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HW9_soln - Homework Set#9 Solutions IE 336 Spring 2011 1(a...

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Homework Set #9 Solutions IE 336 Spring 2011 1. (a) Determine the steady-state probabilities. Since each row in a transition probability matrix must equal 1: p 11 ( t ) = 1 - p 12 ( t ) = 1 - 7 11 ( 1 - e - 11 t ) = 4 11 + 7 11 e - 11 t Using the definition of steady-state probabilities: π j = lim t →∞ p ij ( t ) π 1 = lim t →∞ p 11 ( t ) = lim t →∞ 4 11 + 7 11 e - 11 t = 4 11 π 2 = lim t →∞ p 12 ( t ) = lim t →∞ 7 11 ( 1 - e - 11 t ) = 7 11 (b) Determine the transition rate matrix. The transition diagram is shown in Figure 1. λ ij = d dt p ij ( t ) | t =0 λ 11 = d dt p 11 ( t ) | t =0 = d dt 4 11 + 7 11 e - 11 t t =0 = - 7 λ 12 = d dt p 12 ( t ) | t =0 = d dt 7 11 ( 1 - e - 11 t ) t =0 = 7 Using the relationship 0 = π Λ , we can determine the remaining transition rates: 0 = π Λ = 4 11 7 11 λ 11 λ 12 λ 21 λ 22 λ 21 = - 4 7 λ 11 = 4 λ 22 = - 4 7 λ 12 = - 4 Therefore, the transition rate matrix is: Λ = - 7 7 4 - 4 (c) P ( τ 1 > 5) = e λ 11 t = e - 7 · 5 = e - 35 (d) P ( τ 2 7) = 1 - P ( τ 2 > 7) = 1 - e λ 22 t = 1 - e - 4 · 7 = 1 - e - 28 1
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Figure 1: Transition Diagram for Problem 1b 2. (a) Since the sum of each row in a transition rate matrix must equal 0, p = 2, q = 0, r = - 2, s = - 5, t = 0. Therefore, the transition rate matrix is: Λ = r + 2 0 0 3 q 0 0 - 7 - r 3 p 0 0 q 0 0 2 2 2 p -
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