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# Reliability math and the exponential distribution

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A short review of reliability functions commonly used for life data analysis. Plus a detailed review of the exponential distribution.

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### Reliability math and the exponential distribution

1. 1. Chet Haibel ©2013 Hobbs Engineering Corporation Reliability Math and the Exponential Distribution 0 0
2. 2. Chet Haibel ©2013 Hobbs Engineering Corporation General Reliability Function, R(t) Fraction of a group surviving until a certain time. Probability of one unit surviving until a certain time. Monotonic downward, assuming failed things stay failed. 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 1
3. 3. Chet Haibel ©2013 Hobbs Engineering Corporation General Cumulative Distribution Function, F(t) Fraction of a group failing before a certain time. Probability of one unit failing before a certain time. Monotonic upward, assuming failed things stay failed. What fraction fails at 20 (arbitrary time units)? 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 2
4. 4. Chet Haibel ©2013 Hobbs Engineering Corporation The probability of one unit failing between two times is found by subtracting the CDF for the two times, in this case perhaps 0.49 minus 0.41 equals 0.08 or 8%. 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 General Cumulative Distribution Function, F(t) 3
5. 5. Chet Haibel ©2013 Hobbs Engineering Corporation This is the time derivative (the slope) of F(t). This is a histogram of when failures occur. The height is the fraction failing per unit time. 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0 10 20 30 40 50 General Probability Density Function, f(t) 4
6. 6. Chet Haibel ©2013 Hobbs Engineering Corporation The height is the fraction failing per unit time. The area under the pdf between two times gives the probability of failure during this time interval, in this case perhaps 0.008 high by 10 units long for an area of 8%. 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0 10 20 30 40 50 General Probability Density Function, f(t) 5
7. 7. Chet Haibel ©2013 Hobbs Engineering Corporation General Hazard Function, h(t) The conditional probability density of failure, given that the product has survived up to this point in time. The fraction of the survivors failing per unit time. 0 5 10 15 20 0 10 20 30 40 50 6
8. 8. Chet Haibel ©2013 Hobbs Engineering Corporation 7 Bathtub Curve Operating Time (t) HazardRate-h(t) Useful-Life Failures Early-Life Failures aka infant mortalities Wear-Out Failures aka end-of-life failures Life to the Beginning of Wear-Out Random-in-Time Failures
9. 9. Chet Haibel ©2013 Hobbs Engineering Corporation Summary of Functions Always True R(t) = Reliability function probability of surviving until some time F(t) = Cumulative distribution function probability of failure before some time R(t) + F(t) = 1 f(t) = Probability density function failure rate h(t) = Hazard function = f(t) / R(t) conditional (normalized) failure rate   dt F(t)d  8
10. 10. Chet Haibel ©2013 Hobbs Engineering Corporation Relations Among Functions Always True All functions are not defined for t < 0 All parts, subassemblies, products, and systems are assumed to be working at t = 0 R(t) & F(t) are probabilities and lie between 0 and 1 R(t) & F(t) are monotonic, assuming things “stay failed” f(t) the pdf is the derivative of the CDF where h(t)   dt F(t)d     t 0 dh eR(t)  R(t) f(t)  9
11. 11. Chet Haibel ©2013 Hobbs Engineering Corporation Is the most widely used (and sometimes misused) failure distribution for reliability analysis for complex electronic systems. Is applicable when the hazard rate is constant. The hazard rate is the surviving fraction failing per unit of time or equivalent; such as percent per million cycles, or failures per billion (109) hours (FITs). Requires the knowledge of only one parameter for its application. Is used to describe steady-state failure rate conditions. Models device performance after the Early-Life (infant mortality) period and prior to the Wear-Out (end-of-life) period of the Bathtub Curve. The Exponential Failure Distribution 10
12. 12. Chet Haibel ©2013 Hobbs Engineering Corporation Life of 10 Constant Hazard Rate Devices Life in Hours 11
13. 13. Chet Haibel ©2013 Hobbs Engineering Corporation Sort from First to Last 10 Constant Hazard Rate Devices Life in Hours 12 How do the devices know to fail this way?
14. 14. Chet Haibel ©2013 Hobbs Engineering Corporation Fraction Surviving over Time Exponential Distribution Exponential Distribution  = Constant Hazard Rate t etR  )( 13 Life in Hours
15. 15. Chet Haibel ©2013 Hobbs Engineering Corporation 14 Reliability Function Exponential Distribution R(t) = e-t Extending the initial slope intersects the x-axis at the mean MTTF = 1/ Life in Hours
16. 16. Chet Haibel ©2013 Hobbs Engineering Corporation R(t) = e-t F(t) = 1-e-t f(t) = e-t h(t) =          θ t e         θ t e θ 1         θ t e1 θ 1  Functions Exponential Distribution 15
17. 17. Chet Haibel ©2013 Hobbs Engineering Corporation Functions Exponential Distribution pdf Operating Time t f t e t ( )     CDF Operating Time t PROBABILITY DENSITY FUNCTION CULMULATIVE DISTRIBUTION FUNCTION F t e t ( )    1  t pdf Operating Time t f t e t ( )     CDF Operating Time t PROBABILITY DENSITY FUNCTION CULMULATIVE DISTRIBUTION FUNCTION F t e t ( )    1  R(t) Operating Time t THE RELIABILITY FUNCTION R t e t ( )   h(t) Operating Time t THE HAZARD RATE  Operating Time t Operating Time t PROBABILITY DENSITY FUNCTION CULMULATIVE DISTRIBUTION FUNCTION R(t) Operating Time t THE RELIABILITY FUNCTION R t e t ( )   h(t) Operating Time t THE HAZARD RATE  1 ------  ------ h(t) =  1 ------ 16
18. 18. Chet Haibel ©2013 Hobbs Engineering Corporation For REPAIRABLE products, there is no limit to how many failures one can have! Percent of Failures Over Time (shown for 3%per month failure rate) 0% 20% 40% 60% 80% 100% 120% 140% 160% 180% 200% 0 12 24 36 48 60 72 84 96 Months in Service Repairable Non-Repairable H(t) F(t) Repairable & Non Repairable Systems Exponential Distribution 17
19. 19. Chet Haibel ©2013 Hobbs Engineering Corporation Mean is: Standard deviation is: To get the “point estimate” of  λ 1 θμ  failuresofnumbertotal not)or(failedhoursofnumbertotal or MTBFMTTFθ  λ 1 θσ  Mean & Standard Deviation Exponential Distribution 18