  Uniformization theory of Gromov hypebolic spaces investigated by Bonk,
Heinonen and Koskela, generalizes the case where a classical Poincar\'e ball
type model is used as the starting point. In this paper, we develop this
approach in the case where the underlying domain is unbounded, corresponding to
the classical Poincar\'e half-space model. More precisely, we study conformal
densities via Busemann functions on Gromov hyperbolic spaces and prove that the
deformed spaces are unbounded uniform spaces. Furthermore, we show that there
is a one-to-one correspondence between the bilipschitz classes of proper
geodesic Gromov hyperbolic spaces that are roughly starlike with respect to a
point on Gromov boundary and the quasisimilarity classes of unbounded locally
compact uniform spaces. Our result can be understood as an unbounded
counterpart of the main result of Bonk, Heinonen, and Koskela in "Uniformizing
Gromov Hyperbolic Spaces", Ast\'erisque 270 (2001).
