  We study the asymptotic behavior of the asymptotic translation lengths on the
curve complexes of pseudo-Anosov monodromies in a fibered cone of a fibered
hyperbolic 3-manifold $M$ with $b_1(M) \geq 2$. For a sequence $(\Sigma_n,
\psi_n)$ of fibers and monodromies in the fibered cone, we show that the
asymptotic translation length on the curve complex is bounded above by
$1/\chi(\Sigma_n)^{1+1/r}$ as long as their projections to the fibered face
converge to a point in the interior, where $r$ is the dimension of the
$\psi_n$-invariant homology of $\Sigma_n$ (which is independent of $n$). As a
corollary, if $b_1(M) = 2$, the asymptotic translation length on the curve
complex of such a sequence of primitive elements behaves like
$1/\chi(\Sigma_n)^{2}$. Furthermore, together with a work of E. Hironaka, our
theorem can be used to determine the asymptotic behavior of the minimal
translation lengths of handlebody mapping class groups and the set of mapping
classes with homological dilatation one.
