Failure rate

Failure rate data can be obtained in several ways. The most common means are:

Estimation

From field failure rate reports, statistical analysis techniques can be used to estimate failure rates. For accurate failure rates the analyst must have a good understanding of equipment operation, procedures for data collection, the key environmental variables impacting failure rates, how the equipment is used at the system level, and how the failure data will be used by system designers.

Historical data about the device or system under consideration

Many organizations maintain internal databases of failure information on the devices or systems that they produce, which can be used to calculate failure rates for those devices or systems. For new devices or systems, the historical data for similar devices or systems can serve as a useful estimate.

Government and commercial failure rate data

Handbooks of failure rate data for various components are available from government and commercial sources. MIL-HDBK-217F, Reliability Prediction of Electronic Equipment, is a military standard that provides failure rate data for many military electronic components. Several failure rate data sources are available commercially that focus on commercial components, including some non-electronic components.

Prediction

Time lag is one of the serious drawbacks of all failure rate estimations. Often by the time the failure rate data are available, the devices under study have become obsolete. Due to this drawback, failure-rate prediction methods have been developed. These methods may be used on newly designed devices to predict the device's failure rates and failure modes. Two approaches have become well known, Cycle Testing and FMEDA.

Life Testing

The most accurate source of data is to test samples of the actual devices or systems in order to generate failure data. This is often prohibitively expensive or impractical, so that the previous data sources are often used instead.

Cycle Testing

Mechanical movement is the predominant failure mechanism causing mechanical and electromechanical devices to wear out. For many devices, the wear-out failure point is measured by the number of cycles performed before the device fails, and can be discovered by cycle testing. In cycle testing, a device is cycled as rapidly as practical until it fails. When a collection of these devices are tested, the test will run until 10% of the units fail dangerously.

FMEDA

Failure modes, effects, and diagnostic analysis (FMEDA) is a systematic analysis technique to obtain subsystem / product level failure rates, failure modes and design strength. The FMEDA technique considers:

• All components of a design,
• The functionality of each component,
• The failure modes of each component,
• The effect of each component failure mode on the product functionality,
• The ability of any automatic diagnostics to detect the failure,
• The design strength (de-rating, safety factors) and
• The operational profile (environmental stress factors).

Given a component database calibrated with field failure data that is reasonably accurate[1] , the method can predict product level failure rate and failure mode data for a given application. The predictions have been shown to be more accurate[2] than field warranty return analysis or even typical field failure analysis given that these methods depend on reports that typically do not have sufficient detail information in failure records.[3]

The failure rate can be defined as the following:

The total number of failures within an item population, divided by the total time expended by that population, during a particular measurement interval under stated conditions. (MacDiarmid, et al.)

Although the failure rate, ${\displaystyle \lambda (t)}$, is often thought of as the probability that a failure occurs in a specified interval given no failure before time ${\displaystyle t}$, it is not actually a probability because it can exceed 1. Erroneous expression of the failure rate in % could result in incorrect perception of the measure, especially if it would be measured from repairable systems and multiple systems with non-constant failure rates or different operation times. It can be defined with the aid of the reliability function, also called the survival function, ${\displaystyle R(t)=1-F(t)}$, the probability of no failure before time ${\displaystyle t}$.

${\displaystyle \lambda (t)={\frac {f(t)}{R(t)}}}$, where ${\displaystyle f(t)}$ is the time to (first) failure distribution (i.e. the failure density function).

${\displaystyle \lambda (t)={\frac {R(t_{1})-R(t_{2})}{(t_{2}-t_{1})\cdot R(t_{1})}}={\frac {R(t)-R(t+\Delta t)}{\Delta t\cdot R(t)}}\!}$

over a time interval ${\displaystyle \Delta t}$ = ${\displaystyle (t_{2}-t_{1})}$ from ${\displaystyle t_{1}}$ (or ${\displaystyle t}$) to ${\displaystyle t_{2}}$. Note that this is a conditional probability, where the condition is that no failure has occurred before time ${\displaystyle t}$. Hence the ${\displaystyle R(t)}$ in the denominator.

Hazard rate and ROCOF (rate of occurrence of failures) are often incorrectly seen as the same and equal to the failure rate. To clarify; the more promptly items are repaired, the sooner they will break again, so the higher the ROCOF. The hazard rate is however independent of the time to repair and of the logistic delay time.

Hazard function ${\displaystyle h(t)}$ plotted for a selection of log-logistic distributions.

Calculating the failure rate for ever smaller intervals of time results in the hazard function (also called hazard rate), ${\displaystyle h(t)}$. This becomes the instantaneous failure rate or we say instantaneous hazard rate as ${\displaystyle \Delta t}$ approaches to zero:

${\displaystyle h(t)=\lim _{\Delta t\to 0}{\frac {R(t)-R(t+\Delta t)}{\Delta t\cdot R(t)}}.}$

A continuous failure rate depends on the existence of a failure distribution, ${\displaystyle F(t)}$, which is a cumulative distribution function that describes the probability of failure (at least) up to and including time t,

${\displaystyle \operatorname {Pr} (T\leq t)=F(t)=1-R(t),\quad t\geq 0.\!}$

where ${\displaystyle {T}}$ is the failure time. The failure distribution function is the integral of the failure density function, f(t),

${\displaystyle F(t)=\int _{0}^{t}f(\tau )\,d\tau .\!}$

The hazard function can be defined now as

${\displaystyle h(t)={\frac {f(t)}{1-F(t)}}={\frac {f(t)}{R(t)}}.}$

Exponential failure density functions. Each of these has a (different) constant hazard function (see text).

Many probability distributions can be used to model the failure distribution (see List of important probability distributions). A common model is the exponential failure distribution,

${\displaystyle F(t)=\int _{0}^{t}\lambda e^{-\lambda \tau }\,d\tau =1-e^{-\lambda t},\!}$

which is based on the exponential density function. The hazard rate function for this is:

${\displaystyle h(t)={\frac {f(t)}{R(t)}}={\frac {\lambda e^{-\lambda t}}{e^{-\lambda t}}}=\lambda .}$

Thus, for an exponential failure distribution, the hazard rate is a constant with respect to time (that is, the distribution is "memory-less"). For other distributions, such as a Weibull distribution or a log-normal distribution, the hazard function may not be constant with respect to time. For some such as the deterministic distribution it is monotonic increasing (analogous to "wearing out"), for others such as the Pareto distribution it is monotonic decreasing (analogous to "burning in"), while for many it is not monotonic.

Solving the differential equation

${\displaystyle h(t)={\frac {f(t)}{1-F(t)}}={\frac {F'(t)}{1-F(t)}}}$

for ${\displaystyle F(t)}$, it can be shown that

${\displaystyle F(t)=1-\exp {\left(-\int _{0}^{t}h(t)dt\right)}.}$

A decreasing failure rate (DFR) describes a phenomenon where the probability of an event in a fixed time interval in the future decreases over time. A decreasing failure rate can describe a period of "infant mortality" where earlier failures are eliminated or corrected[4] and corresponds to the situation where λ(t) is a decreasing function.

Mixtures of DFR variables are DFR.[5] Mixtures of exponentially distributed random variables are hyperexponentially distributed.

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