Consider the following situation:
Jobs arrive as a Poisson process at a rate of $\lambda$ per hour. We have two servers A and B with mean processing rate $\mu_A$ and $\mu_B$ respectively. Both servers have their own buffer, and jobs are assigned to server A with probability $\frac{1}{4}$ or to server B otherwise. Thus it may occur that one server is idle while the other one is busy and has jobs in its buffer.
I want to derive the steady state distribution. However, I am stuck.
In previous cases such as M/M/1 and M/M/c queues I was able to define a recursive relation between each state. How would I do that if we want to know the relation between, for example, 1 and 2 jobs in the system? Since there are three ways to have a total of 2 jobs in the system ((A=2,B=0),(A=1,B=1),(A=0,B=0)), how would I define this?