Stochastic Modeling

Download Lévy Processes and Their Applications in Reliability and by Mohamed Abdel-Hameed PDF

By Mohamed Abdel-Hameed

This e-book offers a normal advent to the third-dimensional research and layout of constructions for resistance to the results of fireside and is meant for a normal readership, specifically people with an curiosity within the layout and development of constructions below critical loads.

A significant element of layout for hearth resistance comprises the development components or elements. The emphasis is put on constitution, which has a first-rate function in combating severe harm or structural cave in. a lot of the fabric during this publication examines development structures.The designed examples are in line with 3-dimensional finite elements.

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Extra resources for Lévy Processes and Their Applications in Reliability and Storage

Example text

In Sect. 6, inference about the parameters of some degradation processes and other related results, are discussed. 2 Basic Definitions and Results In this section, we give some basic reliability definitions, and discuss some wellknown results that will be used in the rest of this chapter. Detailed proofs of these results are found in the references indicated at the end on this chapter. Let X be a positive random variable describing the life time of_ a given device. For any x ≥ 0, the survival probability (reliability), by definition, F(x) = P{X > x}.

Hence ∧ R(x, [x, M], ∧) = Q n (x, [x, M], ∧) n=0 ∧ ⊂ F (n) (M) n=0 ∧ (F(M))n ⊂ n=0 1 1 − F(M) < ∧, = and our assertion is proven. 25 For any α, x ≤ R+ , ⎩ E x [e−αTM ,TM = φ] = ↓ Rα (x, dy)[ Q α (y, R+ ) − Q α (y, R+ )] [x,M] Proof For n ≥ 1, we let Gn be the sigma algebra generated by {(X k , Tk ), 1 ⊂ k ⊂ n}. 14 of [23], it follows that the unique solution of the last equation is E x [e−αTM ,TM = φ] = ⎩ ↓ Rα (x, dy)[ Q α (y, R+ ) − Q α (y, R+ )], [x,M] and our assertion is proven. 26 For any x ≤ R+ , ⎩ Px {TM = φ} = R(x, dy, ∧)[1 − Q(y, R+ , ∧)].

Ii) _Assume that the_degradation process is a nonhomogeneous subordinator. Then, F is Weibull if G is exponential and (t) = t ν , ν > 0. ↓ Proof (i)_ =∩ Assume that the degradation process X is a stationary subordinator, _ and G exponential. 3), it follows that H is_exponential. _ G exponential implies that H is exponential. Suppose ↑= Assume that _ _ G(x) = e−ζx , ζ, x ≥ 0. If H is exponential then, for each t, s ≥ 0, _ _ ↓ _ H (t + s) _= H (t) H (s). For t ≥ 0, A ≡ R+ , let Pt (A) = P( X t ≤ A).

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