  We consider a discrete-time two-dimensional process $\{(X_{1,n},X_{2,n})\}$
on $\mathbb{Z}^2$ with a background process $\{J_n\}$ on a finite set $S_0$,
where individual processes $\{X_{1,n}\}$ and $\{X_{2,n}\}$ are both skip free.
We assume that the joint process
$\{\boldsymbol{Y}_n\}=\{(X_{1,n},X_{2,n},J_n)\}$ is Markovian and that the
transition probabilities of the two-dimensional process $\{(X_{1,n},X_{2,n})\}$
vary according to the state of the background process $\{J_n\}$. This
modulation is assumed to be space homogeneous. We refer to this process as a
two-dimensional skip-free Markov modulate random walk. For $\boldsymbol{Y},
\boldsymbol{Y}'\in \mathbb{Z}_+^2\times S_0$, consider the process
$\{\boldsymbol{Y}_n\}_{n\ge 0}$ starting from the state $\boldsymbol{Y}$ and
let $\tilde{q}_{\boldsymbol{Y},\boldsymbol{Y}'}$ be the expected number of
visits to the state $\boldsymbol{Y}'$ before the process leaves the nonnegative
area $\mathbb{Z}_+^2\times S_0$ for the first time. For
$\boldsymbol{Y}=(x_1,x_2,j)\in \mathbb{Z}_+^2\times S_0$, the measure
$(\tilde{q}_{\boldsymbol{Y},\boldsymbol{Y}'}; \boldsymbol{Y}'=(x_1',x_2',j')\in
\mathbb{Z}_+^2\times S_0)$ is called an occupation measure. Our main aim is to
obtain asymptotic decay rate of the occupation measure as the values of $x_1'$
and $x_2'$ go to infinity in a given direction. We also obtain the convergence
domain of the matrix moment generating function of the occupation measures.
