The Weibull Distribution: A Handbook - Index of

1.2 Physical meanings and interpretations of the WEIBULL distribution 19
1.2.2 Two models of data degradation leading to WEIBULL distributed failures
In some reliability studies, it is possible to measure physical degradation as a function of
time, e.g. ,
the
wear of a break–disk or a break–block. In other applications actual physical
degradation cannot be observed directly, but measures of product performance degradation,
e.g., the output of a device, may be available. Both kinds of data are generally referred to
as degradation data and may be available continuously or at specific points in time where
measurements are taken.
Most failures can be traced to an underlying degradation process. In some applications
there may be more than one degradation variable or more than one underlying degradation
process. Using only a single degradation variable the failure would occur when this variable
has reached a certain critical level. MEEKER/ESCOBAR (1998, Chapter 13) give many
examples and models of degradation leading to failure. Generally, it is not easy to derive
the failure distribution, e.g., the DF or the CDF, from the degradation path model. These
authors give an example leading to a lognormal failure distribution. We will give two
examples of degradation processes which will result in the WEIBULL failure distribution.
A promising approach to derive a failure time distribution that accurately reflects the dynamic
dependency of system failure and decay on the state of the system is as follows:
System state or wear and tear is modeled by an appropriately chosen random process —
for example, a diffusion process — and the occurrences of fatal shocks are modeled by a
POISSON process whose rate function is state dependent. The system is said to fail when
either wear and tear accumulates beyond an acceptable or safe level or a fatal shock occurs.
The shot–noise model 27 supposes that the system is subjected to “shots” or jolts according
to a POISSON process. A jolt may consist of an internal component malfunctioning or an
external “blow” to the system. Jolts induce stress on the system when they occur. However,
if the system survives the jolt, it may then recover to some extent. For example, the mortality
rate of persons who have suffered a heart attack declines with the elapsed time since
the trauma. In this case, the heart actually repairs itself to a certain degree. The shot–noise
model is both easily interpretable and analytically tractable.
The system wear and tear is modeled by a BROWNIAN motion with positive drift. The
system fails whenever the wear and tear reaches a certain critical threshold. Under this
modeling assumption, the time to system failure corresponds to the first passage time of the
BROWNIAN motion to the critical level, and this first passage time has an inverse GAUS-
SIAN distribution, which is extremely tractable from the viewpoint of statistical analysis.
Based on the two foregoing models, an appropriate conceptual framework for reliability
modeling is the following: Suppose that a certain component in a physical system begins
operating with a given strength or a given operational age (e.g., extent of wear–and–tear
or stress), denoted by x, that can be measured in physical units. Suppose that, as time
goes on, component wear–and–tear or stress builds up (loss of strength with increasing
age), perhaps in a random way. (The concept of wear–and–tear buildup is dual to that of
27 A reference for this model and its background is COX/ISHAM (1980).
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