Consider a Harris chain given by $\{X_n\}$ with the following transition function, $X_{n+1}=\max \{0,X_n-b\} $ with probability $p$ and $X_{n+1}=\max \{0,a-\tau\} $ with probability $1-p$, where $\tau \sim Exp(\lambda) $ and $Exp()$ is the exponential distribution. Also $a$, $b$ can be considered as positive constants.
Is this chain positive Harris recurrent? How to find the stationary distribution of this chain, if it is positive Harris recurrent?
Any good reference is also useful. Thanks!