Hw5-ORIE3510_S12_solutions

Hw5-ORIE3510_S12_solutions - ORIE 3510 – Homework 5...

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Unformatted text preview: ORIE 3510 – Homework 5 Solutions Instructor: Mark E. Lewis due 2PM, Wednesday February 29, 2012 (ORIE Hallway drop box) 1. The chain is irreducible. Since p (4 n +6 m ) 1 , 1 > 0, n,m ∈ N , the period of state 1 is d (1) = 2. By solidarity property, d (8) = 2. 2. Counterexample. Consider a Markov chain { X n ,n ≥ } with transition matrix given by P = parenleftbigg 0 1 1 0 parenrightbigg . The period of this Markov chain is 2. But { X 2 n ,n ≥ } is not irreducible since P 2 n = parenleftbigg 1 0 0 1 parenrightbigg . 3. (a) Solving the steady state equations ( π 1 ,π 2 ,π 3 ) = ( π 1 ,π 2 ,π 3 ) 1 / 3 2 / 3 1 / 3 2 / 3 1 , together with π 1 + π 2 + π 3 = 1, we obtain ( π 1 ,π 2 ,π 3 ) = ( 9 20 , 3 20 , 8 20 ) . Hence the required proportion is given by π 3 = 8 / 20. (b) the long run earning = 10 9 20 + 30 3 20 + 50 8 20 = \$29 . 4. If suffices to check that summationdisplay k ∈ X summationdisplay l ∈ X π ∗ ( k,l ) q ( k,l ) , ( i,j ) = π ∗ ( i,j...
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Hw5-ORIE3510_S12_solutions - ORIE 3510 – Homework 5...

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