ETL 1110-2-560
30 Jun 01
b. Measures of component reliability.
(1) Reliability function. The continuous probabilistic approach to item reliability is represented by
the reliability function. It is simply the probability that an item has survived to time t. The mathematical
expression can be summarized by
R(t) = P(T > t)
(1)
where
R(t) = reliability of the item, i.e., probability of success
P(T > t) = probability that the time to failure of an item will be greater than or equal to its service time
T = time to item failure
t = the designated period of time for the operation of the item
Conversely, the probability of failure F(t) is simply
F(t) = 1 - R(t)
(2)
(2) Hazard function or failure rate.
(a) The failure rate or hazard function h(t) represents the proneness to failure of a component as a
function of its age or time in operation. It reflects how the reliability of a component changes with time
as a result of various factors such as the environment, maintenance, loading, and operating condition.
From Modarres (1993) it can be shown that
-dR(t)
f (t) =
(3)
dt
f (t)
h(t) =
(4)
R(t)
where f(t) is the probability density function (pdf). This is a mathematical description for the curve
approximation of the number of the probable occurrences of a specific random variable (i.e., the failure of
a component for use in this ETL).
(b) The hazard function or instantaneous failure rate is the instantaneous conditional probability of
failure of an item in the next unit of time given that it has survived up to that time. The hazard function
can increase, decrease, or remain constant. It has been shown that the failure rate behavior of most
mechanical and electrical engineering devices follows that shown in Figure 1. This is known as the
bathtub curve. Region A represents a high initial failure rate, which decreases with time to nearly
constant. This is known as the infant mortality region and is a result of poor workmanship or quality
control. Region B represents the useful life phase. Here, failures occur because of random events.
Region C represents the wear-out phase where failures occur due to complex aging or deterioration.
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