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Reliability R(t) is the probability that an item will perform
Hazard Rate & Failure Rate
R(t) is a decreasing function over
time, because as time passes, more units would fail. Alternatively, if
we focus on a distinct unit, we would interprete R(t) as the
probability of survival of that unit over time.
The next picture shows a representation of
how R(t) could look like:
At time t = 0, no units have failed and the Reliability (probability of
survival) is 1. For t > 0, more and more units would fail,
Therefore, the Reliability would tend towards 0 for t -->00.
- a required function without failure
- under stated conditions
- for a stated period of time.
R(t) is always a monotonously decreasing function over time.
The most powerful tool for determining reliability is the reliability block diagram.
The hazard rate h(t) is the first derivative of R(t). h(t) is the
change in probability of survival over time. Unlike R(t), h(t) is not necessarily a monotonously decreasing
function over time, however, in most practical cases it is.
If we consider two timepoints, t1 and
t2, then the change in probability between t1 and
t2 is the
probability that a unit will fail between t1 and
speaking, the same change in probability can be interpreted as the
fraction of units of the initial
population failing between t1 and
t2. If we make (t1 -
enough, then we obtain the hazard rate h(t).
The negative symbol is neccessary in
order to make h(t) a positive figure.
The next picture shows h(t)
for the same R(t) of the picture before:
In contrast to R(t), the range of h(t) is between 0 and a specific
value determined by the nature of R(t). Furthermore, as mentioned
above, h(t) need not necessarily be monotonous as the picture might
In reliability analysis, the hazard rate is of less interest.
Remember that the hazard rate
is the rate of failure occurences with reference to the initial population size (at t =0).
What we want is the rate of
failure occurences with reference to the current population. This is
called the failure rate Lambda(t). The following exemplary statements
makes the difference between clear:
Failure rate Lambda(t): 1% of all functioning units are
expected to fail within the next hour.
Hazard rate h(t):
all units that were functional at t=0 are expected to fail within
the next hour.
--> In order to obtain the failure
rate we must simply divide h(t) by R(t):
As with the
hazard rate h(t), Lambda (t) need not necessarily be
monotonous. Its range is between 0 and a specific
value determined by the nature of R(t). The next picture shows
Lambda(t) for the same R(t) and h(t) of the pictures before:
that l(t) is constant over time. This is no coincidence, since the
constant failure rate case is by far the most important case in
reliability analysis. More on this can be found on the MTBF page. R(t), h(t) and Lambda(t) for different
cases will be shown in the next paragraph.
How must R(t) look like in order to obtain constant failure rate? It's
quite obvious that the derivative, h(t) must be a multiple of the
primitive, R(t), for every t. This is accomplished by the exponential
function. We don't show this in detail here, instead we will derive
h(t) and lambda(t) for the general case using the Weibull distribution
Furtther reading: Weibull analysis