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

Stability of multidimensional skip-free Markov modulated reflecting  random walks: Revisit to Malyshev and Menshikov's results and application to  queueing networks