  We report the first measurement of the inclusive $e^+e^- \to b\bar{b} \to
D_s^{\pm}X$ and $e^+e^- \to b\bar{b} \to D^0/\bar{D}^0X$ cross sections in the
energy range from $10.63$ to $11.02$. Based on these results, we determine
$\sigma(e^+ e^- \to B_s^0 \bar{B}_s^0 X)$ and $\sigma(e^+ e^- \to B\bar{B}X)$
in the same energy range. We measure the fraction of $B_s^0$ events at
$\Upsilon(10860)$ to be $f_{\rm s}=(22.0^{+2.0}_{-2.1})\%$. We determine also
the ratio of the $B_s^0$ inclusive branching fractions $\mathcal{B}(B_s^0 \to
D^0/\bar{D}^0X)/\mathcal{B}(B_s^0 \to D_s^{\pm} X)=0.416 \pm 0.018 \pm 0.092$.
The results are obtained using the data collected with the Belle detector at
the KEKB asymmetric-energy $e^+e^-$ collider.
