I'm searching for hints on how to analyze the following Markov chain. I can solve for the steady state probabilities numerically by using a finite transition matrix. However, I would like to have an analytical result.
For the special case of only transitions to the adjacent states (birth death process) I have succeeded in using the reversibility property. For this extended case, however, I'm missing a starting point.
The Markov chain continues after state "4" to infinity.
EDIT: The transition matrix $P$ is given by \begin{equation} P = \begin{pmatrix} p_{0,0} & p_{0,1} & p_{0,2} & 0 & 0 & 0 & 0\\ p_{1,0} & p_{1,1} & p_{1,2} & p_{1,3} & 0 & 0 & 0\\ p_{2,0} & p_{2,1} & p_{2,2} & p_{2,3} & p_{2,4} & 0 & 0 & \cdots\\ 0 & p_{2,0} & p_{2,1} & p_{2,2} & p_{2,3} & p_{2,4} & 0\\ 0 & 0 & p_{2,0} & p_{2,1} & p_{2,2} & p_{2,3} & p_{2,4} \\ &&&\vdots&&&&\ddots \end{pmatrix} \end{equation} with a repeating pattern from the third row on. $\pi_v=\sum_{w=0}^\infty p_{w,v}\pi_w$ should hold for the elements $\pi_v$ of the steady state vector.
For every $x\geqslant4$, $$\pi_x=p_{2,4}\pi_{x-2}+p_{2,3}\pi_{x-1}+p_{2,2}\pi_{x}+p_{2,1}\pi_{x+1}+p_{2,0}\pi_{x+2},$$ hence the generating function $$\Pi(s)=\sum_{x\geqslant0}\pi_xs^x,$$ solves the identity $$\Pi(s)=(p_{2,4}s^2+p_{2,3}s+p_{2,2}+p_{2,1}s^{-1}+p_{2,0}s^{-2})\Pi(s)+s^{-2}R(s),$$ for some polynomial $R(s)$ of degree at most $5$, in particular, $$\Pi(s)=\frac{R(s)}{s^2-p_{2,4}s^4-p_{2,3}s^3-p_{2,2}s^2-p_{2,1}s-p_{2,0}}.$$ From here, factor the denominator, decompose the fraction into simple elements, and expand each as a series in $s$, to recover each $\pi_x$ as the coefficient of $s^x$ in $\Pi(s)$.