So, I'm looking at a Birth/Death process $Z$ with an arrival rate of $\frac{1}{n+1}$ and a departure rate of $1$.
I'm trying to show that this process is positive recurrent and find the stationary distribution.
My understanding of positive recurrent is that if there is a non-zero probability that the markov chain will not return to a state.
Additionally, it is a class property, and there only seems to be one communication class as you can access every state from any other state.
Since the departure rate is greater than 0, isn't there always a probability that the markov chain will return to any initial starting state? So, if we take state 1, there is always a non-zero probability that the markov chain will return back to state 1 as the departure rate is nonzero?
I'm a bit confused on how to answer the question (and show that this chain is positive recurrent) since I don't agree with the assumption the question is making (that this process is positive recurrent).
I'd appreciate any help! Thank you so much!