1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Busy barber problem
  1. Business Calculus (Replies: 3)

  2. Problem - (Replies: 7)

Loading...