reliability math and the exponential distribution

Chet Haibel ©2013 Hobbs Engineering Corporation

Reliability

Math and the

Exponential

Distribution

0 0


General Reliability Function, R(t)

Fraction of a group surviving until a certain time.

Probability of one unit surviving until a certain time.

Monotonic downward, assuming failed things stay failed.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

1


General Cumulative Distribution Function, F(t)

Fraction of a group failing before a certain time.

Probability of one unit failing before a certain time.

Monotonic upward, assuming failed things stay failed.

What fraction fails at 20 (arbitrary time units)?

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

2


The probability of one unit failing between two times is found by

subtracting the CDF for the two times, in this case perhaps 0.49

minus 0.41 equals 0.08 or 8%.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

General Cumulative Distribution Function, F(t)

3


This is the time derivative (the slope) of F(t).

This is a histogram of when failures occur.

The height is the fraction failing per unit time.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 10 20 30 40 50

General Probability Density Function, f(t)

4


The height is the fraction failing per unit time.

The area under the pdf between two times gives the probability

of failure during this time interval, in this case perhaps 0.008

high by 10 units long for an area of 8%.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 10 20 30 40 50

General Probability Density Function, f(t)

5


General Hazard Function, h(t)

The conditional probability density of failure, given that the

product has survived up to this point in time.

The fraction of the survivors failing per unit time.

0

5

10

15

20

0 10 20 30 40 50

6

Chet Haibel ©2013 Hobbs Engineering Corporation 7

Bathtub Curve

Operating Time (t)

Haza

rd

Rate

- h

(t)

Useful-Life

Failures

Early-Life

Failures

aka

infant

mortalities

Wear-Out

Failures

aka

end-of-life

failures

Life to the Beginning of Wear-Out

Random-in-Time Failures


Summary of Functions Always True

R(t) = Reliability function

probability of surviving until some time

F(t) = Cumulative distribution function

probability of failure before some time

R(t) + F(t) = 1

f(t) = Probability density function failure rate

h(t) = Hazard function = f(t) / R(t)

conditional (normalized) failure rate

dt

F(t)d

8


Relations Among Functions Always True

All functions are not defined for t < 0

All parts, subassemblies, products, and systems are

assumed to be working at t = 0

R(t) & F(t) are probabilities and lie between 0 and 1

R(t) & F(t) are monotonic, assuming things “stay failed”

f(t) the pdf is the derivative of the CDF

where h(t)

dt

F(t)d

t

0

d h

eR(t)

R(t)

f(t)

9


Is the most widely used (and sometimes misused) failure distribution for

reliability analysis for complex electronic systems.

Is applicable when the hazard rate is constant. The hazard rate is the

surviving fraction failing per unit of time or equivalent; such as percent

per million cycles, or failures per billion (109) hours (FITs).

Requires the knowledge of only one parameter for its application.

Is used to describe steady-state failure rate conditions.

Models device performance after the Early-Life

(infant mortality) period and prior to the Wear-Out

(end-of-life) period of the Bathtub Curve.

The Exponential Failure Distribution

10


Life of 10 Constant Hazard Rate Devices

Life in Hours 11


Sort from First to Last 10 Constant Hazard Rate Devices

Life in Hours 12

How do the

devices know to

fail this way?


Fraction Surviving over Time Exponential Distribution

Exponential Distribution

= Constant Hazard Rate

tetR )(

13 Life in Hours

Chet Haibel ©2013 Hobbs Engineering Corporation 14

Reliability Function Exponential Distribution

R(t) = e-t Extending the initial

slope intersects the

x-axis at the mean

MTTF = 1/

Life in Hours


R(t) = e-t

F(t) = 1-e-t

f(t) = e-t

h(t) =

θ

t

e

θ

t

eθ

1

θ

t

e1

θ

1

Functions Exponential Distribution

15


Functions Exponential Distribution

pdf

Operating

Timet

f t e t( ) CDF

Operating

Timet

PROBABILITY DENSITY FUNCTION CULMULATIVE DISTRIBUTION

FUNCTION

F t e t( ) 1

R(t)

Operating

Timet

THE RELIABILITY FUNCTION

R t e t( ) h(t)

Operating

Timet

THE HAZARD RATE

pdf

Operating

Timet

f t e t( ) CDF

Operating

Timet


FUNCTION

F t e t( ) 1

R(t)

Operating

Timet


R t e t( ) h(t)

Operating

Timet

THE HAZARD RATE

pdf

Operating Timet

f t e t( ) CDF

Operating Timet


FUNCTION

F t e t( ) 1

R(t)

Operating Timet


R t e t( ) h(t)

Operating Timet

THE HAZARD RATE

1 ------

------ h(t) =

1 ------

16


For REPAIRABLE products, there is no limit to how many

failures one can have!

Percent of Failures Over Time(shown for 3% per month failure rate)

0%

20%

40%

60%

80%

100%

120%

140%

160%

180%

200%0

12

24

36

48

60

72

84

96

Months in Service

Repairable

Non-Repairable

H(t)

F(t)

Repairable & Non Repairable Systems Exponential Distribution

17


Mean is:

Standard deviation is:

To get the “point estimate” of

λ

1θμ

failures ofnumber total

not)or (failed hours ofnumber totalor MTBF MTTFθ

λ

1θσ

Mean & Standard Deviation Exponential Distribution

18

reliability math and the exponential distribution

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