I came across the following continuous time Markov chain and would like to know if there is a good way to solve/approximate steady state distribution of the states.
Note that the horizontal right and left arrows has rate $\lambda$ and $k\gamma, k=1,2,3,...$, respectively.
The vertical down and up arrows has rate $\mu$ and $k\delta, k=1,2,3,...$, respectively.
The red and blue arrows made the problem difficult. They have rates $\mu$ and $\lambda$, respectively.
It is easy to see that this Markov chain is positively recurrent. But how to solve/approximate the steady state distribution? Please provide ideas/thoughts/references.
