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 ■ RThe 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.