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oldmidterm2sol

oldmidterm2sol - UCSD ECE153 Prof Young-Han Kim Handout#21...

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UCSD ECE153 Handout #21 Prof. Young-Han Kim Tuesday, May 4, 2010 Solutions to Midterm (Spring 2008) 1. First available teller (20 points). Consider a bank with two tellers. The service times for the tellers are independent exponentially distributed random variables X 1 Exp( λ 1 ) and X 2 Exp( λ 2 ), respectively. You arrive at the bank and find that both tellers are busy but that nobody else is waiting to be served. You are served by the first available teller once he/she becomes free. Let the random variable Y denote your waiting time. Find the pdf of Y . Solution: This problem is very similar to Question 4 in Homework Set #4. First observe that Y = min( X 1 , X 2 ). Since P { Y > y } = P { X 1 > y, X 2 > y } = P { X 1 > y } P { X 2 > y } = e - λ 1 y × e - λ 2 y = e - ( λ 1 + λ 2 ) y for y 0, Y is an exponential random variable with pdf f Y ( y ) = braceleftBigg ( λ 1 + λ 2 ) e - ( λ 1 + λ 2 ) y , y 0 , 0 , otherwise. 2. Sum of packet arrivals (40 points). Consider a network router with two types of incoming packets, wireline and wireless. Let the random variable N 1 ( t ) denote the number of wireline packets arriving during time (0 , t ] and let the random variable N

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