Failure Rate Almost Certainly Increases with Time

Stats for Engineers Lecture 11 Acceptance Sampling Summary

Acceptable quality level: 푝1

(consumer happy, want to accept with high probability)

Unacceptable quality level: 푝2

(consumer unhappy, want to reject with high probability)

Producer’s Risk: reject a batch that has acceptable quality

훼 = 푃 Reject batch when 푝 = 푝1

Consumer’s Risk: accept a batch that has unacceptable quality

훽 = 푃 Accept batch when 푝 = 푝2

One stage plan: can use table to find number of samples and criterion Two stage plan: more complicated, but can require fewer samples

Operating characteristic curve 퐿 푝 : probability of accepting the batch Is acceptance sampling a good way of quality testing?

Problems:

It is too far downstream in the production process; better if you can identify where things are going wrong.

It is 0/1 (i.e. defective/OK) - not efficient use of data; large samples are required.

- better to have quality measurements on a continuous scale: earlier warning of deteriorating quality and less need for large sample sizes.

Doesn’t use any information about distribution of defective rates Reliability: Exponential Distribution revision

Which of the following has an exponential distribution?

1. The time until a new car’s engine 45% develops a leak 2. The number of punctures in a car’s

lifetime 27% 3. The working lifetime of a new hard disk drive 18%

4. 1 and 3 above 9% 5. None of the above 0%

1 2 3 4 5

Exponential distribution gives the time until or between random independent events that happen at constant rate. (2) is a discrete distribution. (1) and (3) are times to random events, but failure rate almost certainly increases with time. Reliability

Problem: want to know the time till failure of parts

E.g. - what is the mean time till failure?

- what is the probability that an item fails before a specified time?

If a product lasts for many years, how do you quickly get an idea of the failure time? accelerated life testing: Compressed-time testing: product is tested under usual conditions but more intensively than usual (e.g. a washing machine used almost continuously)

Advanced-stress testing: product is tested under harsher conditions than normal so that failure happens soon (e.g. refrigerator motor run at a higher speed than if operating within a fridge). - requires some assumptions How do you deal with items which are still working at the end of the test programme?

An example of censored data. – we don’t know all the failure times at the end of the test

Exponential data (failure rate 휈 independent of time)

Test components up to a time 푡0

- assuming a rate, can calculate probability of no failures in 푡0.

- calculate probability of getting any set of failure times (and non failures by 푡0)

- find maximum-likelihood estimator for the failure rate in terms of failure times

For failure times 푡푖, with 푡푖 = 푡0 for parts working at 푡0, and 푛푓 failures

푛푓 ⇒ Estimate of failure rate is 휈 = [see notes for derivation] 푖 푡푖 Example: 50 components are tested for two weeks. 20 of them fail in this time, with an average failure time of 1.2 weeks.

What is the mean time till failure assuming a constant failure rate?

푛푓 Answer: 휈 = 푖 푡푖 푛 = 50, 푛푓 = 20

푡푖 = 20 × 1.2 + 30 × 2 = 84 weeks 푖 푛푓 20 ⇒ 휈 = = = 0.238/week 푖 푡푖 84

1 1 ⇒ mean time till failure is estimated to be = = 4.2 weeks 휈 0.238 Reliability function and failure rate

For a pdf 푓(푥) for the time till failure, define:

Reliability function Probability of surviving at least till age 푡. i.e. that failure time is later than 푡

∞ 푅 푡 = 푃 푇 > 푡 = 푓 푥 푑푥 푡

= 1 − 퐹(푡)

푡 퐹 푡 = 0 푓 푡 푑푡 is the cumulative distribution function.

Failure rate 푓 푡 This is failure rate at time 푡 given that it survived until time 푡: 휙 푡 = 푅 푡 Example: Find the failure rate of the Exponential distribution

Answer:

∞ −휈푥 −휈푡 The reliability is 푅 푡 = 푡 휈푒 푑푥 = 푒

푓 푡 휈푒−휈푡 Failure rate, 휙 푡 = = = 휈 Note: 휈 is a constant 푅 푡 푒−휈푡

The fact that the failure rate is constant is a special “lack of ageing property” of the exponential distribution.

- But often failure rates actually increase with age. Reliability function

Which of the following could be a plot of a reliability function? (푅 푡 : probability of surviving at least till age 푡. i.e. that failure time is later than 푡)

1. 2. 42%

33%

25%

푡 푡 3. 4. 0%

1 2 3 4

푡 푡 If we measure the failure rate 휙 푡 , how do we find the pdf?

푑퐹 푓 푡 푑푡 푑 휙 푡 = = = − ln 1 − 퐹 푡 푅 푡 1 − 퐹 푡 푑푡

푡 퐹 0 = 0 ⇒ ln 1 − 퐹 푡 = − 휙 푡′ 푑푡′ 0

- can hence find 퐹(푡), and hence 푓 푡 , 푅(푡) Example

Say failure rate 휙(푡) measured to be a constant, 휙 푡 = 휈 푡 ⇒ ln 1 − 퐹 푡 = − 휈푑푡 = −휈푡 0 푑퐹 푡 ⇒ 1 − 퐹 푡 = 푒−휈푡 ⇒ 퐹 푡 = 1 − 푒−휈푡 ⇒ 푓 푡 = = 휈푒−휈푡 푑푡 - Exponential distribution The Weibull distribution

- a way to model failure rates that are not constant

Failure rate: 휙 푡 = 푚휈푡푚−1

Parameters 푚 (shape parameter) and 휈 (scale parameter)

푚 = 1: failure rate constant, Weibull=Exponential

푚 > 1: failure rate increases with time

푚 < 1: failure rate decreases with time Failure rate: 휙 푡 = 푚휈푡푚−1

푡 ⇒ ln 1 − 퐹 푡 = − 푚휈푥푚−1푑푥 = −휈푡푚 0 푚 ⇒ 퐹 푡 = 1 − 푒−휈푡

푚 푑퐹 푡 푚 Reliability: 푅 푡 = 푒−휈푡 Pdf: 푓 푡 = = 푚휈푡푚−1푒−휈푡 푑푡 The End!

[Note: as from 2011 questions are out of 25 not 20] Reliability: exam question 푓 푡 = 푘푡−4 (푡 > 1)

1. 1/4 43% 2. 3 3. 4 29% 4. -3 5. 1/3 14% 14%

0%

1 2 3 4 5 Answer

푏 푏 푥푛+1 ∞ ∞ 푥푛 = −4 푛 + 1 푓 푡 = 1 ⇒ 푘푡 푑푡 = 1 푎 푎 −∞ 1

∞ ∞ 푘푡−3 −4 푘 푘 푘푡 푑푡 = =0 − − = −3 3 3 1 1

⇒ 푘 = 3 Answer

Know 푘 = 3 ∞ ∞ ∞ 푘 −푘 −푘 3 푇 = 푡푓 푡 푑푡 = = = 0 − = 푡3 2푡2 −∞ 1 1 2 2 Answer

푓 푡 휙 푡 = 푅 푡 = 1 − 퐹(푡) 푅 푡

푡 푡 푡 푘 푘 푘 푘 1 퐹 푡 = 푓 푥 푑푥 = 푑푥 = = − − − = 1 − 푥4 −3푡3 3 3 0 1 1 3푡 3 푡

1 ⇒ 푅 푡 = 1 − 퐹 푡 = 푡3 푓 푡 푘 3 ⇒ 휙 푡 = = × 푡3 = (푡 > 1), otherwise 0 푅 푡 푡4 푡 Answer

휙 푡