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# lec0405 - IEOR 4106 Spring 2011 Professor Whitt...

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IEOR 4106, Spring 2011, Professor Whitt Introduction to Renewal Theory: Tuesday, April 5 1. Painful Memories: Visit to a Museum Recall: A popular museum is open for 11 hours each day, but only admits new visitors during the first 9 hours. All visitors must leave at the end of the eleven-hour period. Suppose that visitors (which may be individuals or small groups, which we treat as individuals) arrive at the museum according to a Poisson process at rate 100 per hour. Suppose that each visitor, independently of all other visitor, spends a random time in the museum that is uniformly distributed between 0 and 2 hours. Suppose that 25% of these visitors visit the museum gift shop while they are in the museum. (Visiting the gift shop is assumed not to alter the total length of stay in the museum.) Statistics have revealed that the dollar value of the purchases by each visitor to the gift shop has, approximately, a gamma distribution with mean \$40 and standard deviation \$30. I. Formalizing what has been assumed: random variables and stochastic pro- cesses. We can formalize the information problem by defining several stochastic processes, some sequence of random variables and some continuous-time processes. We now review the structure, based on the assumptions above: continuous-time stochastic processes: Let N ( t ) be the number of visitors to come to the museum in the first t hours, i.e., during the time interval [0 , t ]. (Here 0 t 9, but ignore the termination time; think of it as a stochastic process with t 0.) Let M ( t ) be the number of visitors to come to the museum during the time interval [0 , t ] that will go to the gift shop sometime during their visit. Let D ( t ) be the dollar value of the purchases from the gift shop by all the visitors that initially arrived at the museum in the interval [0 , t ]. associated random variables; Let X i be the interarrival time between the ( i - 1) st visitor and the i th visitor to the museum. Let U j be the interarrival time between the ( j - 1) st visitor and the j th visitor to the museum, counting only those that eventually go to the gift shop during their visit.. Let Z i = 1 if the i th visitor to the museum goes to the gift shop sometime during his visit; otherwise Z i = 0. Let Y j be the dollar value of all purchases by the j k rmth visitor to arrive among those that go to the gift shop. discrete-time stochastic processes: For the random variables above, we get the as- sociated stochastic process (sequence of these random variables): X i : i 1 } , U j : j 1 } , Z i : i 1 } , Y j : j 1 } .

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lec0405 - IEOR 4106 Spring 2011 Professor Whitt...

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