Instantaneous failure rate

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):

Some common results

The Exponential distribution

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).

The Weibull distribution

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.

More lifetime distributions

ModelRisk includes the following Lifetime distributions based on different, very flexible functional forms for z(t):

Distribution name	z(t)	Restrictions
Lifetime2	z(t) = a + bt	a≥0,b≥0.MAX(a,b)>0
Lifetime3	z(t) = a + bt + ct2	a>0,c>0,a-b^2/4c>0
LifetimeExp	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
Lifetime2	z(t)=a+bt	z(t)=[a+bT]+b(t-T)
Lifetime3	z(t)=a+bt+ct2	z(t)=[a+bT+cT2 ]+[b+2cT](t-T)+c(t-T)2
LifetimeExp	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.

Instantaneous failure rates for other distributions

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