Editing Failure rate (section)

==Mathematical definition==

The simplest definition of failure rate <math>\lambda</math> is simply the number of failures <math>\Delta n</math> per time interval <math>\Delta t</math>:

:<math>\lambda = \frac{\Delta n}{\Delta t}</math>

which would depend on the number of systems under study, and the conditions over the time period.

===Failures over time===

[[File:Exponential distribution cdf.svg|thumb|300px|Cumulative distribution function for the [[exponential distribution]], often used as the cumulative failure function <math>F(t).</math>]]

To accurately model failures over time, a '''cumulative failure distribution''', <math>F(t)</math> must be defined, which can be any [[cumulative distribution function]] (CDF) that gradually increases from <math>0</math> to <math>1</math>. In the case of many identical systems, this may be thought of as the fraction of systems failing over time <math>t</math>, after all starting operation at time <math>t=0</math>; or in the case of a single system, as the [[probability]] of the system having its failure time <math>T</math> before time <math>t</math>:

:<math>F(t) = \operatorname{P}(T\le t).</math>

As CDFs are defined by integrating a [[probability density function]], the '''failure probability density''' <math>f(t)</math> is defined such that: 

[[File:Exponential pdf.svg|thumb|right|300px|Exponential probability functions, often used as the failure probability density <math>f(t)</math>.]]

:<math>F(t)=\int_{0}^{t} f(\tau)\, d\tau \!</math>

where <math>\tau</math> is a dummy integration variable. Here <math>f(t)</math> can be thought of as the ''instantaneous failure rate'', i.e. the fraction of failures per unit time, as the size of the time interval <math>\Delta t</math> tends towards <math>0</math>:

:<math>f(t) = \lim_{\Delta t \to 0^+} \frac{P(t<T\leq t + \Delta t)}{\Delta t}. </math>

===Hazard rate===

A concept closely-related but different<ref name="todinov">{{cite book |last1=Todinov |first1=MT |date=2007 |title=Risk-Based Reliability Analysis and Generic Principles for Risk Reduction |chapter=Chapter 2.2 HAZARD RATE AND TIME TO FAILURE DISTRIBUTION}}</ref> to instantaneous failure rate <math>f(t)</math> is the '''hazard rate''' (or '''{{visible anchor|hazard function}}'''), <math>h(t)</math>.

In the many-system case, this is defined as the proportional failure rate of the systems ''still functioning'' at time <math>t</math> (as opposed to <math>f(t)</math>, which is the expressed as a proportion of the ''initial number'' of systems). 

For convenience we first define the reliability (or [[survival function]]) as:

:<math>R(t) = 1 - F(t)</math>

then the hazard rate is simply the instantaneous failure rate, scaled by the fraction of surviving systems at time <math>t</math>:

:<math>h(t) = \frac{f(t)}{R(t)}</math>

In the probabilistic sense, for a single system this can be interpreted as how much the [[conditional probability]] of failure time <math>T</math> within the time interval <math>t</math> to <math>t + \Delta t</math> changes, ''given that the system or component has already survived to time <math>t</math>'':

:<math>h(t) = \lim_{\Delta t \to 0^+} \frac{P(t < T \leq t + \Delta t \mid T>t)}{\Delta t}.</math>

====Conversion to cumulative failure rate====

To convert between <math>h(t)</math> and <math>F(t)</math>, we can solve the differential equation

:<math>h(t)=\frac{f(t)}{R(t)}=-\frac{R'(t)}{R(t)}</math>

with initial condition <math>R(0)=1</math>, which yields<ref name="todinov" />

:<math>F(t) = 1 - \exp{\left(-\int_0^t h(\tau) d\tau \right)}.</math>

Thus for a collection of identical systems, only one of hazard rate <math>h(t)</math>, failure probability density <math>f(t)</math>, or cumulative failure distribution <math>F(t)</math> need be defined.

Confusion can occur as the notation <math>\lambda(t)</math> for "failure rate" often refers to the function <math>h(t)</math> rather than <math>f(t).</math><ref>{{cite book| first1=Shaoping | last1=Wang |
title=Comprehensive Reliability Design of Aircraft Hydraulic System | chapter=Chapter 3.3.1.3: Failure Rate λ(t) | date= 2016}}</ref>

===Constant hazard rate model===

There are many possible functions that could be chosen to represent failure probability density <math>f(t)</math> or hazard rate <math>h(t)</math>, based on empirical or theoretical evidence, but the most common and easily-understandable choice is to set

:<math>f(t) = \lambda e^{-\lambda t}</math>,

an [[exponential function]] with scaling constant <math>\lambda</math>. As seen in the figures above, this represents a gradually decreasing failure probability density.

The CDF <math>F(t)</math> is then calculated as: 

:<math>F(t)=\int_{0}^{t} \lambda e^{-\lambda \tau}\, d\tau = 1 - e^{-\lambda t}, \!</math>

which can be seen to gradually approach <math>1</math> as <math>t \to \infty,</math> representing the fact that eventually all systems under study will fail.

The hazard rate function is then:

:<math>h(t) = \frac{f(t)}{R(t)} = \frac{\lambda e^{-\lambda t}}{e^{-\lambda t}} = \lambda .</math>

In other words, in this particular case ''only'', the hazard rate is constant over time.

This illustrates the difference in hazard rate and failure probability density - as the number of systems surviving at time <math>t > 0</math> gradually reduces, the total failure rate also reduces, but the hazard rate '''remains constant'''. In other words, the probabilities of each individual system failing do not change over time as the systems age - they are "[[memorylessness|memory-less]]".

===Other models===

[[File:Loglogistichaz.svg|thumb|right|300px|Hazard function <math>h(t)</math> plotted for a selection of [[log-logistic distribution]]s, any of which could be used as a hazard rate, depending on the system under study.]]

For many systems, a constant hazard function may not be a realistic approximation; the chance of failure of an individual component may depend on its age. Therefore, other distributions are often used.

For example, the [[deterministic distribution]] increases hazard rate over time (for systems where wear-out is the most important factor), while the [[Pareto distribution]] decreases it (for systems where early-life failures are more common). The commonly-used [[Weibull distribution]] combines both of these effects, as do the [[log-normal distribution|log-normal]] and [[Hypertabastic survival models|hypertabastic]] distributions.

After modelling a given distribution and parameters for <math>h(t)</math>, the failure probability density <math>f(t)</math> and cumulative failure distribution <math>F(t)</math> can be predicted using the given equations.