  We consider a neighbourhood random walk on a quadrant,
$\{(X_1(t),X_2(t),\varphi(t)):t\geq 0\}$, 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 $(n^{'},m^{'},j)$ with $j\in\{1,\ldots,k(n^{'},m^{'})\}$ and
$n^{'}\in\{n-1,n,n+1\}$, $m^{'}\in\{m-1,m,m+1\}$, 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 $\{(Z(t),\chi(t)):t\geq 0\}$ 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.
