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Statistics Bernoulli single-server queuing process

  1. Apr 3, 2017 #1
    1. The problem statement, all variables and given/known data
    Suppose your office telephone has two lines, allowing you to talk with someone and have at most one other person on hold. You receive 10 calls per hour and a conversation takes 2 minutes, on average. Use a Bernoulli single-server queuing process with limited capacity and 1-minute frames to compute the proportion of time you spend using the telephone.

    2. Relevant equations


    3. The attempt at a solution

    Found the transition probability matrix as:

    5/6 1/6 0
    5/12 1/2 1/12 = [x y z]
    0 5/12 7/12

    From this matrix, I found the following system of equations
    5/6x + 5/12y = x
    1/6x +1/2y + 5/12z = y
    1/12y + 7/12z = z
    Solving the system of equations from this matrix I got
    x=25/81
    y = 10/27
    z = 26/81

    I thought that the proportion of time you would spend on the telephone is 56/81, which would be the steady state probabilities of y ( One customer on the phone) and z (One customer on the phone and another one on hold), but that answer is wrong.
    I also tried the steady state probability of y = 10/27, but that is also wrong.

    Can you please explain what I am doing wrong?

    Thank you so much.
     
  2. jcsd
  3. Apr 3, 2017 #2
    I realized I made an algebra mistake while computing the system of equations..
    The correct answers to the system of equations were
    x = 25/37 y = 10/37 and z = 2/37
     
  4. Apr 5, 2017 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    Your transition probability matrix is incorrect: its second row adds up to less than 1.
     
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