Reliability Engineering

Ashraf Khalil

Billington and Allen, “Reliability Evaluation of Engineering Systems”, 1983.

David J Smith, “Reliability, Maintainability and Risk”, Butterworth-Heinemann, 1997.

Walpole, Raymond, Sharon and Keying, “Probability and Statistics for Engineers and Scientists”, Prentice-Hall, 2002.

References

Reliability engineering is an engineering field that deals with the study, evaluation, and life-cycle management of reliability

 The ability of a system or component to perform its required functions under stated conditions for a specified period of time.

Reliability is theoretically defined as the probability of failure, the frequency of failures, or in terms of availability

What is Reliability?

The reliability is the probability of success, Mean Time Between Failure (MTBF).

The prevention of system loose function. OR Reliability is the probability of a device

performing its purpose adequately for the period of time intended under the operating conditions encountered." Billington and Allen (1983).

What is Reliability?

Used to estimate the reliabilities of individual devices, such as electronic components,

and the reliabilities of systems constructed of components.

Mathematical – based on probability theory.

Reliability Theory

Maintainability: The ability of an item, under stated conditions of use, to be retained in, or restored to, a state in which it can perform its required function(s), when maintenance is performed under stated conditions and using prescribed procedures and resources. Expressed as Mean Time To Repair (MTTR).

Definitions

Availability: Is the probability that a system is available for use at a given time- a function of reliability and maintainability.

A=Up time/(Up time+ Down time) =MTBF/(MTBF+MDT). Failure: The termination of the ability of an

item to perform its required function. A fault: is "An accidental condition that

causes a functional unit to fail to perform its required function"

Definitions

expected number of failures in a given time period

average time between failures average down time expected revenue loss due to failure expected loss of output due to failure

Reliability Indicators

MTBF = MTTF + MTTR Where MTTF is mean time to failure MTTR is mean time to repair

Mean Time Between Failures

The probability that an item will fail in the interval from 0 to time t is F(t), the reliability is then:

R(t)=1-F(t)

Basic Theory

Assumes that items in the series are independent

All items must work for the system to work

Reliability of a Series System

R1 R3R2

The probability of failure in the interval t to t+dt which is

λ(t) dt where λ(t) is the failure rate

The variation of failure rate of electrical or electronic components.

The Bathtub Distribution

Early Failure

Useful

Life

Wearout

Failure

Failure Rate

The set of all possible outcomes of a statistical experiment is called the Sample Space and is represented by the symbol S.

Example 1 The possible outcomes when a coin is tossed? S = {H, T }

Example 2 The experiment of tossing a die? S={1,2,3,4,5,6}

Probability

Example 3: Tree Diagram Tossing a coin then a die! S = {HH, HT, T1, T2, T3, T4, T5, T6 }

Example 4: Suppose that three items are selected at

random from a manufacturing process. Each item is inspected and classified (D=Defective or N=Nondefective).

S={DDD,DDN,DND,DNN,NDD,NDN,NND,NNN}

Event: An event is a subset of a sample space. Example: The outcome is dividable by 3 (toss a die) A= {3, 6}

Example: The number of defective parts is more than

1. B = {DDN, DND, NDD, DDD}

The complement of an event A with respect to S is the subset of all elements of S that are not in A. We denote the complement of A by the symbol A’.

A={DDD, DDN, DND, DNN} A’={NDD, NDN, NND, NNN}

The intersection of two events A and B, denoted by the symbol A⋂B, is the event containing all elements that are common to A and B.

Example M={a,e,i,o,u} N={r,s,t} M ⋂ N= ∅

Two events are called mutually exclusive if :

A⋂B = ∅

The union of the two events A and B, denoted by the symbol A∪ B, is the event containing all the elements that belong to A or B or Both.

Example A={a,b,c} B={b,c,d,e} A∪ B={a,b,c,d,e}

Example M={x|3<x<9} and N={y|5<y<12}, then M∪N = {z|3<z<12}

A∩B = regions a and 2. B ∩C= regions 1 and 3 A∪C=regions 1,2,3,4,5, and 7, B’∩A= regions 4 and 7, A∩B∩C= region 1, (A∪B) ∩ C’=regions 2,6, and 7.

If A and B are any two events, then P(A∪B)=P(A)+P(B)-P(A∩B)

If A and B are mutually exclusive: P(A∪B)=P(A)+P(B) Why P(A∩B)=0

Additive Rules

Problem 1:The probability that a man will be alive

in 10 years is 0.8 and the probability that his wife will be alive in 10 years is 0.9. Find the probability that in 10 years

A- Both will be alive.B- Only the man will be alive.C- Only the wife will be alive.D- At least one will be alive.

)(72.08.09.0)().()(

)(9.0)().(8.0)(

eventstindependenBPAPBAP

alivebewillwifehisBPalivebewillmanaAP

08.072.08.0

)()()(

BAPAPalivebewillmantheOnlyP

Solution:

18.072.09.0

)()()(

BAPBPalivebewillwifetheOnlyP

A- Both will be alive

B- Only the man will be alive:

C- Only the wife will be alive:

08.072.08.0

)()()(

BAPAPalivebewillmantheOnlyP

D- At least one will be alive:

Two dice are tossed together. Let A be the event that the sum of the faces are odd, B the event that at least one is a one. What is the probability that:(i) Both A and B occur?(ii) Either A or B or both occur?(iii) A and not B occur?(iv) B and not A occur?

Problem 5:

A = The sum is odd B = at least one face is a one.

36;6)(65)4)(63)(62)(61)(6(6

6)5)(54)(53)(52)(51)(5(5

6)5)(44)(43)(42)(41)(4(4

6)5)(34)(33)(32)(31)(3(3

6)5)(24)(23)(22)(21)(2(2

6)(15)(14)(13)(12)(11)(1S

111)(61)1)(51)(41)(3(2

6)(15)(14)(13)(12)(11)(1B

18;5)(63)1)(6(6

6)4)(52)(5(5

5)3)(41)(4(4

6)4)(32)(3(3

5)3)(21)(2(2

6)(14)(12)(1A

Both A and B occur:

1)(61)1)(46)(24)(12)(1(1BA

23

6B)(AP

Either A or B or both occur:

36

23B)(AP

2361118BA

I

II

A and not B occurs:

36

12)BP(A12;618BA

B and not A occurs:

36

5)BP(A5;AB 611

IV

III

P(B|A) = the probability that B occurs given that A occurs.

Conditional Probaility