By Mohamed Abdel-Hameed
ISBN-10: 3642400744
ISBN-13: 9783642400742
This booklet covers Lévy methods and their functions within the contexts of reliability and garage. distinctive cognizance is paid to lifestyles distributions and the upkeep of units topic to degradation; estimating the parameters of the degradation technique is usually mentioned, as is the upkeep of dams topic to Lévy enter.
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Extra info for Lévy Processes and Their Applications in Reliability and Storage
Example text
Life distributions of coherent systems of independent components having respective exponential life times is IFRA. DFR random variables include: mixtures of exponential random variables, the Weibull random variable with shape parameter α ⊂ 1, and the gamma random variable with shape parameter α ⊂ 1. _ A positive random variable, with survival function F (t) = exp(−a[bt]), a, b ≥ 0 is NBU, where [x] is the greatest integer greater than or equal to x. Dependence of Random variables. Components of systems exhibit some degree of dependence between their performances, indicated by their life times.
19 For x, t ≤ R+ , A ≤ ν([x, ∧)), ⎩ _ ↓ Q(x, A, t) = Q(x, dy, t) A Proof For n ≥ 1, let Then, G(y) _ G(x) ↓ n ↓ be the sigma algebra generated by {( X k , Tk ), 0 ⊂ k ⊂ n}. Q(x, A, t) = P{X 1 ≤ A, T1 ⊂ t | X 0 = x} ↓ ↓ ↓ ↓ ↓ = P{ X 1 ≤ A, T1 ⊂ t, L > 1 | X 0 = x, L > 0} ↓ ↓ ↓ = P{ X 1 ≤ A, T1 ⊂ t, X 1 < Y | X 0 = x, X 0 < Y } ↓ _ ↓ ↓ ↓ = [G(x)]−1 P{ X 1 ≤ A, T1 ⊂ t, X 1 < Y | X 0 = x} _ I ↓ I ↓ ] = [G(x)]−1 E x [I ↓ { X 1 0 It follows that ∧ ↓ R α (x, A) = ↓ Q nα (x, A), n=0 where ↓ Q nα (x, ⎩∧ A) = ↓ e−αt Q n (x, A, dt). }. 44 2 Degradation Processes Let λ be the life time of the device, that is ↓ λ = inf{t ≥ 0 : Y t ≥ Y }, and ↓ ↓ T M = inf {t ≥ 0 : Y t ≥ M}. Let TM be the random variable denoting the maintenance time (the minimum of the corrective maintenance and preventative maintenance times), that is ↓ TM = T M ↓ λ . 1 −αW (z) ). α E(1 − e For z, α ≤ R+ , we define h α (z) = ↓ ↓α For α ≤ R+ , we let U be the α-potential of the process Y .
Lévy Processes and Their Applications in Reliability and Storage by Mohamed Abdel-Hameed
by Joseph
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