Reliability
193
Integrating Eq. (4.11) and defining R(0)=
t
-- ~ h(~c)
d~c
R(t) = e 0
(4.12)
Now consider the mathematical models for h(t) in Appendix A and D.
IV. RELIABILITY FOR A CONSTANT RATE OF FAILURE
CURVE
The form of Eq. (4.12) suggests considering h(t) a constant as a simple failure model. This model is frequently called the exponential or constant hazard rate model. In addition to the obvious simplicity, there are sound
physical reasons for seriously considering this model. Figure 4.2 shows a
failure rate versus age (time) curve which is typical of the performance
of many systems and some types of components.
The central portion of the curve Fig. 4.2 represents the useful life of the
system and is characterized by chance or random failures. The high initial
failure rate is due to shakedown or debugging failures and can be reduced
by improving production quality control and/or breaking in equipment
before leaving the factory. Aging failures are minimized by preventative
maintenance-i.e, repair or replacement of parts susceptible to aging. Hence,
in Fig. 4.2 there is a kind of empirical justification for assuming h(t) a constant over a substantial portion of the life if a system, provided measurements are taken to minimize or eliminate the initial and wear out failures.
Break-in
Wear out
~I
Co n sta nt
failure rate
I
I
I
I
I
I
Time
Figure 4.2. Typical bath tub aging curve.
193
Integrating Eq. (4.11) and defining R(0)=
t
-- ~ h(~c)
d~c
R(t) = e 0
(4.12)
Now consider the mathematical models for h(t) in Appendix A and D.
IV. RELIABILITY FOR A CONSTANT RATE OF FAILURE
CURVE
The form of Eq. (4.12) suggests considering h(t) a constant as a simple failure model. This model is frequently called the exponential or constant hazard rate model. In addition to the obvious simplicity, there are sound
physical reasons for seriously considering this model. Figure 4.2 shows a
failure rate versus age (time) curve which is typical of the performance
of many systems and some types of components.
The central portion of the curve Fig. 4.2 represents the useful life of the
system and is characterized by chance or random failures. The high initial
failure rate is due to shakedown or debugging failures and can be reduced
by improving production quality control and/or breaking in equipment
before leaving the factory. Aging failures are minimized by preventative
maintenance-i.e, repair or replacement of parts susceptible to aging. Hence,
in Fig. 4.2 there is a kind of empirical justification for assuming h(t) a constant over a substantial portion of the life if a system, provided measurements are taken to minimize or eliminate the initial and wear out failures.
Break-in
Wear out
~I
Co n sta nt
failure rate
I
I
I
I
I
I
Time
Figure 4.2. Typical bath tub aging curve.