  We consider a version of the forest fire model on graph $G$, where each
vertex of a graph becomes occupied with rate one. A fixed vertex $v_0$ is hit
by lightning with the same rate, and when this occurs, the whole cluster of
occupied vertices containing $v_0$ is burnt out. We show that when $G=Z_{+}$,
the times between consecutive burnouts at vertex $n$, divided by $\log n$,
converge weakly as $n\to\infty$ to a random variable which distribution is
$1-\rho(x)$ where $\rho(x)$ is the Dickman function.
  We also show that on transitive graphs with a non-trivial site percolation
threshold and one infinite cluster at most, the distributions of the time till
the first burnout of {\it any} vertex have exponential tails.
  Finally, we give an elementary proof of an interesting limit:
$\lim_{n\to\infty} \sum_{k=1}^n {n \choose k} (-1)^k \log k -\log\log
n=\gamma$.
