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Unformatted text preview: Homework Set #6 Solutions IE 336 Spring 2011 1. (a) p = 1 / 4, q = 1 / 8, r = 1 / 4. Transition diagram omitted. P = 2 q r 2 p 3 r p 2 q 6 q 2 p 3 r 3 = 1 / 4 1 / 4 1 / 2 3 / 4 1 / 8 1 / 8 3 / 4 1 / 6 1 / 12 (b) Given p (4) = 0 . 2 0 . 6 0 . 2 , the vector at time step 8 is: p (8) = p (4) P 4 = . 4813 0 . 2034 0 . 3153 (c) The steadystate probabilities are uniquely determined by the dependent set of linear equations ( P T I ) π T = 0 and the additional normalizing equation π 1 = 1: π 1 π 2 π 3 =  3 / 4 3 / 4 3 / 4 1 / 4 7 / 8 1 / 6 1 1 1  1 1 = 1 / 2 1 / 5 3 / 10 2. (a) Mean sojourn time in state 2. E ( N 2 ) = 1 1 p 22 = 1 1 1 / 8 = 8 7 = 1 . 1429 (b) If the process begins in state 3, the probability that it will be in state 2 for the first time four steps later is f (4) 32 : f (1) 32 = p 32 = 1 6 f (2) 32 = p (2) 32 f (1) 32 p (1) 22 = 2 9 1 6 · 1 8 = 29 144 f (3) 32 = p (3) 32...
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This note was uploaded on 01/19/2012 for the course IE 230 taught by Professor Xangi during the Spring '08 term at Purdue University.
 Spring '08
 Xangi

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