Some results connected with the expectation process

In the paper we find the necessary and sufficient conditions for the exponentiality of the asymptotics of the average time which is needed for the integer expectation process $W_t(t=0,1,2,\dots)$ to reach the level $N\to\infty$. It is established also that the well-known sufficient condition for the exponentiality of
$$ \mu_N=\mathbf P\{\max_{n\ge 0}S_n=N\} $$
(where $S_n$ is a random walk with negative drift) is a necessary one.