Busy Barber Problem: Proportion of Time Apprentice is Busy

[iikii]

Homework Statement

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 first 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.

1. Construct a continuous-time Markov chain for this problem and explain your assumptions.

2. Write down the infinitesimal generator G of this chain.

3. Using your model nd the proportion of time the apprentice is busy cutting hair.

Homework Equations

The Attempt at a Solution

1. 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
2. I attached a picture of my markov chain.
3. Then, for question 3, I calculated the corresponding equilibrium distribution and got: π0 =0.6, π1=0.6, π2=−0.6, π3=−0.6 which leads to the proportion to π2+ π3=0
So I guess there must be something wrong. I appreciate any hint!

 

[Ray Vickson]

Homework Statement

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 first 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.

1. Construct a continuous-time Markov chain for this problem and explain your assumptions.

2. Write down the infinitesimal generator G of this chain.

3. Using your model nd the proportion of time the apprentice is busy cutting hair.

Homework Equations

The Attempt at a Solution

1. 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
2. I attached a picture of my markov chain.
3. Then, for question 3, I calculated the corresponding equilibrium distribution and got: π0 =0.6, π1=0.6, π2=−0.6, π3=−0.6 which leads to the proportion to π2+ π3=0
So I guess there must be something wrong. I appreciate any hint!

You need $a_{ij} \geq 0$ for $i \neq j$, but your second row has negative values for $a_{10}$ and $a_{12}$.

You should realize that you can NEVER get negative probabilities, so getting $\pi_2 < 0$ and $\pi_3 < 0$ is an immediate signal that you have erred.

Also: in future, please just type out the matrix directly; I found it extremely inconvenient to open the attachment and then navigate back to this panel.