By Mehdi Farshad
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Extra resources for Stability of Structures (Developments in Civil Engineering)
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.
Stability of Structures (Developments in Civil Engineering) by Mehdi Farshad