Chapter 2 Failure Models Part 1: Introduction

Marvin Rausand Department of Production and Quality Engineering Norwegian University of Science and Technology [email protected]

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 1 / 31 Introduction Rel. measures Life distributions State variable Time to failure Distr. function Probability density Distribution of T Reliability funct. Failure rate Introduction Bathtub curve Some formulas MTTF Example 2.1 Median Mode MRL Example 2.2 Discrete distributions

Life distributions

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 2 / 31 Reliability measures

Introduction In this chapter we introduce the following measures: Rel. measures Life distributions State variable Time to failure ■ Distr. function The reliability (survivor) function R(t) Probability density ■ The failure rate function z(t) Distribution of T ■ Reliability funct. The mean time to failure (MTTF) Failure rate ■ The mean residual life (MRL) Bathtub curve Some formulas MTTF Example 2.1 of a single item that is not repaired when it fails. Median Mode MRL Example 2.2 Discrete distributions

Life distributions

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 3 / 31 Life distributions

Introduction The following life distributions are discussed: Rel. measures Life distributions State variable ■ The exponential distribution Time to failure ■ Distr. function The gamma distribution Probability density ■ The Weibull distribution Distribution of T ■ Reliability funct. The normal distribution Failure rate ■ The lognormal distribution Bathtub curve Some formulas ■ The Birnbaum-Saunders distribution MTTF ■ The inverse Gaussian distributions Example 2.1 Median Mode MRL In addition we cover three discrete distributions: Example 2.2 Discrete distributions ■ The binomial distribution Life distributions ■ The Poisson distribution ■ The geometric distribution

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 4 / 31 State variable

Introduction X(t) Failure Rel. measures Life distributions 1 State variable Time to failure Distr. function Probability density Distribution of T 0 Reliability funct. Time to failure, T t Failure rate Bathtub curve Some formulas MTTF 1 if the item is functioning at time t Example 2.1 X(t) = Median  0 if the item is in a failed state at time t Mode MRL Example 2.2 The state variable X(t) and the time to failure T will generally Discrete be random variables. distributions

Life distributions

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 5 / 31 Time to failure

Introduction Different time concepts may be used, like Rel. measures Life distributions State variable Time to failure ■ Calendar time Distr. function ■ Operational time Probability density Distribution of T ■ Number of kilometers driven by a car Reliability funct. ■ Failure rate Number of cycles for a periodically working item Bathtub curve ■ Number of times a switch is operated Some formulas MTTF ■ Number of rotations of a bearing Example 2.1 Median In most applications we will assume that the time to failure T is a Mode MRL continuous random variable (Discrete variables may be Example 2.2 approximated by a continuous variable) Discrete distributions

Life distributions

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 6 / 31 Distribution function

Introduction The distribution function of T is Rel. measures Life distributions t State variable F (t) = Pr(T ≤ t) = f(u) du for t > 0 Time to failure Z Distr. function 0 Probability density Distribution of T Note that Reliability funct. F (t) = Probability that the item will fail within the interval (0,t] Failure rate Bathtub curve Some formulas MTTF Example 2.1 Median Mode MRL Example 2.2 Discrete distributions

Life distributions

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 7 / 31 Probability density function

Introduction The probability density function (pdf) of T is Rel. measures Life distributions State variable d F (t +∆t) − F (t) Pr(t < T ≤ t +∆ Time to failure f(t) = F (t)= lim = lim ∆t→∞ ∆t→∞ Distr. function dt ∆t ∆t Probability density Distribution of T When ∆t is small, then Reliability funct. Failure rate Pr(t < T ≤ t +∆t) ≈ f(t) · ∆t Bathtub curve Some formulas MTTF Example 2.1 Median ∆t Mode MRL 0 t Time Example 2.2 Discrete distributions When we are standing at time t =0 and ask: What is the Life distributions probability that the item will fail in the interval (t,t +∆t]? The answer is approximately f(t) · ∆t

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 8 / 31 Distribution of T

Introduction Rel. measures Life distributions State variable Time to failure Distr. function Probability density Distribution of T Reliability funct. Failure rate Bathtub curve Some formulas MTTF Example 2.1 Median Mode MRL ■ The area under the pdf-curve (f(t)) is always 1, Example 2.2 ∞ Discrete f(t) dt =1 distributions 0 ■ TheR area under the pdf-curve to the left of t is equal to F (t) Life distributions ■ The area under the pdf-curve between t1 and t2 is F (t2) − F (t1) = Pr(t1 < T ≤ t2)

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 9 / 31 Reliability Function

Introduction Rel. measures Life distributions State variable Time to failure Distr. function Probability density Distribution of T Reliability funct. Failure rate Bathtub curve Some formulas MTTF Example 2.1 ∞ Median Mode R(t) = Pr(T >t)=1 − F (t) = f(u) du MRL Zt Example 2.2 Discrete distributions ■ R(t) = The probability that the item will not fail in (0,t] Life distributions ■ R(t) = The probability that the item will survive at least to time t ■ R(t) is also called the survivor function of the item

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 10 / 31 Failure rate function

Introduction Consider the conditional probability Rel. measures Life distributions State variable Time to failure Pr(t < T ≤ t +∆t) Distr. function Pr(t < T ≤ t +∆t | T >t) = Probability density Pr(T >t) Distribution of T F (t +∆t) − F (t) Reliability funct. = Failure rate R(t) Bathtub curve Some formulas MTTF The failure rate function of the item is Example 2.1 Median Pr(t < T ≤ t +∆t | T >t) z(t) = lim Mode → MRL ∆t 0 ∆t Example 2.2 F (t +∆t) − F (t) 1 f(t) = lim · = Discrete → distributions ∆t 0 ∆t R(t) R(t) Life distributions When ∆t is small, we have

Pr(t < T ≤ t +∆t | T >t) ≈ z(t) · ∆t

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 11 / 31 Failure Rate Function (2)

Introduction ∆t Rel. measures Life distributions State variable 0 t Time Time to failure Distr. function Probability density Distribution of T Reliability funct. ■ Note the difference between the failure rate function z(t) and Failure rate Bathtub curve the probability density function f(t). Some formulas ■ When we follow an item from time 0 and note that it is still MTTF Example 2.1 functioning at time t, the probability that the item will fail Median during a short interval of length ∆t after time t is z(t) · ∆t Mode MRL ■ The failure rate function is a “property” of the item and is Example 2.2 sometimes called the force of mortality (FOM) of the item. Discrete distributions

Life distributions

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 12 / 31 Bathtub curve

Introduction Rel. measures z(t) Life distributions State variable Time to failure Distr. function Probability density Distribution of T Reliability funct. Failure rate Burn-in Wear-out Bathtub curve period Useful life period period Some formulas MTTF 0 Time t Example 2.1 Median Mode MRL Example 2.2 Discrete distributions

Life distributions

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 13 / 31 Some formulas

Introduction Expressed Rel. measures Life distributions by F (t) f(t) R(t) z(t) State variable Time to failure t t Distr. function F (t) = – f(u) du 1 − R(t) 1 − exp − z(u) du Probability density Z0 „ Z0 « Distribution of T t Reliability funct. d d Failure rate f(t) = F (t) – − R(t) z(t) · exp − z(u) du dt dt „ 0 « Bathtub curve Z Some formulas ∞ t MTTF − − Example 2.1 R(t)= 1 F (t) f(u) du – exp z(u) du Zt „ Z0 « Median Mode MRL dF (t)/dt f(t) d z(t) = ∞ − ln R(t) – Example 2.2 − 1 F (t) t f(u) du dt Discrete R distributions

Life distributions

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 14 / 31 Mean time to failure The mean time to failure, MTTF, of an item is ∞ Introduction Rel. measures MTTF = E(T ) = tf(t) dt (1) Life distributions Z0 State variable ′ Time to failure Since f(t) = −R (t), Distr. function Probability density ∞ ′ Distribution of T MTTF = − tR (t) dt Reliability funct. Z0 Failure rate Bathtub curve Some formulas By partial integration MTTF ∞ Example 2.1 ∞ Median MTTF = − [tR(t)]0 + R(t) dt Mode Z0 MRL ∞ Example 2.2 If MTTF < ∞, it can be shown that [tR(t)]0 =0. In that case Discrete distributions ∞ Life distributions MTTF = R(t) dt (2) Z0

It is often easier to determine MTTF by (2) than by (1).

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 15 / 31 Example 2.1

Introduction Consider an item with survivor function Rel. measures 1 Life distributions R(t) = for t ≥ 0 State variable (0.2 t + 1)2 Time to failure Distr. function Probability density where the time t is measured in months. The probability density Distribution of T function is Reliability funct. Failure rate ′ 0.4 Bathtub curve f(t) = −R (t) = Some formulas (0.2 t + 1)3 MTTF Example 2.1 Median and the failure rate function is Mode f(t) 0.4 MRL z(t) = = Example 2.2 R(t) 0.2 t +1 Discrete distributions The mean time to failure is: Life distributions ∞ MTTF = R(t) dt =5 months Z0

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 16 / 31 Median

Introduction Rel. measures Life distributions 0,08 Mode State variable Median Time to failure 0,06 Distr. function 0,04 MTTF

Probability density f(t) Distribution of T Reliability funct. 0,02 Failure rate Bathtub curve 0,00 0 5 10 15 20 25 Some formulas The median life tm is defined by MTTF Time t Example 2.1 Median R(tm)=0.50 Mode MRL Example 2.2 The median divides the distribution in two halves. The item will Discrete fail before time tm with 50% probability, and will fail after time distributions tm with 50% probability. Life distributions The mode of a life distribution is the most likely failure time, that is, the time tmode where the probability density function f(t) attains its maximum.: Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 17 / 31 Mode

Introduction The mode of a life distribution is the most likely failure time, Rel. measures Life distributions that is, the time tmode where the probability density function f(t) State variable Time to failure attains its maximum.: Distr. function Probability density Distribution of T Reliability funct. Failure rate Bathtub curve Some formulas MTTF Example 2.1 Median Mode MRL Example 2.2 Discrete distributions

Life distributions

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 18 / 31 Mean residual life

Introduction Consider an item that is put into operation at time t =0 and is Rel. measures Life distributions still functioning at time t. The probability that the item of age t State variable Time to failure survives an additional interval of length x is Distr. function Probability density Pr(T >x + t) R(x + t) Distribution of T R(x | t) = Pr(T >x + t | T >t) = = Reliability funct. Pr(T >t) R(t) Failure rate Bathtub curve R(x | t) is called the conditional survivor function of the item at Some formulas MTTF age t. Example 2.1 Median Mode The mean residual (or, remaining) life, MRL(t), of the item at MRL age t is Example 2.2 ∞ ∞ Discrete 1 distributions MRL(t) = µ(t) = R(x | t) dx = R(x) dx Life distributions Z0 R(t) Zt

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 19 / 31 Example 2.2 Consider an item with failure rate function z(t) = t/(t + 1). The failure rate function is increasing and approaches 1 when t →∞. Introduction The corresponding survivor function is Rel. measures Life distributions t State variable u −t Time to failure R(t) = exp − du =(t + 1) e  Z0 u +1  Distr. function ∞ Probability density −t Distribution of T MTTF = (t + 1) e dt =2 Reliability funct. Z0 Failure rate Bathtub curve The conditional survival function is Some formulas MTTF (t + x + 1) e−(t+x) t + x +1 Example 2.1 R(x | t) = Pr(T >x + t | T >t) = = e Median (t + 1) e−t t +1 Mode MRL Example 2.2 The mean residual life is ∞ Discrete 1 distributions MRL(t) = R(x | t) dx = 1 + Life distributions Z0 t +1 We see that MRL(t) is equal to 2 (= MTTF) when t =0, that MRL(t) is a decreasing function in t, and that MRL(t) → 1 when t →∞. Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 20 / 31 Introduction Discrete distributions Binomial Geometric Poisson process

Life distributions Discrete distributions

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 21 / 31 Binomial distribution The binomial situation is defined by:

Introduction 1. We have n independent trials. ∗ Discrete 2. Each trial has two possible outcomes A and A . distributions Binomial 3. The probability Pr(A) = p is the same in all the n trials. Geometric Poisson process The trials in this situation are sometimes called Bernoulli trials. Life distributions Let X denote the number of the n trials that have outcome A. The distribution of X is

n − Pr(X = x) = px(1 − p)n x for x =0, 1,...,n x

n n! where x = x!(n−x)! is the binomial coefficient.
The distribution is called the binomial distribution (n, p), and we sometimes write X ∼ bin(n, p). The mean value and the variance of X are

E(X) = np var(X) = np(1 − p)

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 22 / 31 Geometric distribution

Assume that we carry out a sequence of Bernoulli trials, and want Introduction to find the number Z of trials until the first trial with outcome A. Discrete ∗ distributions If Z = z, this means that the first (z − 1) trials have outcome A , Binomial and that the first A will occur in trial z. The distribution of Z is Geometric Poisson process z−1 Life distributions Pr(Z = z)=(1 − p) p for z =1, 2,... This distribution is called the geometric distribution. We have that

Pr(Z>z)=(1 − p)z

The mean value and the variance of Z are 1 E(Z) = p 1 − p var(X) = p2

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 23 / 31 The homogeneous Poisson process (1)

Introduction Consider occurrences of a specific event A, and assume that Discrete distributions Binomial 1. The event A may occur at any time in the interval, and the Geometric probability of A occurring in the interval (t,t +∆t] is Poisson process independent of t and may be written as λ · ∆t + o(∆t), Life distributions where λ is a positive constant. 2. The probability of more that one event A in the interval (t,t +∆t] is o(∆t). 3. Let (t11,t12], (t21,t22],... be any sequence of disjoint intervals in the time period in question. Then the events “A occurs in (tj1,tj2],” j =1, 2,..., are independent. Without loss of generality we let t =0 be the starting point of the process.

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 24 / 31 The Homogeneous Poisson Process (2)

Introduction Let N(t) denote the number of times the event A occurs during Discrete distributions the interval (0,t]. The stochastic process {N(t),t ≥ 0} is called Binomial Geometric a Homogeneous Poisson Process (HPP) with rate λ. Poisson process

Life distributions The distribution of N(t) is n (λt) − Pr(N(t) = n) = e λt for n =0, 1, 2,... n! The mean and the variance of N(t) are ∞ E(N(t)) = n · Pr(N(t) = n) = λt nX=0 var(N(t)) = λt

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 25 / 31 Introduction Discrete distributions

Life distributions Exponential Weibull

Life distributions

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 26 / 31 Exponential distribution Consider an item that is put into operation at time t =0. Assume that the time to failure T of the item has probability Introduction Discrete density function (pdf) distributions −λt Life distributions λe for t > 0, λ > 0 Exponential f(t) = Weibull  0 otherwise This distribution is called the exponential distribution with parameter λ, and we sometimes write T ∼ exp(λ).

The survivor function of the item is ∞ − R(t) = Pr(T >t) = f(u) du = e λt for t > 0 Zt The mean and the variance of T are ∞ ∞ − 1 MTTF = R(t) dt = e λt dt = Z0 Z0 λ var(T ) = 1/λ2

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 27 / 31 Exponential distribution (2)

The failure rate function is Introduction −λt Discrete f(t) λe distributions z(t) = = −λt = λ Life distributions R(t) e Exponential Weibull The failure rate function is hence constant and independent of time. Consider the conditional survivor function Pr(T >t + x) R(x | t) = Pr(T >t + x | T >t) = Pr(T >t) −λ(t+x) e − = = e λx = Pr(T >x) = R(x) e−λt A new item, and a used item (that is still functioning), will therefore have the same probability of surviving a time interval of length t. A used item is therefore stochastically as good as new.

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 28 / 31 Weibull distribution The time to failure T of an item is said to be Weibull distributed with parameters α and λ [ T ∼ Weibull(α, λ) ] if the distribution Introduction Discrete function is given by distributions −(λt)α Life distributions 1 − e for t > 0 Exponential F (t) = Pr(T ≤ t) = Weibull  0 otherwise The corresponding probability density function (pdf) is

α d αλαtα−1e−(λt) for t > 0 f(t) = F (t) = dt  0 otherwise

1,5 α = 0.5 α = 3 1,0 α = 2 f(t) 0,5 α = 1

0,0 0,00,5 1,01,5 2,02,5 3,0

Time t Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 29 / 31 Weibull distribution (2)

Introduction The survivor function is Discrete distributions −(λt)α Life distributions R(t) = Pr(T > 0) = e for t > 0 Exponential Weibull and the failure rate function is

f(t) − z(t) = = αλαtα 1 for t > 0 R(t)

2,0 α = 3 α = 2 1,5

α = 1 1,0 z(t)

α = 0.5 0,5

0,0 0,00,2 0,40,6 0,81,0

Marvin Rausand, March 14, 2006 SystemTime t Reliability Theory (2nd ed), Wiley, 2004 – 30 / 31 Weibull Distribution (3)

Introduction The mean time to failure is Discrete ∞ distributions 1 1 Life distributions MTTF = R(t) dt = Γ +1 Exponential Z0 λ α  Weibull The median life tm is

1 1/α R(tm)=0.50 ⇒ tm = (ln2) λ The variance of T is 1 2 1 var(T ) = Γ +1 − Γ2 +1 λ2  α  α 

Note that MTTF/ var(T ) is independent of λ. p

Marvin Rausand, March 14, 2006 System Reliability Theory (2nd ed), Wiley, 2004 – 31 / 31