Parametric Models

0

S(t)dt = ∞

0

(e^−t+ e^−2t− e^−3t)dt = 1 +1 2 −1

3 = 7 6

The mean time to failure of the stronger component is 1 and the mean time to failure of the weaker component is 1/2. The addition of the weaker component in parallel with the stronger only increases the mean time to system failure by 1/6. This is yet another illustration of the law of diminishing returns for parallel systems. 

3.4 Parametric Models

The survival patterns of a machine tool, a fuse, and an aircraft are vastly different. One would certainly not want to use the same failure time distribution with identical parameters to model these diverse lifetimes. This section introduces two distributions that are commonly used to model lifetimes. To adequately survey all the distributions currently in existence would require an entire textbook, so detailed discussion here is limited to the exponential and Weibull distributions.

3.4.1 Parameters

We begin by describing parameters, which are common to all lifetime distributions. The three most common types of parameters used in lifetime distributions are location, scale, and shape. Parameters in a lifetime distribution allow modeling of such diverse applications as light bulb failure time, patient postsurgery survival time, and the failure time of a muffler on an automobile by a single lifetime distribution (e.g., the Weibull distribution).

Location (or shift) parameters are used to shift the distribution to the left or right along the time axis. If c1 and c2 are two values of a location parameter for a lifetime distribution with survivor function S(t; c), then there exists a real constant α such that S(t; c1) =S(α + t; c2). A familiar example of a location parameter is the mean of the normal distribution.

Scale parameters are used to expand or contract the time axis by a factor ofα. If λ1and λ2 are two values for a scale parameter for a lifetime distribution with survivor function S(t; λ), then there exists a real constant α such that S(αt; λ1) =S(t; λ2). A familiar example of a scale parameter isλ in the exponential distribution. The probability density function always has the same shape, and the units on the time axis are determined by the value ofλ.

Shape parameters are appropriately named because they affect the shape of the probabil-ity densprobabil-ity function. Shape parameter values might also determine whether a distribution belongs to a particular distribution class such as IFR or DFR. A familiar example of a shape parameter isκ in the Weibull distribution.

In summary, location parameters translate survival distributions along the time axis, scale parameters expand or contract the time scale for survival distributions, and all other parameters are shape parameters.

3.4.2 Exponential Distribution

Just as the normal distribution plays an important role in classical statistics because of the central limit theorem, the exponential distribution plays an important role in reliability and

lifetime modeling because it is the only continuous distribution with a constant hazard func-tion. The exponential distribution has often been used to model the lifetime of electronic components and is appropriate when a used component that has not failed is statistically as good as a new component. This is a rather restrictive assumption. The exponential distri-bution is presented first because of its simplicity. The Weibull distridistri-bution, a more complex two-parameter distribution that can model a wider variety of situations, is presented sub-sequently. The exponential distribution has a single positive scale parameterλ, often called the failure rate, and the four lifetime distribution representations are

S(t) = e^−λt f(t) = λe^−λt h(t) = λ H(t) = λt for t ≥ 0

There are several probabilistic properties of the exponential distribution that are useful in understanding how it is unique and when it should be applied. In the properties to be outlined below, the nonnegative lifetimeT typically has the exponential distribution with parameterλ, which denotes the number of failures per unit time. The symbol ∼ means “is distributed as.” Proofs of these results are given in most reliability textbooks.

THEOREM 3.1 (Memoryless Property) IfT ∼ exponential(λ), then P [T ≥ t] = P [T ≥ t + s|T ≥ s] t ≥ 0; s ≥ 0

As shown in Figure 3.8 for λ = 1 and s = 0.5, the memoryless property indicates that the conditional survivor function for the lifetime of an item that has survived to time s is identical to the survivor function for the lifetime of a brand new item. This used-as-good-as-new assumption is very strong. The exponential lifetime model should not be applied to mechanical components that undergo wear (e.g., bearings) or fatigue (e.g., structural supports) or electrical components that contain an element that burns away (e.g., filaments) or degrades with time (e.g., batteries). An electrical component for which the exponential lifetime assumption may be justified is a fuse. A fuse is designed to fail when there is a power surge that causes the fuse to burn out. Assuming that the fuse does not undergo any weakening or degradation over time and that power surges that cause failure occur with

0.0 0.0 0.2 0.4 0.6 0.8 1.0

t

S(t)

0.5 1.0 1.5 2.0

FIGURE 3.8 The memoryless property.

Reliability 3-19 equal likelihood over time, the exponential lifetime assumption is appropriate, and a used fuse that has not failed is as good as a new one.

The exponential distribution should be applied judiciously as the memoryless property restricts its applicability. It is usually misapplied for the sake of simplicity as the statistical techniques for the exponential distribution are particularly tractable, or small sample sizes do not support more than a one-parameter distribution.

THEOREM 3.2 The exponential distribution is the only continuous distribution with the memoryless property.

This result indicates that the exponential distribution is the only continuous lifetime distribution for which the conditional lifetime distribution of a used item is identical to the original lifetime distribution. The only discrete distribution with the memoryless property is the geometric distribution.

THEOREM 3.3 If T ∼ exponential(λ), then E[T^s] = Γ(s + 1)

λ^s s > −1 where the gamma function is defined by

Γ(α) = ∞

0

x^α−1e^−xdx

When s is a nonnegative integer, this expression reduces to E[T^s] =s!/λ^s. By setting s = 1 and 2, the mean and variance can be obtained:

E[T ] = 1

λ V [T ] = 1 λ²

THEOREM 3.4 (Self-Reproducing) If T1, T2, . . ., T_n are independent, T_i∼ expo-nential(λ_i), fori = 1, 2, . . ., n, and T = min{T1, T2, . . ., T_n}, then

T ∼ exponential  _n

i=1

λi



This result indicates that the minimum of n exponential random lifetimes also has the exponential distribution, as alluded to in the previous section. This is important in two applications. First, ifn independent exponential components, each with exponential times to failure, are arranged in series, the distribution of the system failure time is also exponential with a failure rate equal to the sum of the component failure rates. When then components have the same failure rateλ, the system lifetime is exponential with failure rate nλ. Second, when there are several independent, exponentially distributed causes of failure competing for the lifetime of an item (e.g., failing by open or short circuit for an electronic item or death by various risks for a human being), the lifetime can be modeled as the minimum of the individual lifetimes associated with each cause of failure.

THEOREM 3.5 If T1, T2, . . ., T_n are independent and identically distributed

whereχ²(2n) denotes the chi-square distribution with 2n degrees of freedom.

This property is useful for determining a confidence interval for aλ based on a data set ofn independent exponential lifetimes. For instance, with probability 1 − α,

χ²_2n,1−α/2< 2λ n i=1

Ti< χ²_2n,α/2

where the left- and right-hand sides of this inequality are the α/2 and 1 − α/2 fractiles of the chi-square distribution with 2n degrees of freedom, that is, the second subscript denotes right-hand tail areas. Rearranging this expression yields a 100(1− α)% confidence interval forλ: exponential(λ) random variables, T(1), T(2), . . ., T(n) are the corresponding order statistics (the observations sorted in ascending order), Gk=T(k)− T(k−1) for k = 1, 2, . . ., n, and if T(0)= 0, then

• P [Gk≥ t] = e^{−(n−k+1)λt};t ≥ 0; k = 1, 2, . . ., n.

• G1, G2, . . ., G_n are independent.

This property is most easily interpreted in terms of a life test of n items with exponential(λ) lifetimes. Assume that the items placed on the life test are not replaced with new items when they fail. The ith item fails at time t_(i), andG_i=t_(i)− t_(i−1) is the time between the (i − 1)st and ith failure, for i = 1, 2, . . ., n, as indicated in Figure 3.9 for n = 4. The result states that these gaps (G_i’s) between the failure times are independent and exponentially distributed. The proof of this theorem relies on the memoryless property and the self-reproducing property of the exponential distribution, which implies that when theith failure occurs the time until the next failure is the minimum of n − i independent exponential random variables.

THEOREM 3.7 If T1, T2, . . ., Tn are independent and identically distributed exponential(λ) random variables and T(r)is the rth order statistic, then

E[T_(r)] = FIGURE 3.9 Order statistics and gap statistics.

Reliability 3-21 and

V [T(r)] = r k=1

1 [(n − k + 1)λ]² forr = 1, 2, . . ., n.

The expected value and variance of therth-ordered failure are simple functions of n, λ, andr. The proof of this result is straightforward, as the gaps between order statistics are independent exponential random variables from the previous theorem, and therth ordered failure on a life test of n items is the sum of the first r gaps. This result is useful in determining the expected time to complete a life test that is discontinued afterr of the n items on test fail.

THEOREM 3.8 If T1, T2, . . . are independent and identically distributed exponential(λ) random variables denoting the interevent times for a point process, then the number of events in the interval [0, t] has the Poisson distribution with parameter λt.

This property is related to the memoryless property and can be applied to a compo-nent that is subjected to shocks occurring randomly over time. It states that if the time between shocks is exponential(λ) then the number of shocks occurring by time t has the Poisson distribution with parameterλt. This result also applies to the failure time of a cold standby system ofn identical exponential components in which nonoperating units do not fail and sensing and switching are perfect. The probability of fewer thann failures by time t (the system reliability) is

n−1

k=0

(λt)^k k! e^−λt

The exponential distribution, for which the item under study does not age in a probabilis-tic sense, is the simplest possible lifetime model. Another popular distribution that arises in many reliability applications is the Weibull distribution, which is presented next.

3.4.3 Weibull Distribution

The exponential distribution is limited in applicability because of the memoryless prop-erty. The assumption that a lifetime has a constant failure rate is often too restrictive or inappropriate. Mechanical items typically degrade over time and hence are more likely to follow a distribution with a strictly increasing hazard function. The Weibull distribution is a generalization of the exponential distribution that is appropriate for modeling lifetimes hav-ing constant, strictly increashav-ing, and strictly decreashav-ing hazard functions. The four lifetime distribution representations for the Weibull distribution are

S(t) = e^−(λt)κ f(t) = κλ^κt^κ−1e^−(λt)κ h(t) = κλ^κt^κ−1 H(t) = (λt)^κ

for allt ≥ 0, where λ > 0 and κ > 0 are the scale and shape parameters of the distribution.

The hazard function approaches zero from infinity forκ < 1, is constant for κ = 1, the expo-nential case, and increases from zero whenκ > 1. Hence, the Weibull distribution can attain hazard function shapes in both the IFR and DFR classes and includes the exponential distri-bution as a special case. One other special case occurs whenκ = 2, commonly known as the Rayleigh distribution, which has a linear hazard function with slope 2λ². When 3< κ < 4,

the shape of the probability density function resembles that of a normal probability density

forr = 1, 2, . . ., the mean and variance for the Weibull distribution are μ = 1

The lifetime of a machine used continuously under known operating conditions has the Weibull distribution withλ = 0.00027 and κ = 1.55, where time is measured in hours. (Esti-mating the parameters for the Weibull distribution from a data set will be addressed subse-quently, but the parameters are assumed to be known for this example.) What is the mean time to failure and the probability the machine will operate for 5000 hours?

The mean time to failure is

μ = E[T ] = 1 The probability that the machine will operate for 5000 hours is

S(5000) = e−[(0.00027)(5000)]^1.55= 0.203  The Weibull distribution also has the self-reproducing property, although the conditions are slightly more restrictive than for the exponential distribution. IfT1, T2, . . ., T_n are inde-pendent component lifetimes having the Weibull distribution with identical shape param-eters, then the minimum of these values has the Weibull distribution. More specifically, if Ti∼ Weibull(λi,κ) for i = 1, 2, . . ., n, then

Although the exponential and Weibull distributions are popular lifetime models, they are limited in their modeling capability. For example, if it were determined that an item had a bathtub-shaped hazard function, neither of these two models would be appropriate. Dozens of other models have been developed over the years, such as the gamma, lognormal, inverse Gaussian, exponential power, and log logistic distributions. These distributions provide further modeling flexibility beyond the exponential and Weibull distributions.