Reliability Measures

Question 1

The time until a component fails is a lifetime T. The probability for lifetime T to be equal to t is given by a PDF f(t). What should this PDF satisfy?

Answer 1

f(t) >= 0, t>=0 and int[0 to inf]f(t)dt = 1

Question 2

How is the cumulative distribution function F(t) defined And what does it represents?

Answer 2

F(t) = Prob{T <= t}

probability that the component fails at some point between 0

and t:

Question 3

How is the reliability function R(t) defined And what does it represents?

Answer 3

R(t) = Prob{T >= t} = 1 - F(t) It represents the probability that a component survives at least until T.

Question 4

What is the Mean time to failure and how is it defined?

Answer 4

The MTTF is the expected value of the life time T. It represents the average run time of a component until it fails. MTTF = E[T] = int[0 to inf]R(t)dt

Question 5

What is the failure rate and how is it defined?

Answer 5

The failure rate gives the rate of components that are expected to die at time t. It is defined as : f(t)/(1 - F(t))

Question 6

Explain the phases of the bathtub curve

Answer 6

Early life - infant mortality due to manufacturing tests. Constant failure period Wear-out phase - failure rate grows due to aging affects

Question 7

Definition of availability

Answer 7

A(t) = MTTF/(MTTF + MTTR) = MTTF/MTBF Average fraction of time over the interval [0, t] that a component is up.

Question 8

Definition of MTBF

Answer 8

Mean time between failures : MTBF = MTTF + MTTR

Question 9

For systems that need high availability, it is desirable to design the system such that some tasks can be executed quickly. Which are these tasks? They are the components of a time to repair a system MTTR.

Answer 9

Fault detection, diagnosis, replacement and validation.

Question 10

If we assume a constant failure rate, then the reliability R(t) has an exponential distribution. Why is that so?

Answer 10

I understand.

Question 11

Why can the Weibull distribution be used to represent the lifetime of a component in the early phase and the wear-out phase.

Answer 11

The Weibull has two parameters: lamda and beta. failure rate is defined as: l.b.t^(b-1).

If b < 1 fail rate decreases

b=1 failure rate constant

b > 1 failure rate increases.

Question 12

The reliability of a component i in a series system is given by: Ri(t).

The reliability of the system is?

Answer 12

Rs(t) = prod[i = 0 to N]Ri(t)

Question 13

A parallel system is operational if at least one of its components is operational.

The reliability of a parallel system Rp(t) is given by?

Each component has reliability Ri(t)

Answer 13

Rp(t) = 1 - prod[i = 1 to N] (1 - Ri(t))

Question 14

How is the reliability of complex structures calculated?

Answer 14

Reliability is calculated by expandign the system about a single module i: Rsys = Ri . Prob{System works | i is faulty free} + (1-Ri) . Prob{System Works | i is faulty}