  In this paper we study the problem of hyperball (hypersphere) packings in
$3$-dimensional hyperbolic space. We introduce a new definition of the
non-compact saturated ball packings and describe to each saturated hyperball
packing, a new procedure to get a decomposition of 3-dimensional hyperbolic
space $\HYP$ into truncated tetrahedra. Therefore, in order to get a density
upper bound for hyperball packings, it is sufficient to determine the density
upper bound of hyperball packings in truncated simplices.
