  Let $\{\boldsymbol{X}_n\}$ be a discrete-time $d$-dimensional process on
$\mathbb{Z}_+^d$ with a supplemental (background) process $\{J_n\}$ on a finite
set and assume the joint process
$\{\boldsymbol{Y}_n\}=\{(\boldsymbol{X}_n,J_n)\}$ to be Markovian. Then, the
process $\{\boldsymbol{X}_n\}$ can be regarded as a kind of reflecting random
walk (RRW for short) in which the transition probabilities of the RRW are
modulated according to the state of the background process $\{J_n\}$; we assume
this modulation is space-homogeneous inside $\mathbb{Z}_+^d$ and on each
boundary face of $\mathbb{Z}_+^d$. Further we assume the process
$\{\boldsymbol{X}_n\}$ is skip free in all coordinates and call the joint
process $\{\boldsymbol{Y}_n\}$ a $d$-dimensional skip-free Markov modulated
reflecting random walk (MMRRW for short). The MMRRW is an extension of an
ordinary RRW and stability of ordinary RRWs have been studied by Malyshev and
Menshikov. Following their results, we obtain stability and instability
conditions for MMRRWs and apply our results to stability analysis of a
two-station network.
