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Statistics Bernoulli single-server queuing process with ATMS

  1. Apr 3, 2017 #1
    1. The problem statement, all variables and given/known data
    Customers arrive at an ATM at a rate of 12 per hour and spend 2 minutes using it, on average. Model this system using a Bernoulli single-server queuing process with 1-minute frames.

    a. Compute the transition probability matrix for the system.

    b. If the ATM is idle now, find the probability distribution of states in 3 minutes. (That is, compute the probabilities that, in 3 minutes, the ATM will be: (i) idle, (ii) serving one customer with no one in queue, (iii) serving one customer with one customer waiting, and (iv) serving one customer with two customers waiting.

    c. Use your answer in part b to compute the expected number of customers in the system 3 minutes after the ATM was idle.

    d. Use your answers in parts b and c to compute the expected number customers waiting in queue 3 minutes after the ATM was idle.

    2. Relevant equations
    3. The attempt at a solution

    I got the correct matrix for part a.
    \begin{pmatrix}
    0.8 & 0.2 & 0& 0 \\ 0.4 & 0.5 & 0.1 & 0 \\ 0 & 0 & 0.4 & 0.5
    \end{pmatrix}

    But I am unable to compute the steady state probabilities for part b, and therefore, my answers for part c and d are also wrong.
    My attempt at b is as follows.
    I found the probability distribution in 3 minutes which would be the transition probability matrix 3,
    \begin{pmatrix}
    0.7200000000000002 & 0.26 & 0.020000000000000004 & 0 \\ 0.52& 0.37& 0.1& 0.010000000000000002\\
    0.16000000000000003 & 0.4 & 0.33000000000000007& 0.1\\ 0& 0.16000000000000003& 0.44 & 0.4
    \end{pmatrix}

    and solving it I got pi = \left< \frac 32 53 \, \frac 16 53, \frac 4 53, \frac 1 53 right>

    Please advise.

     
  2. jcsd
  3. Apr 3, 2017 #2
    Steady State Probabilities I got were
    32/53 , 16/53, 4/53, 1/53
     
  4. Apr 5, 2017 #3
    Realized my mistake!
    My matrix was incorrect, but to solve the problem you should still find the matrix to the 3rd power.
     
  5. Apr 5, 2017 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Your transition probability matrix is wrong: it should have 4 rows, not three, and all its row-sums should be 1.
     
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