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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 spendsexponentially distributed amount of time in $(n,m,i)$ according to someparameter $\lambda_i^{(n,m)}$. Upon leaving state $(n,m,i)$ the process movesto 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 someprobabilities $(p_{n;a}^{m;b})_{i,j}$ with $a,b\in\{+,-,0\}$. We transform thisprocess into a one-dimensional LD-QBD $\{(Z(t),\chi(t)):t\geq 0\}$ with levelvariable $Z(t)$ and phase variable $\chi(t)$. Using this transform we find itstransient and stationary analysis using matrix-analytic methods, as well as thedistribution at first hitting times.

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