  In this paper, we consider the $(1,R)$ state-dependent reflecting random walk
(RW) on the half line, allowing the size of jumps to the right at maximal $R$
and to the left only 1. We provide an explicit criterion for positive
recurrence and the explicit expression of the stationary distribution based on
the intrinsic branching structure within the walk. As an application, we obtain
the tail asymptotic of the stationary distribution in the "near critical"
situation.
