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Download a pdf copy of this help file here |
The principle of the instantaneous
failure rate function
Reliability theory
is much concerned with the probability distribution of the time
a
component or machine will operate before failing. The instantaneous failure
rate, often called the hazard
function, 
of a component or device at time t is
defined as:
where f(t) and
F(t) are the probability density function and cumulative distribution
function respectively for the amount of time the component or machine
will work before failing. In other words, z(t) is the rate of failure
of the component at time F(t) given that it has survived up to time t
with probability 1-F(t).
It can be shown that the expression
in Equation 1 for z(t) results in an equation for f(t):
In a Poisson process, the instantaneous failure rate z(t) is constant i.e. z(t) = l, then
Using 
we have
the equation of the Exponential distribution,
i.e. the exponential distribution describes the distribution of survival
time of a component given that it has a constant failure rate. The alternative
parameter 
is
called the mean time between failures (MTBF).
If z(t) is not assumed to be constant, but rather increases or decreases smoothly with time, we can consider using the equation:
Equation 3
The equation looks unnecessarily complicated: it is in fact just z(t) = atb but the form used above helps in producing a neater equation in the next step. The graph below helps to visualize how this function behaves. If a = 1, the equation for z(t) reduces to: z(t) = l which is the formula that produces the exponential distribution. If a < 1, z(t) decreases with time which typifies the running in period for a component. If 1< a < 2, z(t), increases with time, first rapidly and then more slowly. If a = 2, z(t), increases linearly, and if a > 2, z(t), increases at an ever increasing rate, which typifies the period of the end of a component's useful life.
Putting Equation
3 for z(t) into Equation 2 and then Equation 1,
and using results in the following expression:
which is the distribution function for the Weibull (a, b) distribution.
A limitation of the Weibull's equation for z(t) is that z(0) is either zero or infinite which is unrealistic (ignoring the constant z(t) exception). Also note that a component with a Weibull lifetime when first put into service will never have the same, or any other Weibull-distributed lifetime afterwards because after any amount of service time they have travelled along the z(t) curve, which is now neither zero of infinity.
ModelRisk includes the following Lifetime distributions based on different, very flexible functional forms for z(t):
Distribution name |
z(t) |
Restrictions |
z(t) = a + bt |
a≥0,b≥0.MAX(a,b)>0 |
|
z(t) = a + bt + ct2 |
a>0,c>0,a-b^2/4c>0 |
|
z(t) = exp[a + bt] |
b>0 |
The Lifetime2 distribution has a linearly increasing instantaneous failure
rate that may begin at a non-zero value:
The Lifetime3
distribution has a quadratic instantaneous failure rate that can begin
at a zero or a positive value, can increase constantly or at an increasing
rate, and which can also produce a bathtub curve (b<0):
The LifetimeExp distribution has an exponential form for the instantaneous failure rate, which is always >0 and may increase or decrease with time:
Each of these three distributions can be used at the beginning of a component’s service life and at some later time T (where the lifetime left is now (t-T) ) in a consistent way, as follows:
Distribution name |
Initial z(t) |
z(t) after time T |
z(t)=a+bt |
z(t)=[a+bT]+b(t-T) |
|
z(t)=a+bt+ct2 |
z(t)=[a+bT+cT2 ]+[b+2cT](t-T)+c(t-T)2 |
|
z(t)=exp[a+bt] |
z(t)=exp[[a+bT]+b(t-T)] |
The ability to retain the same functional form for z(t) means that we can apply and reapply these same distribution types throughout the lifetime of a component without contradicting any previous assumptions.
Provided a distribution
is continuous, has a minimum of zero and smooth and calculable density
and distribution functions, we can use it for a lifetime distribution
and investigate its instantaneous failure rate function. The following
distributions comply with these requirements and are often used as lifetime
distributions:
Lognormal
– also justified if one believes that a lifetime is the product of a large
number of random factors
Gamma – if one believes that a lifetime is the sum of a number of exponential events
Fatigue – (with α=0 to have a minimum of zero) the fatigue life distribution is based on a conceptual model of a crack growing to breaking point
Burr – because with its four parameters it has a lot of flexibility of shape
Inverse Gaussian – when a Lognormal has too heavy a right tail
LogGamma – (with λ=0 to have a minimum of zero) if one believes that a lifetime is the product of a number of LogExponential events
Pareto2 – when you want a lifetime distribution with the longest possible right tail
The following z(t) plots illustrate
some of the variety of forms that can be obtained with these families
of distributions.
