Mathematical Reliability Theory: A Concise Overview

About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Mathematical Theory of Reliability: A Précis Daniel Topa daniel.m.topa.ctr@mail.mil Defense Threat Reduction Information Analysis Center (DTRIAC) December 13, 2023 DISTRIBUTION STATEMENT A. Approved for public release: distribution unlimited. DTRIAC Tutorial Mathematical Theory of Reliability 1 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Seminal Work by Barlow and Proschan DTRIAC Tutorial Mathematical Theory of Reliability 2 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Overview 1 About the Book 2 Introduction 3 Failure Distributions 4 Backup: Survey of Basic Theorems 5 References DTRIAC Tutorial Mathematical Theory of Reliability 3 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References From the Publisher First Chapters .. Fine Print... What is Mathematical Reliability? Intended Audience Topical Coverage DTRIAC Tutorial Mathematical Theory of Reliability 4 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References From the Publisher First Chapters .. What is Mathematical Reliability? Mathematical reliability refers to a body of ideas, mathematical models, and methods directed toward the solution of problems in predicting, estimating, or optimizing the probability of survival, mean life, or, more generally, life distribution of components and systems. Few (books) provide as mathematically rigorous a treatment of the required probability background as this one. DTRIAC Tutorial Mathematical Theory of Reliability 5 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References From the Publisher First Chapters .. Audience Although this book primarily addresses the interests of applied mathematicians, it will be useful to applied statisticians and probabilists as well as scientists and engineers in all disciplines ... It is frequently cited and contains original proofs not reproduced elsewhere. DTRIAC Tutorial Mathematical Theory of Reliability 6 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References From the Publisher First Chapters .. Coverage . It includes a detailed discussion of life distributions corresponding to wearout and their use in determining maintenance policies, and covers important topics such as the theory of increasing (decreasing) failure rate distributions, optimum maintenance policies, and the theory of coherent systems. DTRIAC Tutorial Mathematical Theory of Reliability 7 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References From the Publisher First Chapters .. Table of Contents DTRIAC Tutorial Mathematical Theory of Reliability 8 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Primal Error Distribution Weibull proposes distribution for “life length of materials” F (x) = 1 − e−φ(x) DTRIAC Tutorial Mathematical Theory of Reliability 9 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Primal Error Distribution Life length of materials F (x) = 1 − e−φ(x) 0≤x<∞ 0 ≤ φ(x) < ∞ 0 ≤ F (x) ≤ 1 DTRIAC Tutorial Mathematical Theory of Reliability 10 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Primal Error Distribution F (x) = 1 − e −(x−µ)m σ Choose a simple function, test it empirically... DTRIAC Tutorial Mathematical Theory of Reliability 11 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Primal Error Distribution . F (x) = 1 − e −(x−µ)m σ 1 µ=3 σ=2 m=2 0 0 1 DTRIAC Tutorial 2 Mathematical Theory of Reliability 3 12 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Failures of Electronics after discussion with electronics experts...(we turned) to ordered observations drawn from non normal distributions. Specifically, ... to ordered observations drawn from non normal distributions. 1 x f (x; θ) = e− θ , θ DTRIAC Tutorial θ > 0, x ≥ 0 Mathematical Theory of Reliability 13 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Other Distributions . And so begins an era of rich development of distribution functions DTRIAC Tutorial Mathematical Theory of Reliability 14 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Tower of Babel Crucial definitions are not consistent across literature... Working definitions follow... DTRIAC Tutorial Mathematical Theory of Reliability 15 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions System State at Time t System state at time t: X (t) = (X1 (t), . . . , Xm (t)) ∈ Rm 0 = failed, 1 = functioning DTRIAC Tutorial Mathematical Theory of Reliability 16 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Distribution function System state at time t: X (t) = (X1 (t), . . . , Xm (t)) ∈ Rm (1) Is governed by the distribution function F (x1 , . . . , xm ; t) Which is the probability that X1 ≤ x1 , . . . , Xm ≤ xm DTRIAC Tutorial Mathematical Theory of Reliability 17 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Gain For any state System state at time t: x = (x1 , . . . , xm ) There is a gain or payoff g (x). Functioning state x = 1, g(1) = 1 Failed state x = 0, g(0) = 0 DTRIAC Tutorial Mathematical Theory of Reliability 18 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Gain Expected gain: Z G(t) = Eg (X (t)) = Z ··· g (x1 , . . . , xm ) dF (x1 , . . . , xm ; t) (2) Average expected gain with respect to a weight function W (t): Z t1 H (t0 , t1 ) = G(t)dW (t) (3) t0 DTRIAC Tutorial Mathematical Theory of Reliability 19 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Reliability Reliability is the probability of a device performing ... adequately for the period of time intended... DTRIAC Tutorial Mathematical Theory of Reliability 20 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Reliability Reliability is the probability of a device performing ... adequately for the period of time intended... ∴ Probability that devices functions over [0, t]: G(t) = Reliability DTRIAC Tutorial Mathematical Theory of Reliability (4) 21 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Reliability Reliability is sometimes called availability DTRIAC Tutorial Mathematical Theory of Reliability 22 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Interval Availability Interval availability is the expected fraction of a given interval of time that the system will be able to operate within the tolerances. DTRIAC Tutorial Mathematical Theory of Reliability 23 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Interval Availability Interval availability is the expected fraction of a given interval of time that the system will be able to operate within the tolerances. Repair or replacement is allowed. DTRIAC Tutorial Mathematical Theory of Reliability 23 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Interval Availability Using the weight function W (t) = t − t0 t1 − t0 Equation (3) becomes 1 H (t0 , t1 ) = t1 − t0 Z t1 t0 1 G(t)dW (t) = t1 − t0 DTRIAC Tutorial Z t1 Eq (X(t)) dt (5) t0 Mathematical Theory of Reliability 24 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Interval Availability Using the weight function W (t) = t − t0 t1 − t0 Equation (3) becomes 1 H (t0 , t1 ) = t1 − t0 Z t1 t0 1 G(t)dW (t) = t1 − t0 DTRIAC Tutorial Z t1 Eg (X(t)) dt (6) t0 Mathematical Theory of Reliability 25 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Interval Availability With mild regularity conditions (Riemann integrability and a Lipschitz condition on bounded variation), equation (6) becomes Z t1 1 H (t0 , t1 ) = E g (X(t)) dt (7) t1 − t0 t0 which describes the expected fraction of the time interval (t0 , t1 ) that the system is functioning. H (t0 , t1 ) = interval availability DTRIAC Tutorial Mathematical Theory of Reliability (8) 26 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Interval Availability Interval availability is sometimes called efficiency. DTRIAC Tutorial Mathematical Theory of Reliability 27 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Limiting Interval Availability Limiting interval availability is the expected fraction of time of time in the long run that the system operates satisfactorily. Using (7), lim H (0, τ ) τ →∞ DTRIAC Tutorial Mathematical Theory of Reliability 28 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Interval Availability Limiting interval availability is sometimes called limiting efficiency. DTRIAC Tutorial Mathematical Theory of Reliability 29 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Interval Availability Interval reliability is the probability that at a specified time, the system is operating and will continue to operate for an interval of duration, say ∆t. DTRIAC Tutorial Mathematical Theory of Reliability 30 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Interval Availability Interval reliability is the probability that at a specified time, the system is operating and will continue to operate for an interval of duration, say ∆t. Repair or replacement is allowed. DTRIAC Tutorial Mathematical Theory of Reliability 30 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Interval Availability Interval reliability for an interval of ∆t starting at time T : R (∆t, T ) = P [X(t) = 1, T ≤ t ≤ T + ∆t] DTRIAC Tutorial Mathematical Theory of Reliability (9) 31 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References History Other Distributions Definitions Limiting Interval Availability .. Limiting interval reliability: lim R (∆t, T ) T →∞ (10) Also called strategic reliability. DTRIAC Tutorial Mathematical Theory of Reliability 32 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played What are Failure Distributions? “A failure distribution represents an attempt to describe mathematically the length of life of a material, a structure, or a device.” DTRIAC Tutorial Mathematical Theory of Reliability 33 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played What is a Failure Distribution? Modes of possible failure ... will affect the analytic form of the failure distribution DTRIAC Tutorial Mathematical Theory of Reliability 34 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Examples of Failures Mar 6, 2021 DTRIAC Tutorial Mathematical Theory of Reliability 35 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Examples of Failures Tank 610, Bhopal, India DTRIAC Tutorial Mathematical Theory of Reliability 36 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Examples of Failures Piper Alpha DTRIAC Tutorial Mathematical Theory of Reliability 37 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Examples of Failures Three Mile Island DTRIAC Tutorial Mathematical Theory of Reliability 38 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Examples of Failures Chernobyl DTRIAC Tutorial Mathematical Theory of Reliability 39 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Examples of Failure Modes static failure when fracture occurs during a single load instability of a structure caused by stored strain energy chemical corrosion such as hydrogen embrittlement fatigue due to cyclic loading sticking of mechanical assemblies electronic devices may fail with changes in time, temperature, humidity, and altitude DTRIAC Tutorial Mathematical Theory of Reliability 40 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played How to Select a Failure Distribution? “Unfortunately, the choice of a failure distribution on the basis of these physical considerations is still largely an art.” DTRIAC Tutorial Mathematical Theory of Reliability 41 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Which Failure Distribution? . “... differences among the gamma, Weibull, and log normal distribution functions become significant only in the tails of the distribution, but actual observations are sparse in the tails because of limited sample sizes.” DTRIAC Tutorial Mathematical Theory of Reliability 42 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Failure Rate Function “to discriminate among probability functions that cannot be distinguished from each other within the range of actual observation, it is necessary ... to base the differentiation among distribution functions on physical considerations. Such a concept is based on the failure rate function.” DTRIAC Tutorial Mathematical Theory of Reliability 43 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Failure Rate Function Given a failure distribution F with a density f . the failure rate function r(t) is defined over the time domain where F (t) < 1 as r(t) = DTRIAC Tutorial f (t) 1 − F (t) Mathematical Theory of Reliability (11) 44 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Probabilistic Interpretation r(t)dt is the probability that an item of age t will fail during the interval [t, t + dt] DTRIAC Tutorial Mathematical Theory of Reliability 45 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Also Known As... 1 in actuarial mathematics: force of mortality 2 in statistics: (reciprocal for the normal distribution) Mill’s ratio 3 in extreme value theory: intensity function 4 in reliability theory: hazard rate DTRIAC Tutorial Mathematical Theory of Reliability 46 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Conventional Notation Notational simplification: F (t) = 1 − F (t) DTRIAC Tutorial Mathematical Theory of Reliability (12) 47 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Conventional Notation Notational simplification: F (t) = 1 − F (t) (12) Say “F bar” or “F complement”. DTRIAC Tutorial Mathematical Theory of Reliability 47 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Intuition We expect the conditional mean residual life of a used component < new component DTRIAC Tutorial Mathematical Theory of Reliability 48 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Intuition We expect the conditional mean residual life of a used component < new component We can strengthen that condition. DTRIAC Tutorial Mathematical Theory of Reliability 48 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Decreasing Mean Residual Life F has decreasing mean residual life ⇐⇒ the mean residual life of a component of age t R∞ F (t)(τ )dτ t (13) F (t) is decreasing in time DTRIAC Tutorial Mathematical Theory of Reliability 49 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Decreasing Mean Residual Life We can strengthen the concept of decreasing mean residual life by assuming the conditional probability of failure given survival to time t is increasing in t DTRIAC Tutorial Mathematical Theory of Reliability 50 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Increasing Failure Rate Increasing failure rate (IFR): failure rate function r(t) increases with time t DTRIAC Tutorial Mathematical Theory of Reliability 51 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Decreasing Failure Rate Decreasing failure rate (DFR): failure rate function r(t) decreases with time t DTRIAC Tutorial Mathematical Theory of Reliability 52 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Either Failure Rate Both assumptions permit sharp bounds on the survival probability in terms of moments and percentiles. DTRIAC Tutorial Mathematical Theory of Reliability 53 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Either Failure Rate Both assumptions permit sharp bounds on the survival probability in terms of moments and percentiles. Bounds are much tighter than classical Chebyshev bounds. DTRIAC Tutorial Mathematical Theory of Reliability 53 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Failure Distribution Ways to quantify the failure distribution: 1 Using F (t), cumulative probability 2 Using f (t), probability density: F (t) = 3 Rt f (τ )dτ 0 Rt Using r(t), failure rate: F (t) = exp − r(τ )dτ 0 DTRIAC Tutorial Mathematical Theory of Reliability 54 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Failure Distribution Properties of the cumulative probability: lim F (t) = 1 t→∞ lim F (t) = 0 t→0− F (t) is right–continuous. DTRIAC Tutorial Mathematical Theory of Reliability 55 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Useful Failure Distributions 1 exponential 2 gamma 3 Weibull 4 modified extreme value 5 truncated normal 6 log normal (Useful resource: ReliaWiki.) DTRIAC Tutorial Mathematical Theory of Reliability 56 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Useful Failure Densities: Exponential f (t) = λe−λt ; r(t) = λ, λ > 0, t ≥ 0 1.0 0.8 0.6 0.4 0.2 0.0 0 1 DTRIAC Tutorial 2 3 4 Mathematical Theory of Reliability 57 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Useful Failure Densities: Gamma f (t) = λ (λt)α−1 e−λt ; Γ (α) λ, α > 0, t ≥ 0 1.0 0.8 0.6 0.4 0.2 0.0 0 1 DTRIAC Tutorial 2 3 4 Mathematical Theory of Reliability 58 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Useful Failure Densities: Weibull α f (t) = λαtα−1 e−λt ; λ, α > 0, t ≥ 0 4 3 2 1 0 0 1 DTRIAC Tutorial 2 3 4 Mathematical Theory of Reliability 59 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Useful Failure Densities: Modified Extreme Value t e −1 1 +t , f (t) = exp − λ λ r(t) = et ; λ λ > 0, t ≥ 0 2.0 1.5 1.0 0.5 0.0 0 1 DTRIAC Tutorial 2 3 4 Mathematical Theory of Reliability 60 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Useful Failure Densities: Truncated Normal 1 (t − µ)2 f (t) = √ exp − 2σ 2 σ 2π ! ; −∞ < µ < ∞, σ > 0, t ≥ 0 1.0 0.8 0.6 0.4 0.2 0.0 0 1 2 DTRIAC Tutorial 3 4 5 6 Mathematical Theory of Reliability 61 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Useful Failure Densities: Log Normal (ln t − µ)2 f (t) = exp − 2σ 2 tσ 2π 1 √ ! ; −∞ < µ < ∞, σ > 0, t ≥ 0 0.4 0.3 0.2 0.1 0.0 0 1 2 DTRIAC Tutorial 3 4 5 6 Mathematical Theory of Reliability 62 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Discrete Failure Distributions Discrete distributions are used for studies of number of cycles to failure DTRIAC Tutorial Mathematical Theory of Reliability 63 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Discrete Failure Distributions . 1 Geometric pk = p (1 − p)k , 0 < p < 1 2 Pascal pk = k −α α k p (−q) , p = 1 − q > 0, α ≥ 0 3 Binomial pk = nk pk (1 − p)n−k , 0 < p < 1, n ∈ Z+ , k ≤ n 4 Poisson k pk = λk! e−λ , λ > 0 k = 0, 1, 2, . . . DTRIAC Tutorial Mathematical Theory of Reliability 64 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Device Assumptions Consider a complex device with many differing components: DTRIAC Tutorial Mathematical Theory of Reliability 65 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Device Assumptions Consider a complex device with many differing components: 1 components are stochastically independent 2 every component failure causes device failure 3 broken components are immediately replaced 4 failure detection, trouble location, component replacement requires zero time DTRIAC Tutorial Mathematical Theory of Reliability 66 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Distribution of Failures What is the distribution of failures as n → ∞? DTRIAC Tutorial Mathematical Theory of Reliability 67 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Iterated limit Solution method: 1 let time t → ∞ (stationary state) 2 let n → ∞ DTRIAC Tutorial Mathematical Theory of Reliability 68 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Stationary Solution Recall from partial differential equations... DTRIAC Tutorial Mathematical Theory of Reliability 69 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Stationary Solution Fourier equation for diffusion of heat ∂t φ(x, t) = κ∆φ(x, t) (14) Solution relaxes to a steady state or equilibrium. DTRIAC Tutorial Mathematical Theory of Reliability 70 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Stationary Solution Fourier equation for diffusion of heat ∂t φ(x, t) = κ∆φ(x, t) (14) Solution relaxes to a steady state or equilibrium. lim φ(x, t) = φs (x, t) t→∞ (15) This steady state, or stationary solution, satisfies ∂t φ(x, t) = 0 DTRIAC Tutorial Mathematical Theory of Reliability (16) 70 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Stationary Solution The stationary form of the diffusion equation is ∂t φ(x, t) = κ∆φ(x, t) ⇓ 0 DTRIAC Tutorial (17) = κ∆φ(x, t) Mathematical Theory of Reliability 71 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Stationary Solution The stationary form of the diffusion equation is ∂t φ(x, t) = κ∆φ(x, t) ⇓ (17) 0 = κ∆φ(x, t) The stationary solution is described by Laplace’s equation ∆φ(x, t) = 0 DTRIAC Tutorial Mathematical Theory of Reliability (18) 71 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Stationary Solution Looking for the stationary solution is tantamount to saying “I want to find the equilibrium state and I don’t care how we got there.” DTRIAC Tutorial Mathematical Theory of Reliability 72 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Stationary Solution Looking for the stationary solution is tantamount to saying “I want to find the equilibrium state and I don’t care how we got there.” Solve this ∆φ(x, t) = 0 (19) Not this κ∆φ(x, t) = ∂t φ(x, t) DTRIAC Tutorial Mathematical Theory of Reliability (20) 72 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Stationary Probability of Survival . Probability that equipment with n components survives the interval [0, t] ∞  Z n Y  F (t)i (τ ) dτ  Gn (t) = mi i=1 DTRIAC Tutorial (21) t Mathematical Theory of Reliability 73 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Definition: Discrete Distributions A distribution is discrete if it concentrates all probability on, at most, a countable set of points. DTRIAC Tutorial Mathematical Theory of Reliability 74 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Definition: Failure Rate Failure rate: probability of failure in interval ∆t at age t F (t + ∆t) − F (t) F (t) DTRIAC Tutorial Mathematical Theory of Reliability (22) 75 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Definition: Instantaneous Failure Rate Failure rate: probability of failure in interval ∆t at age t F (t + ∆t) − F (t) F (t) (22) Instantaneous failure rate, a.k.a hazard rate: r(t) = lim ∆t→0 F (t + ∆t) − F (t) F (t)∆t DTRIAC Tutorial Mathematical Theory of Reliability (23) 76 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Definition: Failure Rate Behavior (Nondiscrete) A nondiscrete distribution F is IFR (DFR) iff F (t + ∆t) − F (t) F (t) (22) is increasing (decreasing) in ∆t > 0, t ≥ 0 such that F (t) < 1 DTRIAC Tutorial Mathematical Theory of Reliability 77 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Definition: Failure Rate Behavior (Discrete) A discrete distribution {pk }∞ k=0 is IFR (DFR) iff pk ∞ X pi (24) i=k is increasing (decreasing) in k = 0, 1, 2, . . . DTRIAC Tutorial Mathematical Theory of Reliability 78 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Discussion of IFR “We could have defined IFR without restricting t to nonnegative values; however, for DFR, we cannot extend t → −∞. Note that if F is DFR, then F (t) > 0 for t > 0. ” DTRIAC Tutorial Mathematical Theory of Reliability 79 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Discussion of IFR “We could have defined IFR without restricting t to nonnegative values; however, for DFR, we cannot extend t → −∞. Note that if F is DFR, then F (t) > 0 for t > 0. ” “The convenience of this definition is that we need not assume that F has a density. A distribution can be IFR (DFR) and have a jump at the right– (left–) hand end point of its interval of support.” DTRIAC Tutorial Mathematical Theory of Reliability 79 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Definition: Pólya Frequency Function PF2 A function p(x) defined for x ∈ R is a Pólya frequency function of order 2 (PF2 ) iff p(x) ≥ 0 for all x and p(x1 − y1 ) p (x1 − y2 ) p(x2 − y1 ) p (x2 − y2 ) ≥0 (25) whenever −∞ < x1 ≤ x2 < ∞, and −∞ < y1 ≤ y2 < ∞ DTRIAC Tutorial Mathematical Theory of Reliability 80 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Definition: Totally Positive Function TP2 A function p(x, y) defined for x ∈ R, y ∈ R is a totally positive function of order 2 (TP2 ) iff p(x, y) ≥ 0 for all x ∈ X, y ∈ Y and p(x1 , y1 ) p (x1 , y2 ) p(x2 , y1 ) p (x2 , y2 ) ≥0 (26) whenever x 1 ≤ x 2 , y 1 ≤ y2 with x1 , x2 ∈ X; y1 , y2 ∈ Y DTRIAC Tutorial Mathematical Theory of Reliability 81 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Side by Side: PF2 and TP2 Pólya frequency function PF2 totally positive function TP2 p(x) : R 7→ R p(x, y) : R2 7→ R iff p(x) ≥ 0 for all x iff p(x, y) ≥ 0 for all x ∈ X, y ∈ Y p(x1 − y1 ) p (x1 − y2 ) p(x2 − y1 ) p (x2 − y2 ) ≥0 p(x1 , y1 ) p (x1 , y2 ) p(x2 , y1 ) p (x2 , y2 ) ≥0 −∞ < x1 ≤ x2 < ∞, x1 ≤ x2 , y1 ≤ y2 −∞ < y1 ≤ y2 < ∞ x1 , x2 ∈ X; y1 , y2 ∈ Y DTRIAC Tutorial Mathematical Theory of Reliability 82 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Equivalent Statements Given F (0− ) = 0, the following statements are equivalent F is an IFR (DFR) distribution log F (t) is concave (convex) for t in {t : F (t) < 1, t ≥ 0} F (t) is PF2 (F (t)(x + y) is TP2 in x and y for x + y ≥ 0) DTRIAC Tutorial Mathematical Theory of Reliability 83 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Distributions and Percentiles Given ξp represents a pth percentile F which is IFR (DFR) then ( ≥e−at , F (t) ≤e−at , t ≤ ξp t ≥ ξp where α=− DTRIAC Tutorial ( ≤e−at , F (t) ≥e−at , t ≤ ξp t ≥ ξp log(1 − p) ξp Mathematical Theory of Reliability 84 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played If F is IFR with mean µ1 ( e−t/µ1 t < µ1 F (t) ≥ 0 t ≥ µ1 DTRIAC Tutorial Mathematical Theory of Reliability (27) 85 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Components in Series Given m components in series IFR distributions {Fk }m k=1 means {µk }m k=1 !  m X 1  exp −t µm F (t)k (t) ≥ k=1   k=1 0 m Y DTRIAC Tutorial t < min (µ1 , . . . , µm ) (28) elsewhere Mathematical Theory of Reliability 86 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Components in Parallel Given m components in parallel IFR distributions {Fk }m k=1 means {µk }m k=1 m Y k=1 (Q m F (t)k (t) ≤ k=1 1 − e−t/µk 1 DTRIAC Tutorial t < min (µ1 , . . . , µm ) elsewhere Mathematical Theory of Reliability (29) 87 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played System Reliability . system reliability is often better than predicted on a parts count DTRIAC Tutorial Mathematical Theory of Reliability 88 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Best Upper Bound on F (t) . If F is IFR with mean µ ( 1 t≤µ F (t) ≤ −ωt e t>µ (30) where ω satisfies 1 − ωµ = e−ωt DTRIAC Tutorial Mathematical Theory of Reliability (31) 89 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Best Upper Bound on F (t) If F is DFR with mean µ ( e−ωt t ≤ µ F (t) ≤ µ t≥µ et (32) where ω satisfies 1 − ωµ = e−ωt DTRIAC Tutorial Mathematical Theory of Reliability (33) 90 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Dominance Theorem Given F is IFR with mean µ G(x) = e−x/µ ϕ(x) is increasing (decreasing) Z ∞ Z ∞ ϕ(x)F (x)dx ≤ ϕ(x)G(x)dx (34) Mathematical Theory of Reliability 91 / 126 (≥) 0 0 DTRIAC Tutorial About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Sandwich Theorem Example 4.8’ 1 2 3 4 5 6 F is IFR, F (0) = 0 R∞ 0 xdF (x) = µ ϕ(x, y) is convex (concave) in y and ϕyy (x, y) ∃ ϕy (x, y) is nonincreasing (nondecreasing) in x ( 0 x<µ step function D(x) = 1 x≥µ LeibnizRrule satisfied for convexity parameter λ in ∞ I(λ) = 0 ϕ x, λe−x/µ − (1 − λ)F (x) dx Z ∞ Z ∞ ϕ(x, D(x))dx ≤ 0 Z ∞ ϕ(x, F (x))dx ≤ (≥) 0 DTRIAC Tutorial ϕ(x, e−x/µ )dx (35) (≥) 0 Mathematical Theory of Reliability 92 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Sandwich Theorem Example 4.8’ easy R∞ 0 ϕ(x, D(x))dx hard ≤ R∞ 0 ϕ(x, F (x))dx DTRIAC Tutorial easy ≤ R∞ 0 ϕ(x, e−x/µ )dx Mathematical Theory of Reliability 93 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Best Upper Bound on F (t) How the game is played: Characterization DTRIAC Tutorial Mathematical Theory of Reliability 94 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Best Upper Bound on F (t) µ=1 How the game is played: Bounding DTRIAC Tutorial Mathematical Theory of Reliability 95 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Archimedes: How to Compute π How can we compute π? DTRIAC Tutorial Mathematical Theory of Reliability 96 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Archimedes: How to Compute π How can we compute π? Area of disk = πr2 DTRIAC Tutorial Mathematical Theory of Reliability 96 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Archimedes: How to Compute π How can we compute π? Area of disk = πr2 Approximate the unit disk with a sequence of polygons. DTRIAC Tutorial Mathematical Theory of Reliability 96 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Archimedes: How to Compute π Table: Polygon sequence ⇒ ⇒ DTRIAC Tutorial Mathematical Theory of Reliability ⇒ ··· 97 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Archimedes: Sandwich Theorem Strengthen estimate using the Sandwich Theorem: Bound from above and below DTRIAC Tutorial Mathematical Theory of Reliability 98 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Archimedes: Approximate Area of Circle DTRIAC Tutorial Mathematical Theory of Reliability 99 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Archimedes: Approximate Area of Circle outer area = 4 disk area = π inner area = 2 DTRIAC Tutorial Mathematical Theory of Reliability 99 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Archimedes: Squeezing π Out of Polygons ⇒ DTRIAC Tutorial Mathematical Theory of Reliability 100 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Archimedes: Squeezing π Out of Polygons ⇒ 2<π<4 2.8 < π < 3.3 DTRIAC Tutorial Mathematical Theory of Reliability 100 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Sandwich Theorem (a.k.a. Squeeze) Table: Archimedes’ method for computing π k 1 2 3 4 5 6 7 8 9 10 11 v 2 4 8 16 32 64 128 256 512 1024 2048 outer 4 3.3 3.2 3.15 3.144 3.142 3.1417 3.1416 3.14160 3.14160 3.14159 DTRIAC Tutorial <π< <π< <π< <π< <π< <π< <π< <π< <π< <π< <π< inner 2 2.8 3.1 3.12 3.137 3.140 3.1413 3.1415 3.14157 3.14159 3.14159 Mathematical Theory of Reliability 101 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Sandwich Theorem (a.k.a. Squeeze) 4 outer polygon area π 3 inner polygon 2 1 10 100 1000 104 vertices DTRIAC Tutorial Mathematical Theory of Reliability 102 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Convergence of the Method of Archimedes Quadratic Convergence |error| 0.100 0.001 10-5 10-7 10 100 1000 104 vertices DTRIAC Tutorial Mathematical Theory of Reliability 103 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Convergence of the Method of Archimedes Quadratic Convergence |error| 0.100 0.001 10-5 10-7 work × 10 ⇒ precision / 100 10 100 1000 104 vertices DTRIAC Tutorial Mathematical Theory of Reliability 103 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Monotone Convergence Theorem Let R fn be the area of a polygon with n vertices Z Z ? lim fn = lim fn n→∞ DTRIAC Tutorial n→∞ Mathematical Theory of Reliability 104 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Failure Distributions Failure Rate Functions Exponential Failure Rates Monotone Failure Rates How the Game is Played Convergence Concepts .. DTRIAC Tutorial Mathematical Theory of Reliability 105 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates Survey of Basic Theorems Let’s look at powerful theorems which open deep insight into functions without knowing either location or dispersion parameters. In fact, we do not even need to know a functional form. DTRIAC Tutorial Mathematical Theory of Reliability 106 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates Totally Positive Functions “Many of the most useful consequences of increasing failure rate (IFR) and decreasing failure rate (DFR) derive from the happy fact that distributions with monotone failure rate are members of the class of totally positive functions.” DTRIAC Tutorial Mathematical Theory of Reliability 107 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates Total Positivity A function f (x, y) of two variables ranging over linearly ordered one dimensional sets X and Y , respectively, is said to be totally positive of order k (T Pk ) if for all x1 < x2 < · · · < xm , y1 < y2 < · · · < ym where xi ∈ X, yj ∈ Y, and all 1 ≤ m ≤ k DTRIAC Tutorial Mathematical Theory of Reliability 108 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates Total Positivity f x1 , x2 , . . . , xm y1 , y2 , . . . , ym = f (x1 , y1 ) f (x2 , y1 ) .. . f (x1 , y2 ) f (x2 , y2 ) .. . ... ... f (x1 , ym ) f (x2 , ym ) .. . ≥0 f (xm , y1 ) f (xm , y2 ) . . . f (xm , ym ) DTRIAC Tutorial Mathematical Theory of Reliability 109 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates Variation Diminishing Property A function is P F2 iff for all ∆ F (x + ∆) − F (x) f (x) is decreasing in x DTRIAC Tutorial Mathematical Theory of Reliability 110 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates Unimodality If f (u) is P F2 then f (u) is unimodal. DTRIAC Tutorial Mathematical Theory of Reliability 111 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates Unimodality: In Other Words... If f (u) is P F2 then f (u) has a single highest value. DTRIAC Tutorial Mathematical Theory of Reliability 112 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates Unimodality: Application If f is P F2 , and x < y, and r ≥ 1, and ∆ ≥ 0, then F (y) − F (y − r∆) F (x) − F (x − ∆) ≥r , x − r∆ ≥ m, f (y) f (x) F (x + r∆) − F (x) F (y + ∆) − F (y) ≥r , y + r∆ ≤ m f (x) f (y) (36) Where m is the mode of f . DTRIAC Tutorial Mathematical Theory of Reliability 113 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates Test for Increasing Failure Rate Let X1 , X2 , . . . , Xn , be a sample of independent observations from the common distribution F with density f , where f (t) = 0 for t < 0, and failure rate r(t). Distinguish between: 1 2 r is constant: H0 , Null hypothesis r is nondecreasing, but not constant: H1 , Alternative hypothesis DTRIAC Tutorial Mathematical Theory of Reliability 114 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates Test for Increasing Failure Rate The point is that we can distinguish between H0 and H1 with specific confidence levels. DTRIAC Tutorial Mathematical Theory of Reliability 115 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates General Failure Rate What can we do if the failure rate is not monotone? DTRIAC Tutorial Mathematical Theory of Reliability 116 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates General Failure Rate What can we do if the failure rate is not monotone? A great deal! DTRIAC Tutorial Mathematical Theory of Reliability 116 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates General Failure Rate For example, Limiting value of failure rate helps to determine the properties of the failure distribution Failure distributions of interest have finite moments of all orders DTRIAC Tutorial Mathematical Theory of Reliability 117 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates General Failure Rate For 0 < 1/α ≤ r(t) ≤ 1/β ≤ ∞ ∀t, then Z ∞ µs = xs f (x)dx < ∞, s > −1 (37) 0 e−tβ ≤ F (t) ≤ e−tα (38) e−tβ e−tα ≤ f (t) ≤ α β (39) β s ≤ λs ≤ α s (40) inf r(t) ≤ t DTRIAC Tutorial 1 ≤ sup r(t) µ1 t Mathematical Theory of Reliability (41) 118 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates General Failure Rate Properties of moments of IFR distributions are true under weaker conditions DTRIAC Tutorial Mathematical Theory of Reliability 119 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates General Failure Rates If r(x) ≥ α ∀x ≥ 0 and R∞ 0 xf (x)dx = µ1 , Then F (t) ≤ And ( e−αt α1 µ1 e−αt 1−e−αt t ≤ − α1 log (1 − αµ1 ) = t0 t ≥ t0 ( αµ − 1 + e−αt F (t) ≥ 0 DTRIAC Tutorial (42) t ≤ t0 t ≥ t0 (43) Mathematical Theory of Reliability 120 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Totally Positive Functions Variation Diminishing Property Unimodality Testing for Increasing Failure Rate General Failure Rates General Failure Rates Is this format better? αµ − 1 + e−αt ≤ F (t) ≤ 0 ≤ F (t) ≤ DTRIAC Tutorial e−αt α1 µ1 e−αt 1−e−αt t ≤ t0 t ≥ t0 Mathematical Theory of Reliability (44) 121 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Monographs and Texts Mathematical Perspective Online Monographs and Texts 1965 ⇑ You are here 1961 DTRIAC Tutorial 2013 Mathematical Theory of Reliability 2022 122 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Monographs and Texts Mathematical Perspective Online Mathematical Perspective 1977 2011 DTRIAC Tutorial 2013 Mathematical Theory of Reliability 123 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Monographs and Texts Mathematical Perspective Online Open Source Office of Scientific and Technical Information One in a Million, Given the Accident: Assuring Nuclear Weapon Safety, SAND2014-16545J A Robust Approach to Nuclear Weapon Safety, SAND2011-4123C Nuclear Weapon Reliability Evaluation Methodology, SAND2002-8133 DOE Nuclear Weapon Reliability Definition: History, Description, and Implementation, SAND99-8240 Government Accountability Office Senior Leaders Should Emphasize Key Practices to Improve Weapon System Reliability, GAO-20-151 Nuclear Weapons: Technical Exceptions and Limitations Do Not Constrain DOD’s Planning and Operations, GAO-23-1056713 DTRIAC Tutorial Mathematical Theory of Reliability 124 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Monographs and Texts Mathematical Perspective Online Open Source A New View of Weapon System Reliability and Maintainability, RAND, 1989 Electronic Materials Aging, JASON, 2020 Aging Electronics May Limit Nuke Reliability, Federation of American Scientists, 2021 Tutorial: Parametric Reliability Models*, IDA, 2018 Reliability-based Design, Development based Design, Development and Sustainment Sustainment*, ARA, 2007 Software Reliability Growth Approach*, Raytheon, 2010 * Sourced through DTIC website DTRIAC Tutorial Mathematical Theory of Reliability 125 / 126 About the Book Introduction Failure Distributions Backup: Survey of Basic Theorems References Monographs and Texts Mathematical Perspective Online Mathematical Theory of Reliability: A Précis Daniel Topa daniel.m.topa.ctr@mail.mil Defense Threat Reduction Information Analysis Center (DTRIAC) December 13, 2023 DTRIAC Tutorial Mathematical Theory of Reliability 126 / 126

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