Question: Consider a communication channel where two users, A and B are sharing. Their arrival rates are $\lambda_A$ and $\lambda_B$ and service rates are $\mu_A$ and $\mu_B$. Only one user can use the channel at a time and no waiting queues, so the arrival rate is dropped if the channel is busy.
After thinking about it a while I drew this transition diagram, and I wonder if it's correct?
My logic was that, for user B to enter, the channel should go to a state where user A was served and vise versa. So the state n-1 was when user A accessing the system and n+1 would be user B's.
Also what could be the transition matrix for this scenario?
Update: Moved the updated work to an answer.
A (1) is the state when A is being served and B (3) is the state when B is being served. Idle (2) is when channel is empty.
So I came up with the infinitesimal generator of this process as, \begin{equation*} \mathbf{A} =\begin{bmatrix} -\mu_A & \mu_A & 0\\ \lambda_A & -(\lambda_A+\lambda_B) & \lambda_B\\ 0 & \mu_B & -\mu_B \end{bmatrix} \end{equation*}
and the transition matrix as, \begin{equation*} \mathbf{P} =\begin{bmatrix} \gamma_{00} = -\upsilon_0 & \gamma_{01} & \gamma_{02}\\ \gamma_{10} & \gamma_{11} = -\upsilon_1 & \gamma_{12}\\ \gamma_{20} & \gamma_{21} & \gamma_{22}=-\upsilon_2 \end{bmatrix} =\begin{bmatrix} 0 & 1 & 0\\ \lambda_A/(\lambda_A+\mu_A) & 0 & \lambda_B/(\lambda_A+\mu_A)\\ 0 & 1 & 0 \end{bmatrix} \end{equation*}