Reliability theory
Oxford Dictionary of Statistics:
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reliability theory
Theory concerned with determining the probability that a system (with n components) is working. Let xj=1 if the jth component is working and let xj=0 if it has failed. The vector x=(x1 x2...xn) is called the state vector. The function ϕ(x), which takes the value 1 when the system is working and 0 when it has failed, is called the structure function. For n components in series,ϕ(x)=min{x1, x2,..., xn}=x1×x2×...×xn.For n components in parallel,ϕ(x)=max{x1, x2,..., xn}=1-(1-x1)(1-x2)...(1-xn).If ϕ(x)=1 then x is a path vector: it traces a set of connected working components. If failure of any of its working components results in system failure, the vector is a minimal path vector. Correspondingly, if ϕ(x)=0 then x is a cut vector and, if it is the case that repair of any of the failed components in x leads to the system working, then the vector is a minimal cut vector.
If pj denotes the probability that the jth component continues to work during the next unit of time, then the probability that a structure consisting of n components in series continues to work is p1×p2×···×pn with the corresponding probability for n components in parallel being
1-(1-p1)(1-p2)...(1-pn).
Reliability theory. Complicated systems can always be subdivided into combinations of components arranged either in series or in parallel.
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Reliability theory
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Reliability theory
Reliability theory describes the probability of a system completing its expected function during an interval of time. It is the basis of reliability engineering, which is an area of study focused on optimizing the reliability, or probability of successful functioning, of systems, such as airplanes, linear accelerators, and any other product. It developed apart from the mainstream of probability and statistics. It was originally a tool to help nineteenth century maritime insurance and life insurance companies compute profitable rates to charge their customers. Even today, the terms "failure rate" and "hazard rate" are often used interchangeably.
The failure of mechanical devices such as ships, trains, and cars, is similar in many ways to the life or death of biological organisms. Statistical models appropriate for any of these topics are generically called "time-to-event" models. Death or failure is called an "event", and the goal is to project or forecast the rate of events for a given population or the probability of an event for an individual.
When reliability is considered from the perspective of the consumer of a technology or service, actual reliability measures may differ dramatically from perceived reliability. One bad experience can be magnified in the mind of the customer, inflating the perceived unreliability of the product. One plane crash where hundreds of passengers die will immediately instill fear in a large percentage of the flying consumer population, regardless of actual reliability data about the safety of air travel.
Reliability period of any object is measured within the durability period of that object.
See also
- Actuarial science
- Availability
- Catastrophe modeling
- Extreme value theory
- Failure rate
- Fault tree
- Gompertz law
- Gompertz-Makeham law of mortality
- Reliability (statistics)
- Reliability engineering
- Reliability theory of aging and longevity
- Survival analysis
- Weibull distribution
External links
- Gavrilov LA, Gavrilova NS (December 2001). "The reliability theory of aging and longevity". J. Theor. Biol. 213 (4): 527–45. doi:10.1006/jtbi.2001.2430. PMID 11742523. http://linkinghub.elsevier.com/retrieve/pii/S0022-5193(01)92430-0.
- Reliability and Availability Basics
- System Reliability and Availability
- reliability database
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Oxford Dictionary of Statistics
A Dictionary of Statistics. Second edition revised. Copyright © Oxford University Press, 2008. All rights reserved.
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