# Reliability, Weibull's law

## Definitions

Reliability:
Reliability is the ability of an entity to perform a required function, under given conditions, during a given time interval.

Failure:
Failure is the end of a device's ability to perform a function that was expected. It may be partial (impairment of performance) or total (end of function).

R(t)Probability that an entity E is non-faulty over the period [0; t ], assuming that it is not faulty at time t = 0.

F(t)F(t) is the cumulative function of failures 猫 F(t) = 1 - R(t)

Probability density f(t) represents the failure rate of a product, i.e. the probability that a product will fail over the time interval [t,t+dt].

$f(t)=\frac{\text{number of failed elements over time}\Delta t}{\text{number of elements tested}}$

$\lambda(t)=\frac{\text{number of failed elements over time}\Delta t}{\text{number of failed elements still in test}}$

• 位(t) decreases The defect rate decreases over time, and this generally corresponds to early defects. Products with intrinsic defects deteriorate rapidly, whereas other products last much longer.
• 位(t) constant The failure rate does not depend on time. There is just as much risk of a product failing at time t, whatever its lifespan. These are intrinsic failures. This is the type of failure often found in electronic products.
• 位(t) increases The failure rate increases over time. There is an increasing risk that a product will fail as its lifespan increases. This is the end of the product's life.
MTTF (Mean Time To Failure)
The MTTF, which is more commonly used in product reliability, is the average time to first failure.
MTBF (Mean Time Between Failure)
MTBF (Mean Time Between Failures) is the average of the time intervals between two failures.

## Weibull's law

To model a product's failure law, you need to be able to model multiple types of failure:

For this reason, Weibull's law is the main one used, as it allows great variability in shape.

Thanks to its great flexibility, Weibull's law can be used to model the behaviour of many types of failure, such as :

• The breaking strength of components or the effort required to fatigue metals
• The failure time of an electronic component
• Failure time for items used outdoors, such as car tyres
• Systems that fail when the weakest component in the system fails

Weibull's law can also be used to model the behaviour of different life situations for the same component
The Weibull distribution function is as follows :

$R\left( t \right)=e-\left( \frac{t-\delta}{\phi} \right)^{\beta}$

It has 3 parameters:

• $\beta$shape parameter聽
• $\phi聽$scale parameter
• $\delta$delay parameter

## $\beta$shape parameter

It allows the shape of the law to be adapted to be as close as possible to the observed failure rate:

$\beta$=1: The failure rate is constant ($\lambda$ constant)

$\beta$>1: The failure rate increases over time ($\lambda$ increases - end of product life)

$\beta$<1: Failure rate decreases over time ($\lambda$ decreases - youth defect)

## $\phi$scale parameter

The parameter 胃 is used to adjust the scale of the distribution law to the scale of the observed problem, for example :
Failure occurs around t = 90.4
Failure occurs around t = 904

## $\delta$delay parameter

The distribution law can be shifted by a parameter 未

Failure occurs around t = 90.4

Failure occurs around t = 190.4
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