Random walk on a quadrant: mapping to a one-dimensional level-dependent Quasi-Birth-and-Death process (LD-QBD) (2302.02225v1)

Abstract: 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.