A minimal path set is defined to be the minimal set of components whose function

Question

A minimal path set is defined to be the minimal set of components whose functioning ensures the function of the system. According to the system of components shown above, define the sample space for the minimal path set. a. Total reliability is defined to be the reliability of the first half of the system in serial with the second half of the system. Given that the reliability R(t) of components is defined as follows: R(1) = .95, R(2) = .95, R(3) = .99, R(4) = .99 and R(5) = 90, find the total reliability of the system. That is, find the probability that the system does not fail. Assume that all components work independently of one another. b. Find the probability that the system does not fail if it is known that component 2 has failed. c.

Explanation / Answer

a. sample space for minimal path set = (1,3,4), (1,5) , (2,3,4) , (2,5)

b. Total reliability of the system is R can be calculated by taking series sum of two reliability system. Let these two system has reliability RA and RB , where

RA = R1 ll R2 = 1- ( 1 - R1 ) ( 1- R2 )= 1 - ( 1- 0.95) * (1 -0.95) = 0.9975

RB = R3R4 ll R5

R3R4 = 0.99 * 0.99 = 0.9801

R3R4 ll R5 = 1 - ( 1- 0.9801) * (1 - 0.90) = 0.998

so

R = RA RB = 0.998 * 0.9975 = 0.9955

c. here we have known that compnent 2 has failed so we have to find the probability that system does not fail.

so if system 2 has failed than system 1 must be working

so the reliability of new system = R1 * RB = 0.95 * 0.998 = 0.948

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