2004Midterm1sln

2004Midterm1sln - E Y 1 Y 2 = 1 2 (b Let Y be the time it...

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IEOR 161 Operations Research II University of California, Berkeley Spring 2004 Midterm 1 Question 3 (a) (i) Let N(t) be the number of female customers when the first male customer arrives at time t. E [ N ( t )] = λt E [number of customers before first male customer] = Z t =0 ( λt ) μe - λt = λ μ (ii) Let M be the Poisson process for men arrivals, and W be the Poisson process for women arrivals. Pr { M (1) = 0 ,W (1) = 0 } = Pr { M (1) = 0 } Pr { W (1) = 0 } = e - λ · 1 e - μ · 1 = e - ( λ + μ ) (iii) Let S 2 be the 2nd male customer arrival time. E [ S 2 ] = 1 +
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Unformatted text preview: E [ Y 1 + Y 2 ] = 1 + 2 λ (b) Let Y be the time it takes for the first man to arrive and X be that for the first woman to arrive. Y ∼ exp ( λ ), X ∼ exp ( μ ) Pr { Y < t | Y < X } = Pr { Y < t,Y < X } Pr { Y < X } = Pr { Y < X < t } + Pr { Y < t,X > t } Pr { Y < X } = R t x =0 Pr { Y < x } μe-μx dx + (1-e-λt ) e-μt λ/ ( μ + λ ) = R t x =0 (1-e-λx ) μe-μx dx + (1-e-λt ) e-μt λ/ ( μ + λ ) 1...
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This note was uploaded on 09/07/2009 for the course IEOR 160 taught by Professor Hochbaum during the Spring '07 term at Berkeley.

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