By Ilya Gertsbakh, Yoseph Shpungin
This ebook is dedicated to the probabilistic description of a community within the technique of its destruction, i.e. removing of its elements (links, nodes) showing because of technical disasters, common failures or intentional assaults . it really is occupied with a pragmatic method of community reliability, in line with program of Monte Carlo technique to numerical review of community most crucial structural parameters. this enables to procure a probabilistic description of the community within the technique of its destruction, to spot most vital community parts and to increase effective heuristic algorithms for community optimum layout. The technique works with passable accuracy and potency for small –to-medium measurement networks.
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Then, using our second marginal D-spectrum we obtain that K −2 n PN (J ; p) = F (2) (x)q x p (n−x) x =1 J =1 n! (n − x)! 12) we obtain that n PN (K − 1; p) = x =1 F (1) (x) − F (2) (x) q x p (n−x) n! (n − x)! 2 Put F (K +1) (x) = 0. Then for J = K − 1, . . , 0, n PN (J ; p) = x =1 F (K −J ) (x) − F (K −J +1) (x) q x p (n−x) n! (n − x)! g. [17–19], may realize that numerically the signature s = (s1 , s2 , . . , f n ) introduced in Sect. 2. F. Samaniego , p. 21 gives the following definition: Assume that the lifetimes of coherent systems n components are independent and identically distributed according to the (continuous) distribution G.
N + m. , z = 1, . . , n + m. 3) Very similar is the situation with two networks in series connection. We will consider this situation on an example, see Fig. 6a. 1 (Two networks in series) Both networks have three components, see Fig. 6a. e. C 2 (3) = 1. Let Q 1 and Q 2 be the DOWN probabilities of the first and second network, respectively. Then the DOWN probability for the series connection of these networks is Q N = 1 − (1 − Q 1 )(1 − Q 2 ) = 1 − 1 − qp 2 − 3q 2 p − q 3 (1 − q 3 ). Now it remains to bring the expression for Q N to the (qp)-polynomial form.
For π = (2, 1, 3) we have r1 = r2 = 1, r3 = 3; for π = (1, 3, 2) we obtain r1 = 1, r2 = r3 = 2. Finally, π = (1, 2, 3) is the only permutation where the anchors are separated: r1 = 1, r2 = 2, r3 = 3. It is easy to obtain now the cumulative marginal spectra: F (1) (1) = 1; F (2) (1) = 2/3, F (2) (2) = 1; F (3) (1) = 0, F (3) (2) = 2/3, F (3) (3) = 1. Here F (1) (x) is the probability that after x failures, the number of clusters is ≥ 2, F (2) (x) is the probability that after x failures, the number of clusters is ≥ 3, F (3) (x) is the probability that after x failures, the number of clusters is 4.
Network Reliability and Resilience by Ilya Gertsbakh, Yoseph Shpungin