08SimHw8sol - IEOR 4404 Simulation Prof Mariana...

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IEOR 4404 Assignment #8 Solutions Simulation November 20, 2008 Prof. Mariana Olvera-Cravioto Page 1 Assignment #8 Solutions 1. Assuming { Y i , i = 1 , 2 , . . . } are defined as Y i = ( - ρ ) i where ρ (0 , 1), then the steady-state is lim n →∞ Y n = 0 and the steady-state mean is E ( Y n ) = ν = 0 Notice that | ¯ Y ( n ) | ≤ ρ n n < 1 n If we regard that ¯ Y ( m ) does not change appreciably when n N , no matter how larger N is, we can assign ρ close enough to 1, for instance, ρ = ( N - 1 N ) 1 N so thus | Y N | = ρ N = N - 1 N , still needs more warmup before the stabilization lim n →∞ Y n = 0 2. (a) By the definition of regeneration points, we need to find a sequence of random indices 1 < B 1 < B 2 < ... called regeneration points, at which the process starts over, i.e., { Y B j + i : i = 0 , 1 , 2 , ... } = { Y B 1 + i : i = 0 , 1 , 2 , ... } in distribution for all j . So, let B j be the index of the j th time when it is the beginning of a month with the inventory level S . (b) There are no regeneration times if the interarrivals are not exponential. 3. Notice that e U 2 (1 + e 1 - 2 U ) 2 = e U 2 + e 1 - 2 U e U 2 2 = e U 2 + e 1 - 2 U + U 2 2 = e U 2 + e (1 - U ) 2 2
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IEOR 4404, Assignment #8 Solutions 2 Since 1 - U 2 is also
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