Burkes theorem and M/M/1 queue

Question

Burke's theorem says that the output process of an $M/M/1$ queue with arrival rate $\lambda$ and service rate $\mu$ follows a Poisson with parameter $\lambda$. Suppose after service completion the customer enters state 1 with probability $p$ and state 2 with probability $1-p$. Suppose that the current state is $(\text{number of customers in the queue},state\ 1, state \ 2)=(N,n,m)$. I am interested in finding the rate of going from $(N,n,m)$ to $(N-1,n+1,m)$. Initially I thought the rate is $\mu p$. But I guess Burkes theorem says that the rate is $\lambda p$. Which one of the two is correct? Any help will be appreciated.