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Homework Help: Busy barber problem

  1. Nov 17, 2016 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations

    3. 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!

    Attached Files:

  2. jcsd
  3. Nov 17, 2016 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    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.
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