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hmwk8sol

# hmwk8sol - ISyE 3232 Stochastic Manufacturing and Service...

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ISyE 3232 Stochastic Manufacturing and Service Systems Spring 2011 H.Ayhan and J. Dai Solutions to Homework 8 1. (a) The state space is { \$10 , \$20 } . The transition matrix is 0 . 8 0 . 2 0 . 1 0 . 9 . It is irreducible. (b) The state space is { \$10 , \$25 } . The transition matrix is 0 . 9 0 . 1 0 . 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. (a) Yes, it is a Markov Chain. The state space is { 0 , 1 , 2 , 3 , · · · } . a = (1 , 0 , 0 , · · · ) , P i,j = 98 / 100 j = i + 1; 2 / 100 j = 0 . (b) Yes, it is irreducible. Starting from any state i , the probability of visiting state 0 in one step is 2 / 100. And starting from state 0, the probability of visiting state i in i step is (98 / 100) i . Therefore, state 0 communicates with any other state. Thus any state commutes with each other.

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hmwk8sol - ISyE 3232 Stochastic Manufacturing and Service...

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