# Question

Consider a system consisting of five components, labeled 1, 2, 3, 4, 5. The system is able to function satisfactorily as long as at least one of the following three combinations of components has every component in that combination functioning satisfactorily:

(1) Components 1 and 4;

(2) Components 2 and 5;

(3) Components 2, 3, and 4.

For a given amount of time t, let Ri (t) be the known reliability of component i (i = 1, 2, 3, 4, 5), that is, the probability that this component will function satisfactorily for this length of time. Assume that the times until failure of the individual components are independently distributed. Let R(t) be the unknown reliability of the overall system.

(a) Draw a network representation of this system.

(b) Develop an explicit expression for the structure function of this system.

(c) Find R(t) as a function of the Ri(t).

(1) Components 1 and 4;

(2) Components 2 and 5;

(3) Components 2, 3, and 4.

For a given amount of time t, let Ri (t) be the known reliability of component i (i = 1, 2, 3, 4, 5), that is, the probability that this component will function satisfactorily for this length of time. Assume that the times until failure of the individual components are independently distributed. Let R(t) be the unknown reliability of the overall system.

(a) Draw a network representation of this system.

(b) Develop an explicit expression for the structure function of this system.

(c) Find R(t) as a function of the Ri(t).

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