On computing signatures of coherent systems

Component importance in coherent systems with exchangeable components

“…From Theorem 2, the signature of the overall series system is obtained as s = 0, 4 45 , 19 90 , 3 10 , 86 315 , 34 315 , 2 105 , 0, 0, 0, which coincides with the result in Da et al (2012). Substituting (20) and (21) into (4), we obtain s −i = 0, 1 9 , 2 9 , 25 84 , 65 252 , 2 21 , 1 63 , 0, 0, 0 for i = 1, 2, 4, 5, 6, 7, 9, 10 and s −i = 0, 0, 2 9 , 47 126 , 43 126 , 2 63 , 2 63 , 0, 0, 0 for i = 3, 8.…”

Two-Dimensional Signatures

“…, n 2 )) P(E(l, r, n 1 , n 2 )) +1,l+1)P(N ξ 1 = n 1 , N ξ 2 = n 2 ) P(M 1 = l, M 2 = r) = n 1 <n 2 P(N ξ 1 = n 1 , N ξ 2 = n 2 )readily follows, by taking into account Lemma 2 and relation(4).…”

Computing system signatures through reliability functions

“…By (12) we then obtain h R φ (x) = x 7 x −3 (2x − 1) x −4 (2x 2 − 1) = 1 − 2x − 2x 2 + 4x 3 , from which we derive the system signature s = 1 7 , 8 21 , 38 105 , 4 35 , 0, 0, 0 . As an immediate consequence of our analysis we retrieve the fact (already observed in [8]; see [4,5] for earlier references) that the signature always decomposes through modular partitions. 4 We state this property as follows.…”

“…As an immediate consequence of our analysis we retrieve the fact (already observed in [8]; see [4,5] for earlier references) that the signature always decomposes through modular partitions. 4 We state this property as follows.…”