  In this paper, we consider the state-dependent reflecting random walk on a
half-strip. We provide explicit criteria for (positive) recurrence, and an
explicit expression for the stationary distribution. As a consequence, the
light-tailed behavior of the stationary distribution is proved under
appropriate conditions. The key idea of the method employed here is the
decomposition of the trajectory of the random walk and the main tool is the
intrinsic branching structure buried in the random walk on a strip, which is
different from the matrix-analytic method.
