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Unformatted text preview: n = 0 , 1 , 2 , ... Solution 1. This is a birthdeath process with λ n = λ and μ n = μ . Thus, equation in the notes gives P = 1 1 + P ∞ n =1 λ ··· λ n − 1 μ 1 ··· μ n = 1 1 + P ∞ n =1 λ n μ n = 1 1 + λ /μ 1 − λ μ = 1 − λ μ 1 − λ μ + λ /μ = 1 − λ μ , 1 and P n = λ λ 1 · · · λ n − 1 μ 1 · · · μ n P = λ n μ n P = μ λ μ ¶ n μ 1 − λ μ ¶ . Solution 2: This is a birthdeath process with λ n = λ and μ n = μ . Thus, the balance equations are λ P = μP 1 λ P 1 = μP 2 · · · λ P n = μP n +1 · · · Thus, P 1 = λ μ P , P 2 = λ μ P 1 = μ λ μ ¶ 2 P , ..., P n = μ λ μ ¶ n P , ... Therefore, P = (1 + λ μ + · · · + μ λ μ ¶ n + · · · ) − 1 = Ã 1 1 − λ μ ! − 1 = 1 − λ μ , P n = μ λ μ ¶ n P = μ λ μ ¶ n μ 1 − λ μ ¶ . 2...
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This document was uploaded on 10/18/2011.
 Spring '09

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