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Unformatted text preview: ISyE 3232 Stochastic Manufacturing and Service Systems Fall 2011 H. Ayhan Solutions to Homework 8 1. (a) The state space is { $10 , $20 } . The transition matrix is . 8 0 . 2 . 1 0 . 9 . It is irreducible. (b) The state space is { $10 , $25 } . The transition matrix is . 9 . 1 . 15 0 . 85 . It is irreducible. (c) The stationary distribution is ( π X $10 ,π X $20 ) = (1 / 3 , 2 / 3). (d) The stationary distribution is ( π Y $10 ,π Y $25 ) = (3 / 5 , 2 / 5). (e) What you need look at is E ( ∑ 300 i =1 X i ) and E ( ∑ 300 i =1 Y i ). And choose the one with larger expectation. By the stationary distribution obtained in (b) and (c), we have E ( 300 X i =1 X i ) = 300(10 × 1 3 + 20 × 2 3 ) = 5000 , E ( 300 X i =1 Y i ) = 300(10 × 3 5 + 25 × 2 5 ) = 4800 , so you should pick stock 1. 2. First you should solve a few small examples, for instance: . 25 0 . 75 . 75 0 . 25 and . 2 0 . 4 0 . 4 . 3 0 . 3 0 . 4 . 5 0 . 3 0 . 2 In the first case, the stationary distribution is { . 5 , . 5 } , and, in the second case, the sta...
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 Fall '07
 Billings
 Markov chain, stationary distribution, lim pij, πj Pji

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