1. Reliability Engineering - Part 1

2. Product Probability Law of Series Components
If a system comprises a large number of components, the system reliability may be rather low, even though the individual components have high reliabilities, e.g. V-1 missile in WW II.

3. Transition of Component States
Normal
state continues
Component fails
Failed
state
continues N F
Component is repaired

4. The Repair-to-Failure Process

5. Definitions of Reliability
The probability that an item will adequately perform its specified purpose for a specified period of time under specified environmental conditions.
The ability of an item to perform an required function, under given environmental and operational conditions and for a stated period of time. (ISO 8402)

6. Definition of Quality
The totality of features and characteristics of a product or service that bear on its ability to satisfy or implies needs (ISO 8402).
Quality denotes the conformity of the product to its specification as manufactured, while reliability denotes its ability to continue to comply with its specification over its useful life. Reliability is therefore an extension of quality into the time domain.

7. REPAIR -TO-FAILURE PROCESS MORTALITY DATA
t=age in years ; L(t) =number of living at age t
t L(t) t L(t) t L(t) t L(t)
0 1,023, , , ,221
1 1,000, , , ,577
, , , ,011
, , ,
, , ,548
, , ,982
, , ,765
After Bompas-Smith. J.H. Mechanical Survival : The Use of Reliability
Data, McGraw-Hill Book Company, New York , 1971.

8. repair= birth failure = death Meaning of R(t):
HUMAN RELIABILITY
t L(t), Number Living at
Age in Years Age t R(t)=L(t)/N F(t)=1-R(t)
repair= birth
failure = death
Meaning of R(t):
(1) Prob. Of Survival (0.86) of an individual of an individual to age t (40)
(2) Proportion of a population that is expected to Survive to a given age t.
,023,
,000, , , , , , , , , , , , , , , , , , , , , ,

9. 1.0 P
0.9
Survival distribution
0.8
0.7
0.6
0.5
Probability of Survival R(t) and Death F(t)
0.4
0.3
Failure distribution
0.2
0.1

10. Time to Failure
The time elapsing from when the unit is put into operation until it fails for the first time, i.e. a random variable T. It may also be measured by indirect time concepts:
The number of times a switch is operated
The number of kilometers driven by a car
The number of rotations of a bearing
etc.

11. State Variable
The state of the unit at time t can be described by the state variable

12. Reliability, R(t) Unreliability, F(t)
= probability of survival to (inclusive) age t
= the number of surviving at t divided by the total sample
Unreliability, F(t)
= probability of death to age t (t is not included)
=the total number of death before age t divided by the total population

13. Reliability - R(t)
The probability that the component experiences no failure during the the time interval (0,t] or, equivalently, the probability that the unit survives the time interval (0, t] and is still functioning at time t.
Example: exponential distribution

14. Unreliability - F(t)
The probability that the component experiences the first failure during (0,t].
Example: exponential distribution

15. FALURE DENSITY FUNCTION f(t)
No. of Failures
(death)
Age in Years
000012 1 2 3 4 5 10 15 20 25 30 35 40 45 50 60 65 70 75 80 85 90 95 99
100
23,102
5,770
4,116
3,347
2,950
12,013
9,543
10,787
12,286
14,588
18,055
23,212
30,788
41,654
56,709
99,889
123,334
138,566
134,217
103,554
56,634
18,566
2,886
125 -

16. 140
120
100
Number of Deaths (thousands) 80 60 40 20
Age in Years (t) 20 40 60 80
100

17. 0.14
0.12
0.10
Failure Density f (t)
0.8
0.6
0.4
0.2
Age in Years (t) 20 40 60 80
100

18. Failure Density - f(t)
(exponential distribution)

19. CALCULATION OF FAILURE RATE r(t)
Age in Years
No. of Failures
(death)
Age in Years
No. of Failures
(death)
r(t)=
r(t)= 1 2 3 4 5 10 15 20 25 30 35
23,102
5,770
4,116
3,347
2,950
12,013
9,534
10,787
12,286
14,588
18,055
23,212
0.0512 40 45 50 55 60 65 70 75 80 85 90 95 99
30,788
41,654
56,709
76,420
99,889
123,334
138,566
134,217
103,554
56,634
18,566
2,886
125

20. Random failures
Early failures
Wearout failures
0.2
0.15
Failure Rate r(t)
0.1
0.05 20 40 60 80
100
Failure rate r(t) versus t.

21. Period of Approximately Constant failure rate
(faults/time)
Period of Approximately Constant failure rate
Infant Mortality
Old Age
Time
Figure A typical “bathtub” failure rate curve for process hardware. The failure rate is approximately constant over the mid-life of the component.

22. Comments on Bathtub Curve
Often units are tested before they are distributed to the users. Thus, much of the infant mortality will be removed before the units are delivered for use.
For the majority of mechanical units, the failure rate will usually show a slightly increasing tendency in the useful life period.

23. Failure Rate - r(t)
The probability that the component fails per unit time at time t, given that the component has survived to time t.
Example:
The component with a constant failure rate is considered
as good as new, if it is functioning.

24. As Good As New?
This implies that the probability that a unit will be functioning at time t+x, given that it is functioning at time t, is equal to the probability that a new unit has a time to failure longer than x. Hence the remaining life of a unit, functioning at time t, is independent of t. The exponential distribution has no “memory.”

25. Relation Between Reliability and Failure Rate

26. Interpretation of Failure Rate
In actuarial statistics, the failure rate is called the force of mortality (FOM). The failure rate or FOM is a function of the life distribution of a single unit and an indication of the “proneness of failure” of the unit after time t has elapsed.

27. Failure-Rate Experiment
Split the time interval (0, t) into disjoint intervals of equal length dt. Then put n identical units into operation at time t=0. When a unit fails, record the time and leave that unit out. For each interval, determine
The number of units n(i) that fail in interval i.
The functioning times of the individual units in interval i. If a unit has failed before interval i, its functioning time is zero.

28. Failure-Rate Experiment

29. Mean Time to Failure - MTTF
Variance of Time to Failure

30. Failure Rate
Failure Density
Unreliability
Reliability 1 1
Area = 1
1 - F (t)
f (t)
F (t)
R (t) t
(a) t
(b) t
(c) t
(d)
Figure Typical plots of (a) the failure rate (b) the failure density f (t), (c) the unreliability F(t), and (d) the reliability R (t).

31. TABLE 11-1: FAILURE RATE DATA FOR VARIOUS SELECTED PROCESS COMPONENTS1
Instrument Fault/year
Controller
Control valve
Flow measurement (fluids)
Flow measurement (solids)
Flow switch
Gas - liquid chromatograph
Hand valve
Indicator lamp
Level measurement (liquids)
Level measurement (solids)
Oxygen analyzer
pH meter
Pressure measurement
Pressure relief valve
Pressure switch
Solenoid valve
Stepper motor
Strip chart recorder
Thermocouple temperature measurement
Thermometer temperature measurement
Valve positioner
1Selected from Frank P. Lees, Loss Prevention in the Process Industries (London: Butterworths, 1986), p. 343.

32. Example
Consider two independent components with failure ratesλ1andλ2, respectively. Determine the probability that component 1 fails before component 2.
Similarly,

33. A System with n Components in Parallel
Unreliability
Reliability

34. A System with n Components in Series
Reliability
Unreliability

35. Upper Bound of Unreliability for Systems with n Components in Series

36. The Poisson Process

37. Assumptions of Homogeneous Poisson Process (HPP)
Suppose we are studying the occurrence of a certain event A in the course of a given time period. Let us assume
A can occur at any time in the interval. The probability of A occurring in the interval is independent of t and may be written as where is a positive constant.
The probability of more than one event A in this interval is , which is a function with property
Let (t11,t12], (t21,t22],…be any sequence of disjoint intervals in the time period in question. Then the events “A occurs in (tj1,tj2],” j= 1, 2, …, are independent.
The process is said to have intensityλ

38. Probability of No Event Occurring in (0, t]
Let N(t) denotes the number of times the event A occurs during the period (0, t]. Let

39. Exponential Distribution
Let T1 denotes the time point when A occurs for the first time. T1 is a random variable and
Thus, thee waiting time T between consecutive occurrences in a
HPP is exponentially distributed.

40. Probability of n Events Occurring in (0, t]

41. Poisson Distribution Since
This distribution is called the Poisson distribution with parameter and random variable n
Notice that, since the expected number of occurrences of event A per unit time (t=1) is , expresses the intensity of the process.

42. Example 1
Suppose that exactly one event (failure) of a HPP with intensityλis known to have occurred in the interval (0, t0]. Determine the distribution of the time T1 at which this event occurred.

43. Example 1
In other words, the time at which the first failure occurs in
uniformly distributed over (0,t0]. The expected time is thus

44. Example 2
Suppose that the failure of a system are occurring in accordance with a HPP. Some failures develop into a consequence C, and others do not. The probability of this development is p and is assumed to be constant for each failure. The failure consequences are further assumed to be independent of each other. Determine the distribution of the consequences.

45. Example 2 Thus, M(t) is also a HPP with intensity (pλ) .
The mean number of C consequences in (0,t] is (pλt).

46. Gamma Distribution
Consider a unit that is exposed to a series of shocks which occur a HPP with intensity λ. The time intervals T1, T2, T3, …, between consecutive shocks are then independent and exponentially distributed with parameter λ. Assume that the unit fails exactly at the kth shock, and not earlier. The time to failure of the unit
is then gamma distributed (k, λ).

47. Gamma Distribution Consider a homogeneous Poisson distribution,
The waiting time until the kth occurrence of event A in a
HPP with intensity λis gamma distributed. The gamma
distribution (1, λ) is an exponential distribution with
parameter λ.

48. Weibull Distribution
For majority of mechanical units, the failure rates are slightly increasing in the useful life period (not constant). A distribution often used when r(t) is monotonic is the Weibull distribution. The time to failure T of a unit is said to be Weibull distributed with scale parameterλand shape parameterα.

49. Weibull Distribution

50. Weibull Distribution
α=1: The Weibull distribution reduces to exponential distribution.
α>1: The failure rate is increasing.
α<1: The failure rate is decreasing.

51. Characteristic Lifetime.
Weibull Distribution
Hence, the probability of time to failure larger than 1/λ
is independent of α. This quantity is called the
Characteristic Lifetime.

52. Three-Parameter Weibull Distribution

53. Normal Distribution
The normal distribution is sometimes used as a lifetime distribution, even though it allows negative TTF values with positive probability.
The probability density of standard normal distribution is

54. Normal Distribution
The unreliability, reliability and failure rate can be written as

55. Normal Distribution, Left Truncated at 0

56. Log Normal Distribution
The time to failure T of a unit is said to be lognormally distributed if Y=ln(T) is normally distributed.
The lognormal distribution is commonly used as a distribution for repair time. When modeling the repair time, it is natural to assume that the repair rate in increasing in a first phase. When the repair has been going on for a rather long time, this indicates serious problems. It is thus natural to believe that the repair rate is decreasing after a certain period of time.

57. Log Normal Distribution