# 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).

# 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.

# 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.

# Poisson Distribution

This distribution is used quite frequently in reliability analysis. It can be considered an extension of the binomial distribution when n is infinite. It can be used to approximate the binomial distribution when n > 20 and p < 0.05.

If events are Poisson distributed, they occur at a constant average rate and the number of events occurring in any time interval are independent of the number of events occurring in any other time interval. For example, the number of failures in a given time would be given by:

# Binomial Distribution

The binomial distribution is used for those situations in which there are only two outcomes, such as success or failure, and the probability remains the same for all trials. It is very useful in reliability and quality assurance work. The probability density function (pdf) of the binomial distribution is

# Weibull Distribution

1. The Weibull distribution is particularly useful in reliability work since it is a general distribution which, by adjustment of the distribution parameters, can be made to model a wide range of life distribution characteristics of different classes of engineered items. One of the versions of the failure density function is

# Gamma Distribution

The gamma distribution is used in reliability analysis for cases where partial failures can exist, i.e., when a given number of partial failures must occur before an item fails (e.g., redundant systems) or the time to second failure when the time to failure is exponentially distributed. The failure density function is

for t>0

# Exponential Distribution

This is probably the most important distribution in reliability work and is used almost exclusively for reliability prediction of electronic equipment. It describes the situation wherein the hazard rate is constant which can be shown to be generated by a Poisson process. This distribution is valuable if properly used. It has the advantages of:

1. single, easily estimated parameter (λ)
2. mathematically very tractable
3. fairly wide applicability
4. is additive  that is, the sum of a number of independent exponentially distributed variables is exponentially distributed.

Some particular applications of this model include:

1. items whose failure rate does not change significantly with age.
2. complex and repairable equipment without excessive amounts of redundancy.
3. equipment for which the early failures or “infant mortalities” have been eliminated by “burning in” the equipment for some reasonable time period.

The failure density function is

for t > 0, where λ is the hazard (failure) rate, and the reliability function is

the mean life (θ) = 1/λ, and, for repairable equipment the MTBF = θ = 1/λ .

# Normal Distribution

There are two principal applications of the normal (or Gaussian) distribution to reliability. One application deals with the analysis of items which exhibit failure due to wear, such as mechanical devices. Frequently the wearout failure distribution is sufficiently close to normal that the use of this distribution for predicting or assessing reliability is valid.

The other application deals with the analysis of manufactured items and their ability to meet specifications. No two parts made to the same specification are exactly alike. The variability of parts leads to a variability in systems composed of those parts. The design must take this part variability into account, otherwise the system may not meet the specification requirement due to the combined effect of part variability. Another aspect of this application is in quality control procedures.

The basis for the use of normal distribution in this application is the central limit theorem which states that the sum of a large number of identically distributed random variables, each with finite mean and variance, is normally distributed. Thus, the variations in value of electronic component parts, for example, due to manufacturing are considered normally distributed.

The failure density function for the normal distribution is
Equ. 1

where
μ = the population mean
σ = the population standard deviation, which is the square root of
the variance.

# Lognormal Distribution

The lognormal distribution is the distribution of a random variable whose natural logarithm is distributed normally; in other words, it is the normal distribution with In t as the variate. The density function is

for t ≥ 0