416-8sol - Denker Spring 2010 416 Stochastic Modeling -...

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Denker Spring 2010 416 Stochastic Modeling - Assignment 8 Solution Problem 1: (Problem 1, p. 346) The time required to repair a machine is an exponentially distributed random variable with mean 0.5 hours. What is the probability that the repair takes at least 12.5 hours given that its duration exceeds 12 hours? Solution: Let X denote the repair time of the machine. X is exponential with rate 2. Then we are asked to compute P ( X 12 . 5 | X > 12) = P ( X > 0 . 5) = e - 2(0 . 5) = e - 1 , where we also used the memoryless property. Problem 2: (Problem 2, p. 346) Suppose you arrive at a single-teller bank to Fnd Fve other customers in the bank, one being served, the other four waiting in line. You join the end of the line. If the service ti- mes are all exponential with rate μ , what is the expected time you will spend in the bank? Solution: Let S 1 , ..., S 6 denote the service times of the Fve customer in the bank and yours. S 1 is the remaining service time of the customer just being served. This remaining time has
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416-8sol - Denker Spring 2010 416 Stochastic Modeling -...

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