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# lec0310 - IEOR 4106 Introduction to Operations Research...

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IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2011, Professor Whitt, March 10 1. Money Withdrawn from an ATM Machine Customers arrive at an automated teller machine (ATM) at the times of a Poisson process with a rate of λ = 10 per hour. Suppose that the amount of money withdrawn on each transaction has a mean of \$30 and a standard deviation of \$20. (a) Find the mean and variance of the total amount of dollars withdrawn in 8 hours. (b) What is the approximate probability that the total amount of money withdrawn in the first 8 hours exceeds \$3 , 400? (c) How do the answers change if the Poisson arrival process is a nonhomogeneous Poisson process witharrival rate function λ ( t ) = 4 t , t 0? ———————————————————————- (a) Assuming that successive withdrawals are IID, this is a compound Poisson process ; see Section 5.4.2. Let X ( t ) be the total amount withdrawn in the time interval [0 , t ]. Let N ( t ) be the number of customers to come to the ATM in the interval [0 , t ]. Let Y n be the amount of the n th withdrawal. Then X ( t ) can be represented as the following random sum of random variables X ( t ) = N ( t ) X i =1 Y i . Hence, from § 5.4 (p. 346 in the last edition), E [ X ( t )] = λtE [ Y 1 ] and var ( X ( t )) = λtE [ Y 2 1 ] , (1) so that E [ X (8)] = 10 × 8 × 30 = 2400 and var ( X ( t )) = 10 × 8 × ((30 2 + (20) 2 ) = 104 , 000 . The standard deviation is 104 , 000 322 . 49. But why are those the correct formulas? To see why, look at Examples 3.10 and 3.17 in Chapter 3. We discuss the harder variance formula. Let Y = N X i =1 X i , where N is a nonnegative-integer-valued random variable and X i are IID random variables. (We are using new notation here.) Then, by the conditional variance formula in Proposition 3.1, V ar ( Y ) = E [ V ar ( Y | N )] + V ar ( E [ Y | N ]) = E [ N ] V ar ( X 1 ) + E [ X 1 ] 2 V ar ( N ) .

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lec0310 - IEOR 4106 Introduction to Operations Research...

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