Download PDF by Mehdi Farshad: Stability of Structures (Developments in Civil Engineering)

By Mehdi Farshad

ISBN-10: 0444816984

ISBN-13: 9780444816986

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0106  . 0106. 2 Some Key Information Provided by First Passage Define fi j = ∑∞ n=1 f i j as the probability of the system ever passing from state i to state j. It is clear that (n) • fi j = 1 implies that state j is reachable from state i. • fi j = f ji = 1 implies that states i and j do communicate (n) • fii is the probability that the first recurrence time for state i is n. Letting fii = (n) ∑∞ n=1 f ii then • fii = 1 for all recurrent states • fii < 1 for all transient states. Sometimes we are interested mainly in the first moment of the passage times.

Also, given that the buffer is now full, what is the probability that the next time that the buffer is full again is at the fifth transition from now? I let the reader figure this out as an excercise. What we need to calculate is F (4) and select its (2, 4) element. 0144  . 0106  . 0106. 2 Some Key Information Provided by First Passage Define fi j = ∑∞ n=1 f i j as the probability of the system ever passing from state i to state j. It is clear that (n) • fi j = 1 implies that state j is reachable from state i.

We shall discuss the case of infinite space Markov chains later. e. , the matrix P(n) is the nth order transition matrix of the (n) system. In fact, let pi j be the (i, j)th element of P(n) then we have (2) pi j = (3) pi j = ∑ piv pv j ∑ piv pv j = ∑ piv v∈I v∈I (2) ∑ v∈I (2) piv pv j .. =. 4 Discrete Time Markov Chains 21 Since P(2) = P2 from the first of above equation, by a simple induction, we will know P(n) = Pn holds for any non-negative integer n. This is the well known Chapman-Kolmogorov equation.

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Stability of Structures (Developments in Civil Engineering) by Mehdi Farshad


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