8.3.2.3 __Reliability Function
(Survival Curves)__

A survival curve or reliability function, R(t), is a graphic
representation of the relationship between the probability of survival and
time. Here, probability of survival is synonymous with probability of
nonfailure or probability of satisfactory performance. Three types of survival
curves are of primary interest. The first is a discrete or point-type curve
derived from observed data by nonparametric or distribution-free methods. The
second type is a continuous curve based on an assumption as to the form of the
distribution (Gaussian, exponential, etc.) and on values of the distribution
parameters estimated from the observed data. The third type of curve is the
true reliability function of the population from which the sample observations
were drawn. This last function can only be estimated (i.e., not determined
precisely), although the limits within which it will fall a given percentage
of the time can be defined.

Figure 8.3-6 presents a frequency distribution of failures in a
fixed population of 90 items, over a 6-hour period. To obtain a survival curve
from these data, the following simplified method is used.

During the first period of observation, from 0 to 1 hour, 4 of
the original 90 items failed. The failure rate during this period was 4/90, or
0.0444, which is equivalent to a survival rate of 1 - 0.0444, or 0.9556. In
the second period of observation, 21 of the 86 remaining items failed. The
failure rate was 21/86, or 0.244, and the survival rate was 1 - 0.244, or
0.756. The tabulation above Figure 8.3-7 gives the failure rates and survival
rates for the remaining periods of observation. It will be noted that the
failure rate increases with time.

To obtain a survival curve, which is the cumulative probability
of survival with time, the probability of survival in each time period is
multiplied by the survival rate in the succeeding time period. Thus, 0.9555 x
0.756 = 0.723; 0.723 x 0.538 = 0.388, etc. The probability values are plotted
versus the centers of the time periods as shown at the bottom of 8.3-7.

Figure 8.3-8 presents a frequency distribution of failures for a
population of 90 items in which the removal rate is constant with time. The
approach described in connection with the normal curve yields the tabulation
and exponential survival curve shown in Figure 8.3-9. (Note in this example,
only 83 of 90 items failed in six hours).

Survival curves for most electronic equipment/systems are of the
exponential form. Survival curves for mechanical parts, on the other hand, are
frequently of the normal or Weibull form. As parts wear out, their failure
rate increases and their probability of survival decreases. A large number of
such parts, all having normal or Weibull survival curves but each having a
different mean life and variance, will produce a system malfunction rate which
is essentially constant, since the mean lives of the parts will be randomly
distributed.

To determine what type of
population gives rise to a particular survival curve, the theoretical
reliability function most closely resembling the curve is computed from sample
parameters. The theoretical function is then matched to the observed curve by
statistical techniques. If this procedure establishes that there is no
significant difference between the observed and theoretical curves, the
theoretical curve is usually employed for all additional
calculations.

Figures 8.3-10 and 8.3-11 portray
observed and theoretical probability of survival curves for the case of
exponential and normal distributions of time to failure. Note that the mean
life for the exponential case has R(t) = 0.368, whereas for the normal case,
R(t) = 0.5. This is due to the symmetrical characteristic of the normal
distribution, versus the skewed characteristic of the
exponential.