416-9 - is served Frst by server 1, then by server 2, and...

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Denker Spring 2010 416 Stochastic Modeling - Assignment 9 Due Date: Monday, March 29, 2010 Problem 1: (Problem 12, p. 347) If X i , i = 1 , 2 , 3, are independent exponential random variables with rates λ i , respectively, Fnd 1. P ( X 1 < X 2 < X 3 ), 2. P ( X 1 < X 2 | max( X 1 ,X 2 ,X 3 ) = X 3 ), 3. E [max X i | X 1 < X 2 < X 3 ], 4. E [max X i ]. Problem 2: (Problem 15, p. 347) One hundred items are simultaneously put into a life test. Suppose the lifetimes of the individual items are independent and exponentially distributed with mean 200 hours. The test will end if there have been a total of 5 failures. If T denotes the time when the test ends, Fnd the expectation and variance of T . Problem 3: (Problem 10, p. 347) Let X and Y be independent exponential random variables with rates λ and μ respectively.Let M = min( X,Y ) denote their minimum. ±ind: 1. E [ MX | M = X ], 2. E [ MX | M = Y ], 3. Cov ( X,M ) Problem 4: (Problem 20, p. 348) Consider a two-server system in which a customer
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Unformatted text preview: is served Frst by server 1, then by server 2, and then departs. The service times are independent and exponential random variables with rates i . When you arrive, you Fnd server 1 free and two customers at server 2, one (say customer A) being serviced the other (say customer B) waiting in line. 1. ind the probability that A is still in service when you join the line at server 2. 2. ind the probability that B is still in service when you join the line at server 2. 3. ind E [ T ] where T is the time you spend in the system. Problem 5: (Problem 18, p. 348) Let X 1 and X 2 be independent random variables which are exponentially distribu-ted with rates i , i = 1 , 2, respectively. ind the expectation and the variance of their minimum and their maximum....
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