and let $\rho, \rho_1,
\rho_2, \ldots$ be i.i.d. positive random variables independent of
the point process. Let $C:=\cup_{i \geq 1} \{\xi_i + [0, \rho_i]^d
\}$. If, for some $t >0$, $ (t,\infty)^d \subseteq C,$ then we say
that $(0,\infty)^d$ is eventually covered. We show that the eventual
coverage of $(0,\infty)^d$ depends on the behaviour of $x P(\rho
>x)$ as $x \to \infty$ as well as on whether $d=1$ or $d \geq 2$.
These results are quite dissimilar to those known for complete
coverage of $\mathbb R^d$ by such Poisson Boolean models (Hall
\cite{h88}).
In addition, we consider the region $C:= \cup_{\{i \geq 1:\, X_i =
1\}}[i, i+\rho_i]$, where $X_1, X_2, \ldots$ is a $\{0,1\}$ valued
Markov chain and $ \rho, \rho_1, \rho_2, \ldots$ are i.i.d. positive
integer valued random variables independent of the Markov chain. We
study the eventual coverage properties of this random set $C$.