Weibull Prediction of Future Failures

This is an example of a recently published in the Reliability Analytics Toolkit called Weibull Prediction of Future Failures. This tool is based on work described in references 1 and 2. For a population of N items placed on test, this tool calculates the expected number of failures for some future time interval based on the following two inputs:
1. the estimated Weibull shape parameter and
2. some number of failures (X>=1) during the initial time interval (t1).

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Reliability Allocation

The first step in the design process is to translate the overall system reliability requirement into reliability requirements for each of the subsystems. This process is known as reliability allocation. The allocation of system reliability involves solving the basic inequality:


is the is the allocation reliability parameter for the ith subsystem

R* is the system reliability requirement parameter

f is the functional relationship between subsystem and system

For a simple series system in which the R’s represent probability of
survival for t hours, the above equation becomes

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Reliability Specification

The first step in the reliability engineering process is to specify the required reliability that the equipment/system must be designed to achieve. The essential elements of a reliability specification are:

  1. a quantitative statement of the reliability requirement.
  2. a full description of the environment in which the equipment/system will be stored, transported, operated and maintained.
  3. the time measure or mission profile.
  4. a clear definition of what constitutes failure.
  5. a description of the test procedure with accept/reject criteria that will be used to demonstrate the specified reliability.

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Maintainability Theory

In reliability, one is concerned with designing an item to last as long as possible without failure; in maintainability, the emphasis is on designing an item so that a failure can be corrected as quickly as possible. The combination of high reliability and high maintainability results in high system availability. Maintainability, then, is a measure of the ease and rapidity with which a system or equipment can be restored to operational status following a failure. It is a function of the equipment design and installation, personnel availability in the required skill levels, adequacy of maintenance procedures and test equipment, and the physical environment under which maintenance is performed. As with reliability, maintainability parameters are also probabilistic and are analyzed by the use of continuous and discrete random variables, probabilistic parameters, and statistical distributions. An example of a discrete maintainability parameter is the number of maintenance actions completed in some time t, whereas an example of a continuous maintainability parameter is the time to complete a maintenance action.

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Reliability Modeling: k out of n Configutations

A system consisting of n components or subsystems, of which only k need to be functioning for system success, is called a “k-out-of-n” configuration. For such a system, k is less than n. An example of such a system might be an air traffic control system with n displays of which k must operate to meet the system reliability requirement.

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Reliability Modeling: Series Configuration

The reliability functions of some simple, well known structures will be derived. These functions are based upon the exponential distribution of time to failure.

Series Configurations

The simplest and perhaps most commonly occurring configuration in reliability mathematical modeling is the series configuration. The successful operation of the system depends on the proper functioning of all the system components. A component failure represents total system failure. A series reliability configuration is represented by the block diagram as shown below with n components.

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Bathtub Curve

Figure 1 shows a typical time versus failure rate curve for equipment. This is the well known “bathtub curve,” which, over the years, has become widely accepted by the reliability community.

It has proven to be particularly appropriate for electronic equipment and systems. Note that it displays the three failure rate patterns, a decreasing failure rate (DFR), constant failure rate (CFR), and an increasing failure rate (IFR).

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Failure Modeling

Failure modeling is a key to reliability engineering. Validated failure rate models are essential to the development of prediction techniques, allocation procedures, design and analysis methodologies, test and demonstration procedures, control procedures, etc. In other words, all of the elements needed as inputs for sound decisions to insure that an item can be designed and manufactured so that it will perform satisfactorily and economically over its useful life.

Inputs to failure rate models are operational field data, test data, engineering judgment, and physical failure information. These inputs are used by the reliability engineer to construct and validate statistical failure rate models (usually having one of the distributional forms described previously) and to estimate their parameters.

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