digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

Failure rate is the frequency with which an engineered system or component fails, expressed, for example, in failures per hour. It is often denoted by the Greek letter λ (lambda) and is important in reliability engineering.

The failure rate of a system usually depends on time, with the rate varying over the life cycle of the system. For example, an automobile's failure rate in its fifth year of service may be many times greater than its failure rate during its first year of service. One does not expect to replace an exhaust pipe, overhaul the brakes, or have major transmission problems in a new vehicle.

In practice, the mean time between failures (MTBF, 1/λ) is often reported instead of the failure rate. This is valid and useful if the failure rate may be assumed constant – often used for complex units / systems, electronics – and is a general agreement in some reliability standards (Military and Aerospace). It does in this case only relate to the flat region of the bathtub curve, also called the "useful life period". Because of this, it is incorrect to extrapolate MTBF to give an estimate of the service life time of a component, which will typically be much less than suggested by the MTBF due to the much higher failure rates in the "end-of-life wearout" part of the "bathtub curve".

The reason for the preferred use for MTBF numbers is that the use of large positive numbers (such as 2000 hours) is more intuitive and easier to remember than very small numbers (such as 0.0005 per hour).

The MTBF is an important system parameter in systems where failure rate needs to be managed, in particular for safety systems. The MTBF appears frequently in the engineering design requirements, and governs frequency of required system maintenance and inspections. In special processes called renewal processes, where the time to recover from failure can be neglected and the likelihood of failure remains constant with respect to time, the failure rate is simply the multiplicative inverse of the MTBF (1/λ).

A similar ratio used in the transport industries, especially in railways and trucking is "mean distance between failures", a variation which attempts to correlate actual loaded distances to similar reliability needs and practices.

Failure rates are important factors in the insurance, finance, commerce and regulatory industries and fundamental to the design of safe systems in a wide variety of applications.

## Failure rate in the discrete sense

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, $\lambda (t)$, is often thought of as the probability that a failure occurs in a specified interval given no failure before time $t$, 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, $R(t)$, the probability of no failure before time $t$.

$\lambda(t) = \frac{f(t)}{R(t)}$, where $f(t)$ is the time to (first) failure distribution (i.e. the failure density function) and $R(t)=1-F(t)$.
$\lambda(t) = \frac{R(t_1)-R(t_2)}{(t_2-t_1) \cdot R(t_1)} = \frac{R(t)-R(t+\triangle t)}{\triangle t \cdot R(t)} \!$

over a time interval $(t_2-t_1)$ from $t_1$ (or $t$) to $t_2$ and $\Delta t$ is defined as $(t_2-t_1)$. Note that this is a conditional probability, hence the $R(t)$ in the denominator.

The $\lambda(t)$ function is a CONDITIONAL probability of the failure DENSITY function. The condition is that the failure has not occurred at time $t$.

Hazard rate and ROCOF (rate of occurrence of failures) are often incorrectly seen as the same and equal to the failure rate. And literature is even contaminated with inconsistent definitions. The hazard rate is in contrast to the ROCOF the same a failure rate. ROCOF is used for repairable systems only. In practice not many serious errors are made due to this confusion (although this statement is hard to validate...).

## Failure rate in the continuous sense

Exponential failure density functions

Calculating the failure rate for ever smaller intervals of time, results in the hazard function (loosely and incorrectly called hazard rate), $h(t)$. This becomes the instantaneous failure rate as $\scriptstyle\Delta t$ tends to zero:

$h(t)=\lim_{\triangle t \to 0} \frac{R(t)-R(t+\triangle t)}{\triangle t \cdot R(t)}.$

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

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

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

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

The hazard function can be defined now as

$h(t)=\frac{f(t)}{R(t)}. \!$

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,

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

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

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

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[citation needed] 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.

## Failure rate data

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

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.
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.

### Units

Failure rates can be expressed using any measure of time, but hours is the most common unit in practice. Other units, such as miles, revolutions, etc., can also be used in place of "time" units.

Failure rates are often expressed in engineering notation as failures per million, or 10−6, especially for individual components, since their failure rates are often very low.

The Failures In Time (FIT) rate of a device is the number of failures that can be expected in one billion (109) device-hours of operation. (E.g. 1000 devices for 1 million hours, or 1 million devices for 1000 hours each, or some other combination.) This term is used particularly by the semiconductor industry.

The relationship of FIT to MTBF may be expressed as: MTBF = 1,000,000,000 x 1/FIT.

Under certain engineering assumptions (e.g. besides the above assumptions for a constant failure rate, the assumption that the considered system has no relevant redundancies), the failure rate for a complex system is simply the sum of the individual failure rates of its components, as long as the units are consistent, e.g. failures per million hours. This permits testing of individual components or subsystems, whose failure rates are then added to obtain the total system failure rate.[citation needed]

### Example

Suppose it is desired to estimate the failure rate of a certain component. A test can be performed to estimate its failure rate. Ten identical components are each tested until they either fail or reach 1000 hours, at which time the test is terminated for that component. (The level of statistical confidence is not considered in this example.) The results are as follows:

Estimated failure rate is

$\frac{6\text{ failures}}{7502\text{ hours}} = 0.0007998 \frac{\text{failures}}{\text{hour}} = 799.8 \times 10^{-6} \frac{\text{failures}}{\text{hour}},$

or 799.8 failures for every million hours of operation.

## Estimation

The Nelson–Aalen estimator can be used to estimate the cumulative hazard rate function.

## References

### Print

• Blanchard, Benjamin S. (1992), Logistics Engineering and Management, Fourth Ed., pp 26–32, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.
• Ebeling, Charles E., (1997), An Introduction to Reliability and Maintainability Engineering, pp 23–32, McGraw-Hill Companies, Inc., Boston.
• Federal Standard 1037C
• Kapur, K.C., and Lamberson, L.R., (1977), Reliability in Engineering Design, pp 8–30, John Wiley & Sons, New York.
• Knowles, D.I.,(1995), Should We Move Away From "Acceptable Failure Rate", Communications in Reliability Maintainability and Supportability, Vol. 2, No. 1, P. 23, International RMS Committee, USA
• MacDiarmid, Preston; Morris, Seymour; et al., (no date), Reliability Toolkit: Commercial Practices Edition, pp 35–39, Reliability Analysis Center and Rome Laboratory, Rome, New York.
• Rausand, M. and Hoyland, A., (2004), System Reliability Theory; Models, Statistical methods, and Applications, John Wuiley & Sons, New York. NTNU.no
• Turner, T., Hockley, C., and Burdaky, R., (1997), The Customer Needs A Maintenance-Free Operating Period, 1997 Avionics Conference and Exhibition, No. 97-0819, P. 2.2, ERA Technology Ltd., Leatherhead, Surrey, UK

### Online

 1000000 videos foundNext >
 Reliability 2 - MTTR, MTTF, MTBF, Failure rateThis part of the presentation describes practical reliability metrics: MTTF, MTBF, MTTR, and failure rate. See also my blog: http://dependablesystem.blogspot... Microsoft Confirmed 100% Xbox 360 failure rateMicrosoft Confirmed 100% Xbox 360 failure rate, everyones xbox will get the ring of death "RRoD" there is no safe xbox , My xbox 360 Rant. How to Lower Failure Rate? by Mark Fasciano, Managing Director at Canrock Ventureshttp://www.sardertv.com | http://www.russellsarder.com http://www.netcomlearning.com | http://www.learninggg.com Mark Fasciano, Serial Entrepreneur, co-found... XBOX 360 99.9% FAILURE RATE!IM DONE WITH YOU MICROSOFT, YOUR CONSOLE SUCKS. ELPRESADOR, I COME TO YOU A HUMBLE MAN.. FOR I HAVE BEEN A 360 WEEBOL ALL MY LIFE... PLEASE LET ME IN THE TRE... Xbox 360 vs PS3 failure rateXbox 360 vs PS3 failure rate Which is more reliable? Watch and find out ! VEMMA SCAM | How to Beat the 95% Failure Rate | VEMMAVEMMA SCAM | Beat the 95% Industry Failure Rate | Vemma http://mlmbizreviews.com/vemma Is Vemma a scam? I have read several Vemma reviews, which questioned t... Smaller banks -- bigger failure ratehttp://www.facethefactsusa.org Community banks bore the brunt of recession-related bank failures. 85 percent of failures 2008-2011 were smaller banks, many i... Relationship Failure RateIf you have multiple failed relationships, does that mean you are less likely to have a successful long-term relationship in the future? Check out who brough... New Study Shows Hip Resurfacing has a High Failure Rate - Gerard Ryan of Parker Waichmanhttp://www.yourlawyer.com/topics/overview/DePuy-Hip-Implant-Recall-Johnson-and-Johnson Gerard Ryan of Parker Waichman discusses the possible danger you may b... why the business failure rate is so high?http://new-billionaires.com/building-an-online-business/influence-in-business-chronicles-using-liking-to-influence/ Why the Small Business Failure Rate Is 90...
 1000000 videos foundNext >
 1461 news items
 High Failure Rate In New SUV Crash Test ABC News (blog) Thu, 16 May 2013 04:18:49 -0700 High Failure Rate In New SUV Crash Test. Email Smaller Font Text Larger Text | Print. By ABC News. May 16, 2013 7:17am. By Jeanette Torres. For the first time, the Insurance Institute for Highway Safety (IIHS) has conducted crash tests on small SUVs to ... Small businesses failure rate high Sowetan Thu, 16 May 2013 01:44:17 -0700 Davies, however, dismissed criticism that the high failure rate of small businesses was exacerbated by the amount of red tape that they are burdened with, saying government imposed regulations that should not necessarily be confused with red tape. He ... New Institute aims to reduce failure rate of clinical trials PHG Foundation Tue, 14 May 2013 04:55:43 -0700 A new Target Discovery Institute based at the University of Oxford aims to identify better targets for drugs to tackle common diseases such as cancer, diabetes and Alzheimer's disease. Eight international pharmaceutical companies - including Pfizer, ... Publicize failure rate of condoms Pacific Daily News Sat, 11 May 2013 10:00:16 -0700 Perhaps the most effective thing our Department of Public Health and Social Services can do to encourage HIV/AIDS testing is to work just as hard at increasing awareness of condom effectiveness as they do at condom distribution. Here's why. After many ... Press Herald Why do failure rates vary across the state? Press Herald Sat, 18 May 2013 21:01:03 -0700 That's a failure rate of 0.3 percent. But in Sagadohoc County, 25 out of 193 inspections failed. That's a 13 percent failure rate -- the highest in the state. Cumberland County, meanwhile, is not far behind with a failure rate of nearly 11 percent ... Sedgwick County officials seek ways to reduce probation failure rate Kansas.com Sat, 04 May 2013 15:39:32 -0700 Sedgwick County officials seek ways to reduce probation failure rate. By Deb Gruver; The Wichita Eagle. Published Saturday, May 4, 2013, at 5:37 p.m.; Updated Sunday, May 5, 2013, at 10:52 a.m.. comments; Email · Print · Reprints · Tweet ... Oakville car seat clinic yields 78 per cent failure rate. InsideHalton.com Mon, 06 May 2013 14:23:39 -0700 Halton police are reporting a 78 per cent failure rate following a spot check car seat clinic, which was held Saturday at the Speers Road and Kerr Street intersection. During the clinic a total of 141 vehicles with car seats and boosters were checked ... 680 News Halton police find high failure rate during weekend car-seat check 680 News Tue, 07 May 2013 08:49:40 -0700 Thanks to a weekend car-seat clinic in Oakville, Halton police say they have helped avert some potentially dangerous situations. During a spot check police examined nearly 150 cars with car seats, finding nearly 80 per cent of the seats were improperly ...
 Limit to books that you can completely read online Include partial books (book previews) .gsc-branding { display:block; }

Oops, we seem to be having trouble contacting Twitter