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Reliability Assessment

Reliability is a critical attribute for the modern technological components and systems. Uncertainty exists on the failure occurrence of a component or system, and proper mathematical methods are developed and applied to quantify such uncertainty. The ultimate goal of reliability engineering is to quantitatively assess the probability of failure of the target component or system [1]. In general, reliability assessment can be carried out by both parametric or nonparametric techniques. This chapter offers a basic introduction to the related definitions, models and computation methods for reliability assessments.

1.1 Definitions of Reliability

According to the standard ISO 8402, reliability is the ability of an item to perform a required function, under given environmental and operational conditions and for a stated period of time without failure. The term "item" refers to either a component or a system. Under different circumstances, the definition of reliability can be interpreted in two different ways:

1.1.1 Probability of Survival

Reliability of an item can be defined as the complement to its probability of failure, which can be estimated statistically on the basis of the number of failed items in a sample. Suppose that the sample size of the item being tested or monitored is n0. All items in the sample are identical, and subjected to the same environmental and operational conditions. The number of failed items is nf and the number of the survived ones is ns, which satisfies

The percentage of the failed items in the tested sample is taken as an estimate of the unreliability, ,

Complementarily, the estimate of the reliability, R^(t), of the item is given by the percentage of survived components in the sample:

Example 1.1

A valve fabrication plant has an average output of 2,000 parts per day. Five hundred valves are tested during a reliability test. The reliability test is held monthly. During the past three years, 3,000 valves have failed during the reliability test. What is the reliability of the valve produced in this plant according to the test conducted?

Solution

The total number of valves tested in the past three years is

The number of failed components is

According to Equation 1.3, an estimate of the valve reliability is

1.1.2 Probability of Time to Failure

Let random variable T denote the time to failure. Then, the reliability function at time t can be expressed as the probability that the component does not fail at time t, that is,

Denote the cumulative distribution function (cdf) of T as F(t). The relationship between the cdf and the reliability is

Further, denote the probability density function (pdf) of failure time T as f(t). Then, equation (1.5) can be rewritten as

In all generality, the expected value or mean of the time to failure T is called the mean time to failure (MTTF), which is defined as

It is equivalent to

Another related concept is the mean time between failures (MTBF). MTBF is the average working time between two consecutive failures. The difference between MTBF and MTTF is that the former is used only in reference to a repairable item, while the latter is used for non-repairable items. However, MTBF is commonly used for both repairable and non-repairable items in practice.

The failure rate function or hazard rate function, denoted by h(t), is defined as the conditional probability of failure in the time interval [t, t+?t] given that it has been working properly up to time t, which is given by

Furthermore, the cumulative failure rate function, or cumulative hazard function, denoted by H(t), is given by

Example 1.2 The failure time of a valve follows the exponential distribution with parameter h(t) (in arbitrary units of time-1). The value is new and functioning at time h(t). Calculate the reliability of the valve at time h(t) (in arbitrary units of time).

Solution

The pdf of the failure time of the valve is

The reliability function of the valve is given by

At time, the value of the reliability is

1.2 Component Reliability Modeling

As mentioned in the previous section, in reliability engineering, the time to failure of an item is a random variable. In this section, we briefly introduce several commonly used discrete and continuous distributions for component reliability modeling.

1.2.1 Discrete Probability Distributions

If random variable X can take only a finite number k of different values x1,x2,.,xk or an infinite sequence of different values x1,x2,., the random variable X has a discrete probability distribution. The probability mass function (pmf) of X is defined as the function f such that for every real number x,

If x is not one of the possible values of X, then f(x)=0. If the sequence x1,x2,. includes all the possible values of X, then Sif(xi)=1. The cdf is given by

1.2.1.1 Binomial Distribution

Consider a machine that produces a defective item with probability p (0<p<1) and produces a non-defective item with probability 1-p. Assume the events of defects in different items are mutually independent. Suppose the experiment consists of examining a sample of n of these items. Let X denote the number of defective items in the sample. Then, the random variable X follows a binomial distribution with parameters n and p and has the discrete distribution represented by the pmf in (1.14), shown in Figure 1.1. The random variable with this distribution is said to be a binomial random variable, with parameters n and p,

Figure 1.1 The pmf of the binomial distribution with n=5, p=0.4.

The pmf of the binomial distribution is

For a binomial distribution, the mean, µ, is given by

and the variance, s2, is given by

1.2.1.2 Poisson Distribution

Poisson distribution is widely used in quality and reliability engineering. A random variable X has the Poisson distribution with parameter ?, ?>0, the pmf (shown in Figure 1.2) of X is as follows:

Figure 1.2 The pmf of the Poisson distribution with ?=0.6.

The mean and variance of the Poisson distribution are

1.2.2 Continuous Probability Distributions

We say that a random variable X has a continuous distribution or that X is a continuous random variable if there exists a nonnegative function f, defined on the real line, such that for every interval of real numbers (bounded or unbounded), the probability that X takes a value in an interval [a, b] is the integral of f over that interval, that is,

If X has a continuous distribution, the function f will be the probability density function (pdf) of X. The pdf must satisfy the following requirements:

The cdf of a continuous distribution is given by

The mean, µ, and variance, s2, of the continuous random variable are calculated by

1.2.2.1 Exponential Distribution

A random variable T follows the exponential distribution if and only if the pdf (shown in Figure 1.3) of T is

Figure 1.3 The pdf of the exponential distribution with ?=1.

where  ?>0 is the parameter of the distribution. The cdf of the exponential distribution is

If T denotes the failure time of an item with exponential distribution, the reliability function will be

The hazard rate function is

The mean, µ, and variance, s2 are

1.2.2.2 Weibull Distribution

A random variable T follows the Weibull distribution if and only if the pdf (shown in Figure 1.4) of T is

Figure 1.4 The pdf of the Weibull distribution with ß=1.79, ?=1.

where  ß>0 is the shape parameter and ?>0 is the scale parameter of the distribution. The cdf of the Weibull distribution is

If T denotes the time to failure of an item with Weibull distribution, the reliability function will be

The hazard rate function is

The mean, µ, and variance, s2, are

1.2.2.3 Gamma Distribution

A random variable T follows the gamma distribution if and...