Summary: 1CV40
1
,Table of Contents
An introduction to Reliability and Maintainability Engineering ............................................................. 5
Chapter 1................................................................................................................................................. 5
Probability primer ............................................................................................................................... 5
Binomial distribution....................................................................................................................... 6
Poisson distribution ........................................................................................................................ 6
Chapter 2................................................................................................................................................. 7
2.1 The reliability function .................................................................................................................. 7
2.2 Mean time to failure ..................................................................................................................... 7
2.3 Hazard rate function ..................................................................................................................... 8
2.4 Bathtub curve................................................................................................................................ 8
2.5 Conditional reliability .................................................................................................................... 9
Table of integrals ................................................................................................................................ 9
Chapter 3............................................................................................................................................... 10
3.1 The exponential reliability function ............................................................................................ 10
3.4 The two-parameter exponential function .................................................................................. 11
3.5 Poisson process ........................................................................................................................... 11
3.6 Redundancy and the CFR model ................................................................................................. 12
Chapter 4............................................................................................................................................... 12
4.1 The Weibull distribution ............................................................................................................. 12
Identical Weibull Components:..................................................................................................... 13
The three-parameter Weibull: ...................................................................................................... 13
Redundancy with Weibull failures: ............................................................................................... 13
4.2 The Normal distribution .............................................................................................................. 14
Central limit theorem:................................................................................................................... 14
4.3 The Lognormal distribution......................................................................................................... 14
4.4 The Gamma distribution ............................................................................................................. 15
Chapter 5............................................................................................................................................... 15
5.1 Serial configuration ..................................................................................................................... 15
5.2 Parallel configuration .................................................................................................................. 16
5.3 Combined series-parallel systems .............................................................................................. 16
Decomposition .............................................................................................................................. 17
Enumeration ................................................................................................................................. 17
Chapter 6............................................................................................................................................... 17
6.1 Markov analysis........................................................................................................................... 17
6.2 Load-sharing system ................................................................................................................... 19
2
, 6.3 Standby systems ......................................................................................................................... 19
Chapter 7............................................................................................................................................... 21
7.2 Static models ............................................................................................................................... 21
Random stress and constant strength .......................................................................................... 21
Constant stress and random strength .......................................................................................... 22
Random stress and random strength ........................................................................................... 22
Exponential case ........................................................................................................................... 22
Normal case .................................................................................................................................. 23
Lognormal case ............................................................................................................................. 23
Mixed distributions with analytical solutions ............................................................................... 23
7.3 Dynamic models .......................................................................................................................... 24
Periodic loads ................................................................................................................................ 24
Random loads ............................................................................................................................... 24
Random fixed stress and strength ................................................................................................ 25
Chapter 9............................................................................................................................................... 25
Exponential repair times ............................................................................................................... 26
Lognormal repair times ................................................................................................................. 26
9.6 State-dependent systems with repair......................................................................................... 26
Standby system with repair .......................................................................................................... 27
Chapter 11............................................................................................................................................. 28
11.3 System availability..................................................................................................................... 28
Availability with standby systems ................................................................................................. 29
Steady-state availability ................................................................................................................ 29
Matrix approach............................................................................................................................ 30
Chapter 12............................................................................................................................................. 30
12.1 Data collection .......................................................................................................................... 30
A taxonomy of data....................................................................................................................... 30
12.2 Empirical methods .................................................................................................................... 31
Ungrouped complete data ............................................................................................................ 32
Grouped complete data ................................................................................................................ 33
Ungrouped censored data ............................................................................................................ 34
Grouped censored data ................................................................................................................ 35
Chapter 15............................................................................................................................................. 36
15.1 Identifying candidate distributions ........................................................................................... 36
15.2 Probability plots and least-square curve-fitting ....................................................................... 37
Exponential plots .......................................................................................................................... 37
3
, Weibull plots ................................................................................................................................. 38
Normal plots.................................................................................................................................. 39
Lognormal plots ............................................................................................................................ 39
Multiple censored time plots ........................................................................................................ 39
Applied statistics and probability for Engineers ................................................................................... 39
Chapter 15............................................................................................................................................. 39
15.1 Quality improvement and statistics .......................................................................................... 39
Statistical quality control .............................................................................................................. 40
Statistical process control ............................................................................................................. 40
15.2 Introduction to control charts .................................................................................................. 40
Basic principles .............................................................................................................................. 40
Design of a control chart ............................................................................................................... 41
Rational subgroups ....................................................................................................................... 42
Analysis of patterns on control charts .......................................................................................... 42
15.5 Process capability...................................................................................................................... 42
15.6 Attribute control charts ............................................................................................................ 43
P chart (control chart for proportions) ......................................................................................... 43
U chart (control chart for defects per unit) .................................................................................. 44
15.7 Control chart performance ....................................................................................................... 44
15.11 Implementing SPC .................................................................................................................. 45
4
, An introduction to Reliability and Maintainability Engineering
Chapter 1
Reliability and maintainability engineering attempts to study, characterize, measure and analyze the
failure and repair of systems to improve their operational use by increasing their design life,
eliminating or reducing the likelihood of failures and safety risk and reducing down time, and
thereby increasing available operating time. The primary reason for reliability and maintainability
engineering is to improve the reliability and availability of the product or system being developed
and thereby add to its value. Reliability and maintainability should be practiced throughout the
entire product life cycle.
Reliability is defined to be the probability that a component or system will perform a required
function for a given period of time when used under stated operating conditions. It is the probability
of a nonfailure over time.
Failures should be defined relative to the function being performed by the system. Next, the unit of
time (or usage) should be identified. The system should be observed under normal performance.
Maintainability is defined to be the probability that a failed component or system will be restored or
repaired to a specified condition within a period of time when maintenance is performed in
accordance with prescribed procedures.
Usually maintainability refers to the inherent repair time, which only includes hands-on repair of the
failed unit and not any administrative or resource delay times.
Availability is defined as the probability that a component or system is performing its required
function at a given point in time when used under stated operating conditions. It might also be
interpreted as the percentage of time a component or system is operating over a specified time
interval.
The availability is the probability that the component is currently in a nonfailure state even though it
might have been restored. Therefore, system availability can never be less than system reliability.
Reliability is closely associated with the quality of a product. Quality can be defined qualitatively as
the amount by which the product satisfies the users’ requirements. Quality assurance is a planned
set of processes and procedures necessary to achieve high product quality.
Probability primer
A random variable is a variable that takes on certain values in accordance with specified
probabilities. A random event E will occur with some probability P(E) where 0≤P(E)≤1. P(E) = 0
describes an impossible event, P(E)=1 denotes a certain event. The collection of all possible
outcomes relative to a random process is called the sample space S = {𝐸𝐸1 , 𝐸𝐸2 , … , 𝐸𝐸𝑘𝑘 }. Every event E is
associated with a complementary event 𝐸𝐸 𝐶𝐶 .
The intersection of two events, A and B, is denoted with A∩B. Then the union of two event, A and B,
is denoted with A ∪ B. If A and B are mutually exclusive:
P(A ∪ B) = P(A) + P(B) P(A ∩ B) = 0
If two events A and B are independent:
P(A∩B) = P(A)P(B)
5
1
,Table of Contents
An introduction to Reliability and Maintainability Engineering ............................................................. 5
Chapter 1................................................................................................................................................. 5
Probability primer ............................................................................................................................... 5
Binomial distribution....................................................................................................................... 6
Poisson distribution ........................................................................................................................ 6
Chapter 2................................................................................................................................................. 7
2.1 The reliability function .................................................................................................................. 7
2.2 Mean time to failure ..................................................................................................................... 7
2.3 Hazard rate function ..................................................................................................................... 8
2.4 Bathtub curve................................................................................................................................ 8
2.5 Conditional reliability .................................................................................................................... 9
Table of integrals ................................................................................................................................ 9
Chapter 3............................................................................................................................................... 10
3.1 The exponential reliability function ............................................................................................ 10
3.4 The two-parameter exponential function .................................................................................. 11
3.5 Poisson process ........................................................................................................................... 11
3.6 Redundancy and the CFR model ................................................................................................. 12
Chapter 4............................................................................................................................................... 12
4.1 The Weibull distribution ............................................................................................................. 12
Identical Weibull Components:..................................................................................................... 13
The three-parameter Weibull: ...................................................................................................... 13
Redundancy with Weibull failures: ............................................................................................... 13
4.2 The Normal distribution .............................................................................................................. 14
Central limit theorem:................................................................................................................... 14
4.3 The Lognormal distribution......................................................................................................... 14
4.4 The Gamma distribution ............................................................................................................. 15
Chapter 5............................................................................................................................................... 15
5.1 Serial configuration ..................................................................................................................... 15
5.2 Parallel configuration .................................................................................................................. 16
5.3 Combined series-parallel systems .............................................................................................. 16
Decomposition .............................................................................................................................. 17
Enumeration ................................................................................................................................. 17
Chapter 6............................................................................................................................................... 17
6.1 Markov analysis........................................................................................................................... 17
6.2 Load-sharing system ................................................................................................................... 19
2
, 6.3 Standby systems ......................................................................................................................... 19
Chapter 7............................................................................................................................................... 21
7.2 Static models ............................................................................................................................... 21
Random stress and constant strength .......................................................................................... 21
Constant stress and random strength .......................................................................................... 22
Random stress and random strength ........................................................................................... 22
Exponential case ........................................................................................................................... 22
Normal case .................................................................................................................................. 23
Lognormal case ............................................................................................................................. 23
Mixed distributions with analytical solutions ............................................................................... 23
7.3 Dynamic models .......................................................................................................................... 24
Periodic loads ................................................................................................................................ 24
Random loads ............................................................................................................................... 24
Random fixed stress and strength ................................................................................................ 25
Chapter 9............................................................................................................................................... 25
Exponential repair times ............................................................................................................... 26
Lognormal repair times ................................................................................................................. 26
9.6 State-dependent systems with repair......................................................................................... 26
Standby system with repair .......................................................................................................... 27
Chapter 11............................................................................................................................................. 28
11.3 System availability..................................................................................................................... 28
Availability with standby systems ................................................................................................. 29
Steady-state availability ................................................................................................................ 29
Matrix approach............................................................................................................................ 30
Chapter 12............................................................................................................................................. 30
12.1 Data collection .......................................................................................................................... 30
A taxonomy of data....................................................................................................................... 30
12.2 Empirical methods .................................................................................................................... 31
Ungrouped complete data ............................................................................................................ 32
Grouped complete data ................................................................................................................ 33
Ungrouped censored data ............................................................................................................ 34
Grouped censored data ................................................................................................................ 35
Chapter 15............................................................................................................................................. 36
15.1 Identifying candidate distributions ........................................................................................... 36
15.2 Probability plots and least-square curve-fitting ....................................................................... 37
Exponential plots .......................................................................................................................... 37
3
, Weibull plots ................................................................................................................................. 38
Normal plots.................................................................................................................................. 39
Lognormal plots ............................................................................................................................ 39
Multiple censored time plots ........................................................................................................ 39
Applied statistics and probability for Engineers ................................................................................... 39
Chapter 15............................................................................................................................................. 39
15.1 Quality improvement and statistics .......................................................................................... 39
Statistical quality control .............................................................................................................. 40
Statistical process control ............................................................................................................. 40
15.2 Introduction to control charts .................................................................................................. 40
Basic principles .............................................................................................................................. 40
Design of a control chart ............................................................................................................... 41
Rational subgroups ....................................................................................................................... 42
Analysis of patterns on control charts .......................................................................................... 42
15.5 Process capability...................................................................................................................... 42
15.6 Attribute control charts ............................................................................................................ 43
P chart (control chart for proportions) ......................................................................................... 43
U chart (control chart for defects per unit) .................................................................................. 44
15.7 Control chart performance ....................................................................................................... 44
15.11 Implementing SPC .................................................................................................................. 45
4
, An introduction to Reliability and Maintainability Engineering
Chapter 1
Reliability and maintainability engineering attempts to study, characterize, measure and analyze the
failure and repair of systems to improve their operational use by increasing their design life,
eliminating or reducing the likelihood of failures and safety risk and reducing down time, and
thereby increasing available operating time. The primary reason for reliability and maintainability
engineering is to improve the reliability and availability of the product or system being developed
and thereby add to its value. Reliability and maintainability should be practiced throughout the
entire product life cycle.
Reliability is defined to be the probability that a component or system will perform a required
function for a given period of time when used under stated operating conditions. It is the probability
of a nonfailure over time.
Failures should be defined relative to the function being performed by the system. Next, the unit of
time (or usage) should be identified. The system should be observed under normal performance.
Maintainability is defined to be the probability that a failed component or system will be restored or
repaired to a specified condition within a period of time when maintenance is performed in
accordance with prescribed procedures.
Usually maintainability refers to the inherent repair time, which only includes hands-on repair of the
failed unit and not any administrative or resource delay times.
Availability is defined as the probability that a component or system is performing its required
function at a given point in time when used under stated operating conditions. It might also be
interpreted as the percentage of time a component or system is operating over a specified time
interval.
The availability is the probability that the component is currently in a nonfailure state even though it
might have been restored. Therefore, system availability can never be less than system reliability.
Reliability is closely associated with the quality of a product. Quality can be defined qualitatively as
the amount by which the product satisfies the users’ requirements. Quality assurance is a planned
set of processes and procedures necessary to achieve high product quality.
Probability primer
A random variable is a variable that takes on certain values in accordance with specified
probabilities. A random event E will occur with some probability P(E) where 0≤P(E)≤1. P(E) = 0
describes an impossible event, P(E)=1 denotes a certain event. The collection of all possible
outcomes relative to a random process is called the sample space S = {𝐸𝐸1 , 𝐸𝐸2 , … , 𝐸𝐸𝑘𝑘 }. Every event E is
associated with a complementary event 𝐸𝐸 𝐶𝐶 .
The intersection of two events, A and B, is denoted with A∩B. Then the union of two event, A and B,
is denoted with A ∪ B. If A and B are mutually exclusive:
P(A ∪ B) = P(A) + P(B) P(A ∩ B) = 0
If two events A and B are independent:
P(A∩B) = P(A)P(B)
5
