6. Reliability engineers often work with systems having components connected in

Question

6. Reliability engineers often work with systems having components connected in parallel. In this problem, we will interpret the phrase “in parallel” as follows: The system is reliable (i.e., it is functioning) if at least one of the components is functioning. As a frame of reference, consider a two-engine aircraft.

If both engines are functioning, the aircraft is functioning (at least in lieu of non- engine related problems).

If only one engine is functioning, the aircraft still functions (although losing one engine may warrant an immediate landing).

• If both engines are not functioning, the aircraft is not functioning.
In this problem, we will denote by n the number of components in a parallel system.

(a) Suppose n = 2. If the two components are functioning independently, each with probability p, show that the system reliability r2 is given by

r2 =1(1p)2.
Hint: Let A1 = {component 1 is functioning} and A2 = {component 2 is functioning}.

Then r2 = P(A1 A2).

(b) Generalize the result in part (a) to consider a parallel system with n components (each functioning independently with probability p). That is, show that the system reliability is

rn =1(1p)n.
Hint: rn = P(A1 A2 ···An) = 1P(A1 A2 ···An). Now use the fact that

the components are independent.

(c) I have a parallel system with n = 5 components (each functioning independently with probability p). How unreliable can the individual components be and still have a system with reliability r5 = 0.9999?

(d) So far in this problem, we have made two critical assumptions: (A1) the components function independently

(A2) the components each function with the same probability p.

Give a real-life example where Assumption (A1) is likely violated. Give a real-life example where Assumption (A2) is likely violated. Do not use the plane example I used at the outset. Explain your examples sufficiently so I can understand.

Explanation / Answer

(a)

System reliability r2 = Probability that atleast one component is functioning out of 2 components

= 1 - Probability that no component is functioning

= 1 - (1-p) * (1-p)

= 1 - (1-p)2

(b)

System reliability rn = Probability that atleast one component is functioning out of n components

= 1 - Probability that none of 'n' component is functioning

= 1 - (1-p) * (1-p) * (1-p) * ..... n times

= 1 - (1-p)n

(c)

For parallel system with n = 5 components,

Reliability , r5 = 1 - (1-p)5

=> 0.9999 = 1 - (1-p)5

=> (1-p)5 = 1 - 0.9999

=> (1-p)5 = 0.0001

=> (1-p) = 0.1585

So, the individual components can be unreliable with probability 0.1585

d)

Real-life example where Assumption (A1) is likely violated. - Two electric wiring system in the house taking connections share the load of the electrical equipments in the house. When one wiring system fails, the full load comes on the second working wiring system which makes it more susceptible to failure. So, the function of one wiring system depends on whether the other wiring system is working or not.

Real-life example where Assumption (A2) is likely violated. - If in earlier exampple, we assume that one wiring system is of high quality and other is of low quality, Assumption (A2) is likely violated. That is the probability of both wiring system working are not same.

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