10SimHw5Solution

# 10SimHw5Solution - IEOR E4404.001 Assignment #5 Solutions...

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Unformatted text preview: IEOR E4404.001 Assignment #5 Solutions SIMULATION February 19, 2010 Prof. Mariana Olvera-Cravioto Assignment #5 Solutions 1. • P ( N = n ) = P ( S n ≤ 1 ,S n +1 > 1) = integraldisplay ∞ P ( S N ≤ 1 ,S n +1 > 1 | S n = y ) Gamma ( n,λ ) dy = integraldisplay 1 P ( X n +1 > 1- y ) Gamma ( n,λ ) dy = integraldisplay 1 e- λ (1- y ) y n- 1 e- λy λ n Γ( n ) dy = λ n e- λ Γ( n ) integraldisplay 1 y n- 1 dy = λ n e- λ Γ( n ) 2. In order to simulate the service time, we are trying to use Acceptance-Rejection algorithm. Consider g 1 ( x ) = cλe- λx , subject to g ( x ) ≤ g 1 ( x ) for any x > i.e., 20 e- 40 x (40 x ) 2 ≤ cλe- λx for any x , which implies 20 λ e ( λ- 40) x (40 x ) 2 ≤ c for any x . The first-order-condition of the left side implies that x = 2 40- λ achieves the maximum. So, c ≥ 20 λe 2 ( 80 40- λ ) 2 . If we set λ = 20 ,c = 16 /e 2 , then, h ( x ) = g 1 ( x ) /c = 20 e- 20 x . Thus, we can apply the Acceptance-Rejection algorithm to generate the service time w.r.tapply the Acceptance-Rejection algorithm to generate the service time w....
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## This note was uploaded on 11/17/2010 for the course IEOR IEOR 4404 taught by Professor C during the Spring '10 term at Columbia.

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10SimHw5Solution - IEOR E4404.001 Assignment #5 Solutions...

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