1. An Application of Probability to
Systems Engineering Program
Department of Engineering Management, Information and Systems
EMIS 7370/5370 STAT 5340 :
PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS
An Application of Probability to
Reliability Modeling and Analysis
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering

2. What is Reliability? Reliability is performance over time, probability
Reliability is defined as the probability that an item will perform its intended function for a specified interval under stated conditions. In the simplest sense, reliability means how long an item (such as a machine) will perform its intended function without a breakdown.
Reliability: the capability to operate as intended, whenever used, for as long as needed.
Reliability is performance over time, probability
that something will work when you want it to.

3. Reliability Figures of Merit
Basic or Logistic Reliability
MTBF - Mean Time Between Failures
measure of product support requirements
Mission Reliability
Ps or R(t) - Probability of mission success
measure of product effectiveness

4. “If I had only one day left to live,
Reliability Humor: Statistics
“If I had only one day left to live,
I would live it in my statistics class --
it would seem so much longer.”
From: Statistics A Fresh Approach
Donald H. Sanders
McGraw Hill, 4th Edition, 1990

5. The Reliability Function
The Reliability of an item is the probability that the item will
survive time t, given that it had not failed at time zero, when
used within specified conditions, i.e.,

6. Reliability Function Probability Distribution Function Weibull
Exponential

7. 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

8. Percentiles, tp
Weibull
and, in particular
Exponential

9. Relationship Between h(t), f(t), F(t) and R(t)
Remark: The failure rate h(t) is a measure of proneness to
failure as a function of age, t.

10. Failure Rates - Exponential
Note:
Only for the Exponential Distribution
Cumulative Failure

11. Failure Rates - Weibull
a decreasing function of t if  < 1 Notice that h(t) is a constant if  = 1
an increasing function of t if  > 1
Cumulative Failure Rate
The Instantaneous and Cumulative Failure Rates, h(t)
and H(t), are straight lines on log-log paper.

12. 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

13. Mean Time to Failure and Mean Time Between Failures
Mean Time to Failure (or Between Failures) MTTF (or MTBF)
is the expected Time to Failure (or Between Failures)
Remarks:
MTBF provides a reliability figure of merit for expected failure
free operation
MTBF provides the basis for estimating the number of failures in
a given period of time
Even though an item may be discarded after failure and its mean
life characterized by MTTF, it may be meaningful to
characterize the system reliability in terms of MTBF if the
system is restored after item failure.

14. Mean Time Between Failure - MTBF
Weibull
Exponential

15. The Binomial Model - Example
Problem -
Four Engine Aircraft
Engine Reliability R(t) = p = 0.9
Mission success: At least two engines survive
Find RS(t)

16. The Binomial Model - Example
Solution –
1) X1 = number of engines surviving in time t
Then X1 ~ B (4,0.9)
RS(t) = P(x1 ≥ 2) = b(2) + b(3) + b(4)
= =
2) X2 = number of engines failing in time t
Then X2 ~ B (4,0.1)
RS(t) = P(x2  2) = b(0) + b(1) + b(2)
= =

17. The Weibull Model
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?

18. The Weibull Model –Solution
(a)
(b)
failures per hour

19. The Weibull Model –Solution continued
From the Gamma Function table:

20. Example – Pump life
The design life of a given type of pump for a given operating environment has a Lognormal distribution. If t0.10 = 2000 hours and the median life is 3748 hours. What is the mean life and the the 50th & 90th percentile? A system uses five pumps of this type. What is the probability of at least one of these pumps failing in 3000 hours?

21. Pump Life Solution
Given that t0.10 = 2000 hours and the median is 3748, we need to first find the values for  and .
Since the median life is
=ln 3748
=8.2290
And since t0.1= =2000,
-1.28 = ln 2000
1.28  = –
 =

22. Pump Life Solution
Now t0.9 is the value of t for which
But
So that
and t0.9=7023.8
Probability of one pump failing within 3000 hours

23. Pump Life - solution
Now t0.5 is the value of t for which
But
So that
And t0.5=3748
Mean life is

24. Pump Life Solution
Probability of at least one pump failing in 3000 hours
Y = # of pumps failing in 3000 hours
y = 0, 1, 2, 3, 4, 5
Y has a binomial distribution with
n = 5 and p = 0.326
or you could work this using a probability tree

25. Series Reliability Configuration
Simplest and most common structure in reliability analysis.
Functional operation of the system depends on the successful operation of all system components Note: The electrical or mechanical configuration may differ from the reliability configuration
Reliability Block Diagram
Series configuration with n elements: E1, E2, ..., En
System Failure occurs upon the first element failure E1 E2 En

26. Series Reliability Configuration with Exponential Distribution
Reliability Block Diagram
Element Time to Failure Distribution
with failure rate , for i=1, 2,…, n
System reliability
where E1 E2 En
is the system failure rate
System mean time to failure

27. Series Reliability Configuration
System mean time to failure
Note that q/n is the expected time to the first failure, E(T1), when n identical items are put into service

28. Parallel Reliability Configuration – Basic Concepts
Definition - a system is said to have parallel reliability configuration if the system function can be performed by any one of two or more paths
Reliability block diagram - for a parallel reliability configuration consisting of n elements, E1, E2, ... En E1 E2 En

29. Parallel Reliability Configuration
Redundant reliability configuration - sometimes called a redundant reliability configuration. Other times, the term ‘redundant’ is used only when the system is deliberately changed to provide additional paths, in order to improve the system reliability
Basic assumptions
All elements are continuously energized starting at time t = 0
All elements are ‘up’ at time t = 0
The operation during time t of each element can be described
as either a success or a failure, i.e. Degraded operation or
performance is not considered

30. Parallel Reliability Configuration
System success - a system having a parallel reliability configuration operates successfully for a period of time t if at least one of the parallel elements operates for time t without failure. Notice that element failure does not necessarily mean system failure.

31. Parallel Reliability Configuration Block Diagram
System reliability - for a system consisting of n elements, E1, E2, ... En E1 E2 En
if the n elements operate independently of each other and where Ri(t) is the reliability of element i, for i=1,2,…,n

32. System Reliability Model - Parallel Configuration
Product rule for unreliabilities
Mean Time Between System Failures

33. Parallel Reliability Configurations
p=R(t)

34. Parallel Reliability Configuration with Exponential Distribution
Element time to failure is exponential with failure rate 
Reliability block diagram:
Element Time to Failure Distribution
with failure rate for I=1,2. E1 E2
System reliability
System failure rate

35. Parallel Reliability Configuration with Exponential Distribution
System Mean Time Between Failures:
MTBFS = 1.5 

36. Example - 3 Configurations
Compare the following reliability configurations I, II, and III in terms of (a) system reliability, (b) system failure rate, (c) system mean time between failures and (d) system mean time between maintenance, assuming that a failure requires maintenance.
Element E has an exponential distribution of time to failure, T, with failure rate  = State all ground rules and assumptions, show all work and present the results graphically to convey your results.
I. II. III. E E E E E

37. Example - 3 Configurations - Solution
For the baseline system:
and E

38. Example - 3 Configurations – Solution
For alternative A:
and E

39. Example - 3 Configurations – Solution
For alternative B:
and E

40. Example - 3 Configurations – Solution
Now we compare Alternatives A & B to the baseline system.
In terms of reliability,
and

41. Example - 3 Configurations – Solution
In terms of failure rate,
and

42. Example - 3 Configurations – Solution
In terms of MTBF,
and
In terms of MTBM,

43. Example - 3 Configurations – Solution

44. Example - 3 Configurations – Solution

45. Example A system consists of five components connected as shown.
Find the system reliability, failure rate, MTBF, and MTBM if Ti~E(λ) for i=1,2,3,4,5 E2 E1 E3 E4 E5

46. Solution
This problem can be approached in several different ways. Here is one approach:
There are 3 success paths, namely,
Success Path Event
E1E2 A
E1E3 B
E4E5 C
Then Rs(t)=Ps=
=P(A)+P(B)+P(C)-P(AB)-P(AC)-P(BC)+P(ABC)
=P(A)+P(B)+P(C)-P(A)P(B)-P(A)P(C)-P(B)P(C)+
P(A)P(B)P(C)
=P1P2+P1P3+P4P5-P1P2P3-P1P2P4P5 -P1P3P4P5+P1P2P3P4P5
assuming independence and where Pi=P(Ei) for i=1, 2, 3, 4, 5

47. Since Pi=e-λt for i=1,2,3,4,5
Rs(t)
hs(t)

48. MTBFs