  Let $X_1$, $X_2$, $...$ be a sequence of independently and identically
distributed random variables with $\mathsf{E}X_1=0$, and let $S_0=0$ and
$S_t=S_{t-1}+X_t$, $t=1,2,...$, be a random walk. Denote
$\tau={cases}\inf\{t>1: S_t\leq0\}, &\text{if} \ X_1>0, 1, &\text{otherwise}.
{cases}$ Let $\alpha$ denote a positive number, and let $L_\alpha$ denote the
number of level-crossings from the below (or above) across the level $\alpha$
during the interval $[0, \tau]$. Under quite general assumption, an inequality
for the expected number of level-crossings is established. Under some special
assumptions, it is proved that there exists an infinitely increasing sequence
$\alpha_n$ such that the equality $\mathsf{E}L_{\alpha_n}=c\mathsf{P}\{X_1>0\}$
is satisfied, where $c$ is a specified constant that does not depend on $n$.
The result is illustrated for a number of special random walks. We also give
non-trivial examples from queuing theory where the results of this theory are
applied.
