  Let $\textbf{a}_1,\dots, \textbf{a}_r$ be vectors in a half-space of
$\mathbb{R}^n$. We call $$C=\textbf{a}_1\mathbb{R}^++\cdots+\textbf{a}_r
\mathbb{R}^+$$ a convex polyhedral cone, and call $\{\textbf{a}_1,\dots,
\textbf{a}_r\}$ a generator set of $C$. A generator set with the minimal
cardinality is called a frame. We investigate the translation tilings of convex
polyhedral cones.
  Let $T\subset \mathbb{R}^n$ be a compact set such that $T$ is the closure of
its interior, and $\mathcal{J}\subset \mathbb{R}^n$ be a discrete set. We say
$(T,\mathcal{J})$ is a translation tiling of $C$ if $T+\mathcal{J}=C$ and any
two translations of $T$ in $T+\mathcal{J}$ are disjoint in Lebesgue measure.
  We show that if the cardinality of a frame of $C$ is larger than $\dim C$,
the dimension of $C$, then $C$ does not admit any translation tiling; if the
cardinality of a frame of $C$ equals $\dim C$, then the translation tilings of
$C$ can be reduced to the translation tilings of $(\mathbb{Z}^+)^n$. As an
application, we characterize all the self-affine tiles possessing
  polyhedral corners, which generalizes a result of Odlyzko [A. M. Odlyzko,
\textit{Non-negative digit sets in positional number systems}, Proc. London
Math. Soc., \textbf{37}(1978), 213-229.].
