Hw7_ORIE3510_slns

# Hw7_ORIE3510_slns - ORIE3510 Introduction to Engineering...

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ORIE3510 Introduction to Engineering Stochastic Processes Spring 2010 Homework 7: CTMCs and the Poisson Process Not to be handed in. 1. (a) Let S n be the time that the n th rider departs, then S n is the sum of n independent exponentials with rate λ and thus it has a gamma distribution with parameters n and λ . This is given by f S n ( t ) = λe - λt ( λt ) n - 1 ( n - 1)! for t 0. (b) Recall that if { N ( t ) ,t 0 } is a Poisson process with rate λ and points { S i ,i 0 } , then { ( S 1 ,S 2 ,...,S n ) | N ( t ) = n ) } is equal in distribution to { U (1) ,U (1) ,...,U ( n ) } . Therefore, each of these riders will still be walking at time t with probability p = R t 0 e - μ ( t - s ) ds t =. Hence the probability that none of the riders are walking at time t is (1 - p ) n - 1 . 2. X n = Number of customers in the system immediately before the n-th arrival Y n = Number of customers that remain the system when the n -th customer departs If X n = i then X n +1 ∈ { i + 1 - j ; j 0 } (j is the number of departures between the arrivals)

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Hw7_ORIE3510_slns - ORIE3510 Introduction to Engineering...

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