Suppose I have three patient arrivals $A_1,A_2,A_3$. Suppose that each of these patient arrivals occur (independent from one another) according to an exponential distribution with respective rates $a_1,a_2,a_3$.
Now for each of these 3 types of arrivals, suppose we have a corresponding service time which also takes an exponential amount of time to service with respective rates $b_1,b_2,b_3$. (after the service we can assume the patient leaves the system).
Now suppose that $(X_t)_{t\ge 0}$ is the markov process that models the state of the system. Here $X_t=0$ when no one is in the system, and $X_t= j$ where $j=1,2,3$ denotes the a patient is getting treated for service $j$.
I am trying to figure out what the generator is of this Markov process, and what are the forward equations for the transition probabilities.
Thanks!