Random walk on a quadrant: mapping to a one-dimensional level-dependent Quasi-Birth-and-Death process (LD-QBD)

We consider a neighbourhood random walk on a quadrant, $\{(X_1(t),X_2(t),\phi (t)):t\ge 0\}$, with state space \begin{eqnarray*} $\mathcal{S}$&=&{(n,m,i):n,m=0,1,2,\ldots;i=1,2,\ldots,k(n,m)}. \end{eqnarray*} Assuming start in state $(n,m,i)$, the process spends exponentially distributed amount of time in $(n,m,i)$ according to some parameter ${\lambda }_i^{(n,m)}$. Upon leaving state $(n,m,i)$ the process moves to some state $(n^{{}^{\mathrm{\prime }}},m^{{}^{\mathrm{\prime }}},j)$ with $j\in \{1,\dots ,k(n^{{}^{\mathrm{\prime }}},m^{{}^{\mathrm{\prime }}})\}$ and $n^{{}^{\mathrm{\prime }}}\in \{n-1,n,n+1\}$, $m^{{}^{\mathrm{\prime }}}\in \{m-1,m,m+1\}$, according to some probabilities $(p_{n;a}^{m;b}{)}_{i,j}$ with $a,b\in \{+,-,0\}$. We transform this process into a one-dimensional LD-QBD $\{(Z(t),\chi (t)):t\ge 0\}$ with level variable $Z(t)$ and phase variable $\chi (t)$. Using this transform we find its transient and stationary analysis using matrix-analytic methods, as well as the distribution at first hitting times.