I am trying to find the mean first passage time in a birth-death process to get from state 2 to state 1.
Here is the question in detail:
We are in a birth-death system, so the state space corresponds to the natural numbers. (The states are 0,1,2,3,...) Additionally, the system can only move from its current state to an adjacent state (If in state n, the next state can only be n-1 or n+1), unless its current state is 0, in which case it can only go up to 1.
We have a birthrate $\lambda$ and deathrate $\mu$ for the current state n.
$\lambda=A$
$\mu=nB$
Where A and B are real numbers.
What I want to find is the mean first passage time to get from state 2 to state 1.
I'm not highly familiar with Markov processes or statistics in general, but I am math-literate and would like to understand the derivation. If someone could explain it in a way that is thorough, but doesn't assume too much foreknowledge, that would be greatly appreciated.
Thanks!