

MILHDBK338B: Electronic Reliability Design Handbook 
 

5.2.1 Basic Concepts
5.2.1 Basic Concepts
The cumulative distribution function F(t) is defined as the
probability in a random trial that the random variable is not greater than t
(see note), or


(5.1) 
where f(t) is the probability density function of the random
variable, time to failure. F(t) is termed
the “unreliability function” when speaking of failure. It can be thought of as
representing the probability of failure prior to some time t. If the random
variable is discrete, the integral is replaced by a summation. Since F(t) is
zero until t=0, the integration in Equation 5.1 can be from zero to t.
NOTE: Pure mathematicians
object to the use of the same letter in the integral and also in the limits of
the integral. This is done here, and in the rest of this section in spite of
the objection in order to simplify the reference to time as the variable in
such functions as F(t), R(t), M(t), f(t), etc.
The reliability function, R(t), or
the probability of a device not failing prior to some time t, is given
by


(5.2) 
By differentiating Equation (5.2)
it can be shown that


(5.3) 
The probability of failure in a given time interval between
t_{1} and t_{2} can be
expressed by the reliability function


(5.4) 
The rate at which failures occur in the interval t_{1} to t_{2} , the failure rate,
l(t), is defined
as the ratio of probability that failure occurs in the interval, given that it
has not occurred prior to t_{1}, the start of the interval, divided by the interval
length. Thus,


(5.5) 
or the
alternative form


(5.6) 
where t = t_{1} and t_{2} = t + Dt. The hazard rate, h(t), or instantaneous failure
rate, is defined as the limit of the failure rate as the interval length
approaches zero, or





(5.7) 
But it was previously shown, Eq. (5.3), that
Substituting this into Eq. (5.7) we get
h(t) = 

(5.8) 
This is one of the fundamental relationships in reliability
analysis. For example, if one knows the density function of the time to
failure, f(t), and the reliability function, R(t), the hazard rate function
for any time, t, can be found. The relationship is fundamental and important
because it is independent of the statistical distribution under
consideration.
The differential equation of Eq. (5.7) tells us, then, that the
hazard rate is nothing more than a measure of the change in survivor rate per
unit change in time.
Perhaps some of these concepts can be seen more clearly by use
of a more concrete example.
Suppose that we start a test at time, t_{0} , with N_{O} devices. After some time t, N_{f} of the original devices will have failed, and
N_{S} will have survived (N_{O} = N_{f} + N_{s} ). The reliability, R(t), is given at any time t,
by:


(5.9) 


(5.10) 
From
Eq. (5.3)


(5.11) 
Thus, the failure density function represents the proportion of
the original population, (N_{O}), which fails in the
interval (t, t + Dt).
On the other hand, from Eqs. (5.8),
(5.9) and (5.11)


(5.12)

Thus, h(t) is inversely
proportional to the number of devices that survive to time t, (N_{s} ), which fail in the interval (t, t + Dt).
Although, as can be seen by
comparing Eqs. (5.6) and (5.7), failure rate, l(t), and hazard rate, h(t),
are mathematically somewhat different, they are usually used synonymously in
conventional reliability engineering practice. It is not likely that this
handbook will change firmly entrenched conventional practice, so the reader
should be aware of this common deviation from exact mathematical
accuracy.
Perhaps the simplest explanation of
hazard and failure rate is made by analogy. Suppose a family takes an
automobile trip of 200 miles and completes the trip in 4 hours. Their average
rate was 50 mph, although they drove faster at some times and slower at other
times. The rate at any given instant could have been determined by reading the
speed indicated on the speedometer at that instant. The 50 mph is analogous to
the failure rate and the speed at any point is analogous to the hazard
rate.
In Eq. (5.8), a general expression
was derived for hazard (failure) rate. This can also be done for the
reliability function, R(t). From Eq. (5.7)


(5.13) 



Integrating both sides of Eq.
(5.13)







but R(0) = 1, ln R(0) =
0, and 



(5.14) 
Eq. (5.14) is the general
expression for the reliability function. If h(t) can be considered a constant
failure rate (l),
which is true for many cases for electronic equipment, Eq. (5.14)
becomes


(5.15) 
Eq. (5.15) is used quite
frequently in reliability analysis, particularly for electronic equipment.
However, the reliability analyst should assure himself that the constant
failure rate assumption is valid for the item being analyzed by performing
goodness of fit tests on the data. These are discussed in Section
8.
In addition to the concepts of
f(t), h(t), l(t),
and R(t), previously developed, several other basic, commonlyused reliability
concepts require development. They are: meantimetofailure (MTTF), mean life
(q), and
meantimebetweenfailure (MTBF).
MeanTimeToFailure (MTTF)
MTTF is nothing more than the expected value of time to failure
and is derived from basic statistical theory as follows:





(5.16) 
Integrating by parts and applying “Hopital's rule,” we arrive at
the expression


(5.17) 
Eq. (5.17), in many cases, permits the simplification of MTTF
calculations. If one knows (or can model from the data) the reliability
function, R(t), the MTTF can be obtained by direct integration of R(t) (if
mathematically tractable), by graphical approximation, or by Monte Carlo
simulation. For repairable equipment MTTF is defined as the mean time to first
failure.
Mean Life (q)
The mean life (q) refers to the total
population of items being considered. For example, given an initial population
of n items, if all are operated until they fail, the mean life (q) is merely the arithmetic
mean time to failure of the total population given by:


(5.18) 
where:
t_{i} = time to failure
of the i^{th} item in the
population n = total number of items
in the population
MeanTimeBetweenFailure
(MTBF)
This concept appears quite frequently in reliability literature;
it applies to repairable items in which failed elements are replaced upon
failure. The expression for MTBF is:


(5.19) 
where:
T(t) = total operating time r = number of
failures
It is important to remember that MTBF only has meaning
for repairable items, and, for that case, MTBF represents exactly the same
parameter as mean life (q). More important is the fact that a constant failure
rate is assumed. Thus, given the two assumptions of replacement upon failure
and constant failure rate, the reliability function is:


(5.20) 
and (for this case)


(5.21) 
Figure 5.21 provides a convenient summary of the basic concepts
developed in this section.
FIGURE 5.21: SUMMARY OF BASIC RELIABILITY
CONCEPTS




 
 