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Math Help - exponentials

  1. #1
    Super Member Anonymous1's Avatar
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    exponentials

    In a hardware store you must first go to server 1 to get your goods and then to server 2 to pay for them. Suppose that the times for the times for the two activities are exponentially distributed with means 6 min and 3 min. Compute the average amount of time it takes Bob to get his goods and pay if when he comes in there is a customer named Al with server 1 and no one at server 2.
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  2. #2
    Super Member Random Variable's Avatar
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    I remember doing this problem in a stochastic models class (although I don't think it was a hardware store and I don't think the customers had names).

    Here's what I did:


    Let  T = the total amount of time Bob spends in the hardware store

    Let  T_{1} = the amount of time Bob spends waiting in line for Al to get his goods from server 1

    Let  T_{2} = the amount of time it takes for Bob to get his goods from server 1

    Let  T_{3} = the amount of time Bob spends waiting in line for Al to pay server 2 for his goods

    and Let  T_{4} = the amount of time it takes for Bob to pay server 2 for his goods


    Then  T= T_{1}+T_{2}+T_{3}+T_{4}

    and  E[T] = E[T_{1}]+E[T_{2}]+E[T_{3}]+E[T_{4}]


     E[T_{1}] = 6 because of memoryless property of the exponential distribution


     E[T_{2}] = 6


    To find  E[T_{3}] , condition on the availability of server 2 after Bob is done with server 1

     E[T_{3}] = E[T_{3}|\text{server 2 is free}]P(\text{server 2 is free}) + E[T_{3}|\text{server 2 is not free}] P(\text{server 3 is not free})

     = 0 + 3 \cdot P(\text{Bob is done with server 1 before Al is done with server 2})

     = 3 \cdot \frac{1/6}{1/6+1/3} = 1 (I used the probability than one exponential random variable is smaller than another exponential random variable.)


    and  E[T_{4}] = 3


    so E[T] = 6+6+1+3= 16 minutes
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  3. #3
    Super Member Anonymous1's Avatar
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    Thanks so much. I tried to construct something similar, but got lost around E(T3). And your final answer matches with the back of my book btw.
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    Super Member Random Variable's Avatar
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