1. Reliability Models & Applications Leadership in Engineering
Systems Engineering Program
Department of Engineering Management, Information and Systems
EMIS 7305/5305
Systems Reliability, Supportability and Availability Analysis
Reliability Models & Applications
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering

2. Reliability Model
Mathematical Model: a set of rules or equations which represent
the behavior of a physical system or process
Reliability Math Model: a mathematical relationship which
describes the reliability of an item in terms of its
components and their associated failure rates.
Objective: to represent the real world by expressing physical
or functional relationships in mathematical form
Approaches:
Analytical
Simulation

3. Comparison of Reliability Modeling Approaches
TYPE MODEL
Analytical
Monte Carlo
Advantages
Gives exact results (given the assumptions of the model).
Once the model is developed, output will generally be rapidly obtained.
It need not always be implemented on a computer; paper analyses may suffice.
Very flexible. There is virtually no limit to the analysis. Empirical distributions can be handled.
Can generally be easily extended and developed as required.
Easily understood by non-mathematicians.
Disadvantages
Generally requires restrictive assumptions to make the problem trackable.
Because of (A) it is less flexible than Monte Carlo. In particular, the scope for expending or developing a model may be limited.
Model might only be understood by mathematicians. This may cause credibility problems if output conflicts with preconceived ideas of designers or management.
Usually requires a computer; can require considerable computer time to achieve the required accuracy.
Calculations can take much longer than analytical models.
Solutions are not exact but depend on the number of repeated runs used to produce the output statistics. That is, all outputs are ‘estimates’.

4. Reliability Modeling Development Stages
System Analysis
Model output requirements
Input data requirements
Strike a balance between model complexity and accuracy
Model Development
Formulate the problem in mathematical terms
Specify assumptions and ground rules
Determine modeling approach; analytical or simulation
Model Validation
Use test cases to validate the mathematical formulation
Perform sensitivity analysis to confirm realism
Identify modeling constraints

5. Reliability Model Selection
Model selection is basic to reliability modeling and analysis
Selection criteria should be based on
Physical laws
Experience
Statistical goodness of fit
System configuration and complexity
Type of analysis

6. Reliability Models:
Time to Failure Models:
The random variable T is time to (or between) failure.
Number of Successes Model:
The random variable X is the number of successes that
occur in N trials.
Number of Failures Model:
The random variable Y is the number of failures that occur
in a period of time, t.

7. Statistical Models
Time to Failure Models:
The Exponential Model
The Weibull Model
The Normal (or Gaussian) Model
The Lognormal Model
Discrete Event Models:
The Binomial Model
The Poisson Model

8. f (t) = The Exponential Model: Definition
A random variable T is said to have the Exponential
Distribution with parameters , where  > 0, if the
failure density of T is:
, for t  0
, elsewhere
f (t) =

9. The Exponential Model:
Remarks
The Exponential Model is most often used in
Reliability applications, partly because of mathematical
convenience due to a constant failure rate.
The Exponential Model is often referred to as the
Constant Failure Rate Model.
The Exponential Model is used during the ‘Useful Life’
period of an item’s life, i.e., after the ‘Infant Mortality’
period before Wearout begins.
The Exponential Model is most often associated with
electronic equipment.

10. Properties of the Exponential Model:
Probability Distribution Function
Reliability Function
MTBF (Mean Time Between Failure)

11. Properties of the Exponential Model:
Standard Deviation of Time to Failure:
Failure Rate
Cumulative Failure Rate
P(T > t1 + t2 | T > t1) = P (T > t2), i.e.,
the Exponential Distribution is said to be without memory

12. The Exponential Model - Example
θ=1,000 Hours
Failure Density
f(t)
, , ,000

13. The Exponential Model – Example (continued)
θ=1,000 Hours
R(t)
Reliability Function
1.0
0.8
0.6
0.4
0.2
0.3679
0.1353
0.0498
, , ,000

14. f (t) = The Weibull Model: Definition
A random variable T is said to have the Weibull
Probability Distribution with parameters  and ,
where  > 0 and  > 0, if the failure density of T is:
, for t  0
, elsewhere
Remarks
 is the Shape Parameter
 is the Scale Parameter (Characteristic Life)
f (t) =

15. The Weibull Model - Distributions: Probability Density Function
f(t)
1.8
β=5.0
1.6
β=0.5
1.4
β=3.44
1.2
β=1.0
1.0
β=2.5
0.8
0.6
0.4
0.2 t
t is in multiples of 

16. Properties of The Weibull Model:
Probability Distribution Function
, for t  0
Where F(t) is the Fraction of Units Failing in Time t
Reliability Function

17. The Weibull Model - Distributions: Reliability Functions
R(t) t
t is in multiples of 
β=5.0
β=1.0
β=0.5
1.0
0.8
0.6
0.4
0.2

18. The Weibull Model - Weibull Probability Paper (WPP):
Derived from double logarithmic transformation of the
Weibull Distribution Function.
Of the form
where
Any straight line on WPP represents a Weibull Distribution with
Slope =  & Intercept = - ln 

19. The Weibull Model - Weibull Probability Paper (WPP):
Weibull Probability Paper links

20. Use of Weibull Probability Paper:

21. Properties of the Weibull Model:
p-th Percentile
and, in particular

22. Properties of the Weibull Model:
MTBF (Mean Time Between Failure)
Standard Deviation

23. The Gamma Function 
Values of the
Gamma Function

24. Properties of the Weibull Model:
Failure Rate
Notice that h(t) is a decreasing function of t if  < 1
a constant if  = 1
an increasing function of t if  > 1

25. Properties of the Weibull Model:
Cumulative Failure Rate
The Instantaneous and Cumulative Failure Rates, h(t)
and H(t), are straight lines on log-log paper.
The Weibull Model with  = 1 reduces to the Exponential
Model.

26. The Weibull Model - Distributions: Failure Rates
h(t) t
t is in multiples of  h(t) is in multiples of 1/  3 2 1
β=5
β=1
β=0.5

27. Properties of the Weibull Model:
Conditional Probability of Surviving Time t2, given
survival to time t1, where t1 < t2,
, if  > 1
Mode - The value of time (age) that maximizes the
failure density function.

28. The Weibull Model - Expected Time to First Failure:
n identical items are put on test or into service under
identical conditions and at age of zero time. The failure
distribution of each item is Weibull with parameter  and .
The expected time to first failure is:
The expected time to second failure is:

29. Turbine Spacer Life Expectancy
The Weibull Model
Turbine Spacer Life Expectancy
WEIBULL FIT - A
100 99 98 97 96 95
Based on 2 failures (Excludes 128 hr failure)
New Spacers New Engine Build - Undisturbed
Percent Surviving at 1 Flight Hours
Based on 3 failures
WEIBULL FIT - B
Age (Flight Hours)

30. The Weibull Model – Example 1
Time to failure of an item has a Weibull distribution
with characteristic life  = 1000 hours. Formulate the
reliability function, R(t), and the Failure Rate, h(t), as
a function of time (age) and plot, for:
(a)  = 0.5
(b)  = 1.0
(c)  = 1.5

31. The Weibull Model – Example 1 Solution

32. The Weibull Model – Example 1 Solution continued

33. The Weibull Model – Example 2
Time to failure of an item follows a Weibull
distribution with  = 2 and  = 1000 hours.
(a) What is the reliability, R(t), for t = 200 hours?
(b) What is the hazard rate, h(t), (instantaneous
failure rate) at that time?
(c) What is the Mean Time To Failure?

34. The Weibull Model – Example 2 Solution
failures per hour

35. The Weibull Model – Example 2 Solution continued
From the Gamma Function table: