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HW8_soln - Homework Set#8 Solutions IE 336 Spring 2011 1(a...

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Homework Set #8 Solutions IE 336 Spring 2011 1. (a) To estimate the mean time, let n ij be the number of transitions from i to j , and let T ij ( k ) be the k th observed value of a transition time from i to j : E ( T ij ) n ij k =1 T ij ( k ) n ij E ( T 12 ) 3 + 2 + 4 + 3 + 2 + 2 6 = 16 6 = 8 3 E ( T 21 ) 4 + 3 + 3 + 3 + 2 5 = 15 5 = 3 The estimated transition rates are then the reciprocal of the mean times: λ ij = 1 E ( T ij ) λ 12 = 3 8 and λ 21 = 1 3 Λ = - 3 / 8 3 / 8 1 / 3 - 1 / 3 (b) The system of differential equations is: d dt p 11 ( t ) = p 11 ( t ) λ 11 + p 12 ( t ) λ 21 = - 3 8 p 11 ( t ) + 1 3 p 12 ( t ) d dt p 12 ( t ) = p 11 ( t ) λ 12 + p 12 ( t ) λ 22 = 3 8 p 11 ( t ) - 1 3 p 12 ( t ) d dt p 21 ( t ) = p 21 ( t ) λ 11 + p 22 ( t ) λ 21 = - 3 8 p 21 ( t ) + 1 3 p 22 ( t ) d dt p 22 ( t ) = p 21 ( t ) λ 12 + p 22 ( t ) λ 21 = 3 8 p 21 ( t ) - 1 3 p 22 ( t ) (c) E ( T 21 ) = 3 (d) E ( T 12 ) = 8 / 3. A continuous time Markov process is memoryless. 2. (a) To determine p , q , and r , we solve the following system of equations simultaneously (recall that the sum of each row in a transition rate matrix must equal 0): p q r = 1 2 0 2 0 2 0 1 4 - 1 5 3 4 = 1 2 1 / 2 Λ = - 5 2 q p 2 r - 3 2 p 4 r q - 4 = - 5 4 1 1 - 3 2 2 2 - 4 1
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(b) The system of differential equations is: d dt p 11 ( t ) = - 5 p
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