So the question says:
A barbershop has two barbers: an experienced owner and an apprentice. The owner cuts hair at the rate of 4 customers/hour, while the apprentice can only do 2 customers/hour. The owner and the apprentice work simultaneously, however any new customer will always go rst to the owner, if the latter is available. The barbershop has waiting room for only 1 customer (in case both barbers are busy), any additional customers are turned away. Suppose customers walk by the barbershop at the rate of 6 customers/hour. Construct a model to find the proportion of time the apprentice is busy cutting hair.
Construct a continuous-time Markov chain for this problem and explain your assumptions.
Write down the innitesimal generator G of this chain.
Using your model and the proportion of time the apprentice is busy cutting hair.
So my approach :
For the Markov chain, I don't know how to do it here but I guess is $P(0,1)=1, P(1,2)=0.6, P(2,3)=0.5; P(1,0)=0.4, P(2,1)=0.5,P(3,2)=1$
$\textbf{A} = \matrix{~ & 0 & 1 & 2 &3 \cr 0 & -6 & 6 & 0&0 \cr 1 & -4 & -10 & -6 & 0 \cr 2 & 0 & 6& -12 & 6 \cr 3 &0 & 0& 6 & -6 \cr } $
Then, for question 3, I calculated the corresponding equilibrium distribution and got: $\pi_0$=0.6, $\pi_1=0.6$,$\pi_2=-0.6$, $\pi_3=-0.6$ which leads to the proportion to $\pi_2$+ $\pi_3=0$ So I guess there must be something wrong. I appreciate any hint!