I have a two dimensional finite Markov chain with $(m+1)^2$ states, and with transition rates:
$q_x((x,y)\to (x+1,y))=(m-x)\lambda,\quad 0\leq x< m, 0\leq y \leq m$,
$q_x((x,y)\to (x-1,y))=x\mu,\quad 0 <x\leq m, 0\leq y \leq m$,
$q_y((x,y)\to (x,y+1))=(m-y)\lambda,\quad 0\leq x\leq m, 0\leq y < m$,
$q_y((x,y)\to (x,y-1))=y\mu,\quad 0\leq x< m, 0< y \leq m$,
I need to find the first and second moment of the recurrence time for the set of states $\mathcal{A}$, where $\mathcal{A}=\{ (x,0)\cup (0,y): 0\leq x,y \leq m \}$.
Any help is greatly appreciated!
For the corresponding 1D Markov process, i.e. the Markov process with transition rates: $q(x,x+1)=(m-x)\lambda, 0\leq x<m$ and $q(x,x-1)=x\mu, 0<x\leq m$, the first and the second moments of the recurrence time to state $x=0$ is known. Now I am stuck in finding the recurrence time to set of states $\mathcal{A}$ in the two dimensional Markov process described above.