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Consider a birth and death process with states {0,1}. Let the birth and death rates be Ao a > 0, A₁ = 0, μo = 0, and μ₁ => 0. (a) Show that the infinitesimal generator matrix A = Thus, for n = 1,2,3,..., A" = (-a - b)-1A holds. -a a b satisfies the property A² = (a + b)A. (b) Using the result from part (a) to compute the matrix exponential etA in closed form and thus find the transition probability matrix P(t) = e¹A, t≥ 0. (c) Compute lim∞ P(t) and use it to find the stationary distribution.

Consider a birth and death process with states {0,1}. Let the birth and death rates be Ao a > 0, A₁ = 0, μo = 0, and μ₁ => 0. (a) Show that the infinitesimal generator matrix A = Thus, for n = 1,2,3,..., A" = (-a - b)-1A holds. -a a b satisfies the property A² = (a + b)A. (b) Using the result from part (a) to compute the matrix exponential etA in closed form and thus find the transition probability matrix P(t) = e¹A, t≥ 0. (c) Compute lim∞ P(t) and use it to find the stationary distribution.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Consider a birth and death process with states {0,1}. Let the birth and death rates be Ao = a > 0, A₁ = 0, μo = 0, and µ₁ = b > 0. = ko = A1 (a) Show that the infinitesimal generator matrix A = [ a a satisfies the property A² = - 2 -(a + b)A. b Thus, for n = 1, 2, 3,..., A = (a - b)-1A holds. n (b) Using the result from part (a) to compute the matrix exponential et in closed form and thus find the transition probability matrix P(t) = e¹A, t≥ 0. (c) Compute lim →∞ P(t) and use it to find the stationary distribution.
Transcribed Image Text:Consider a birth and death process with states {0,1}. Let the birth and death rates be Ao = a > 0, A₁ = 0, μo = 0, and µ₁ = b > 0. = ko = A1 (a) Show that the infinitesimal generator matrix A = [ a a satisfies the property A² = - 2 -(a + b)A. b Thus, for n = 1, 2, 3,..., A = (a - b)-1A holds. n (b) Using the result from part (a) to compute the matrix exponential et in closed form and thus find the transition probability matrix P(t) = e¹A, t≥ 0. (c) Compute lim →∞ P(t) and use it to find the stationary distribution.
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