  We consider a two dimensional skip-free reflecting random walk on a
nonnegative integer quadrant. We are interested in the tail asymptotics of its
stationary distribution, provided its existence is assumed. We derive exact
tail asymptotics for the stationary probabilities on the coordinate axis. This
refines the asymptotic results in the literature, and completely solves the
tail asymptotic problem on the stationary marginal distributions in the
coordinate and diagonal directions. For this, we use the so-called analytic
function method in such a way that either generating functions or moment
generating functions are suitably chosen. The results are exemplified by a two
node network with simultaneous arrivals.
