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Reliability Application

Systems Engineering Program. Department of Engineering Management, Information and Systems. EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS. Reliability Application. Dr. Jerrell T. Stracener, SAE Fellow. Leadership in Engineering.

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Reliability Application

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  1. Systems Engineering Program Department of Engineering Management, Information and Systems EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Reliability Application Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering

  2. An Application of Probability to Reliability Modeling and Analysis

  3. Reliability Definitions and Concepts • Figures of merit • Failure densities and distributions • The reliability function • Failure rates • The reliability functions in terms of the failure rate • Mean time to failure (MTTF) and mean time between failures (MTBF)

  4. What is Reliability? • Reliability is defined as the probability that an item will perform its intended function for a specified interval under stated conditions. In the simplest sense, reliability means how long an item (such as a machine) will perform its intended function without a breakdown. • Reliability: the capability to operate as intended, whenever used, for as long as needed. Reliability is performance over time, probability that something will work when you want it to.

  5. Reliability Figures of Merit • Basic or Logistic Reliability • MTBF - Mean Time Between Failures • measure of product support requirements • Mission Reliability • Ps or R(t) - Probability of mission success • measure of product effectiveness

  6. Reliability Humor: Statistics “If I had only one day left to live, I would live it in my statistics class -- it would seem so much longer.” From: Statistics A Fresh Approach Donald H. Sanders McGraw Hill, 4th Edition, 1990

  7. The Reliability of an item is the probability that the item will survive time t, given that it had not failed at time zero, when used within specified conditions, i.e., The Reliability Function

  8. Reliability Relationship between failure density and reliability

  9. Relationship Between h(t), f(t), F(t) and R(t) Remark: The failure rate h(t) is a measure of proneness to failure as a function of age, t.

  10. The reliability of an item at time t may be expressed in terms of its failure rate at time t as follows: where h(y) is the failure rate The Reliability Function

  11. Mean Time to Failure and Mean Time Between Failures • Mean Time to Failure (or Between Failures) MTTF (or MTBF) • is the expected Time to Failure (or Between Failures) • Remarks: • MTBF provides a reliability figure of merit for expected failure • free operation • MTBF provides the basis for estimating the number of failures in • a given period of time • Even though an item may be discarded after failure and its mean • life characterized by MTTF, it may be meaningful to • characterize the system reliability in terms of MTBF if the • system is restored after item failure.

  12. Relationship Between MTTF and Failure Density If T is the random time to failure of an item, the mean time to failure, MTTF, of the item is where f is the probability density function of time to failure, iff this integral exists (as an improper integral).

  13. Relationship Between MTTF and Reliability

  14. Reliability “Bathtub Curve”

  15. Reliability Humor

  16. The Exponential Model: (Weibull Model with β = 1) • Definition • A random variable T is said to have the Exponential • Distribution with parameters, where  > 0, if the • failure density of T is: • , for t  0 • , elsewhere

  17. Probability Distribution Function • Weibull W(b, q) • , for t  0 • Where F(t) is the population proportion failing in time t • Exponential E(q) = W(1, q)

  18. The Exponential Model Remarks The Exponential Model is most often used in Reliability applications, partly because of mathematical convenience due to a constant failure rate. The Exponential Model is often referred to as the Constant Failure Rate Model. The Exponential Model is used during the ‘Useful Life’ period of an item’s life, i.e., after the ‘Infant Mortality’ period before Wearout begins. The Exponential Model is most often associated with electronic equipment.

  19. Reliability Function • Probability Distribution Function • Weibull • Exponential

  20. 1.0 0.8 0.6 0.4 0.2 0 β=5.0 R(t) β=1.0 β=0.5 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 t t is in multiples of The Weibull Model - Distributions Reliability Functions

  21. Mean Time Between Failure - MTBF Weibull Exponential

  22. The Gamma Function Values of the Gamma Function

  23. Weibull • and, in particular • Exponential Percentiles, tp

  24. Failure Rate • a decreasing function of t if < 1Notice that h(t) is a constant if = 1 • an increasing function of t if > 1 • Cumulative Failure Rate • The Instantaneous and Cumulative Failure Rates, h(t) • and H(t), are straight lines on log-log paper. Failure Rates - Weibull

  25. Failure Rates - Exponential • Failure Rate • Note: • Only for the Exponential Distribution • Cumulative Failure

  26. 3 2 1 0 β=5 h(t) β=1 β=0.5 0 1.0 2.0 t t is in multiples of h(t) is in multiples of 1/  The Weibull Model - Distributions Failure Rates

  27. The Binomial Model - Example Application 1 • Problem - • Four Engine Aircraft • Engine Unreliability Q(t) = p = 0.1 • Mission success: At least two engines survive • Find RS(t)

  28. The Binomial Model - Example Application 1 • Solution - • X = number of engines failing in time t • RS(t) = P(x  2) = b(0) + b(1) + b(2) • = 0.6561 + 0.2916 + 0.0486 = 0.9963

  29. E1 E2 En Series Reliability Configuration • Simplest and most common structure in reliability analysis. • Functional operation of the system depends on the successful operation of all system components Note: The electrical or mechanical configuration may differ from the reliability configuration • Reliability Block Diagram • Series configuration with n elements: E1, E2, ..., En • System Failure occurs upon the first element failure

  30. E1 E2 En Series Reliability Configuration with Exponential Distribution • Reliability Block Diagram • Element Time to Failure Distribution • with failure rate , for i=1, 2,…, n • System reliability • where is the system failure rate • System mean time to failure

  31. E1 E2 En Series Reliability Configuration • Reliability Block Diagram • Identical and independent Elements • Exponential Distributions • Element Time to Failure Distribution • with failure rate • System reliability

  32. System mean time to failure Note thatq/n is the expected time to the first failure, E(T1), when n identical items are put into service Series Reliability Configuration

  33. E1 E2 En • Parallel Reliability Configuration – Basic Concepts • Definition - a system is said to have parallel reliability configuration if the system function can be performed by any one of two or more paths • Reliability block diagram - for a parallel reliability configuration consisting of n elements, E1, E2, ... En

  34. Parallel Reliability Configuration • Redundant reliability configuration - sometimes called a redundant reliability configuration. Other times, the term ‘redundant’ is used only when the system is deliberately changed to provide additional paths, in order to improve the system reliability • Basic assumptions • All elements are continuously energized starting at time t = 0 • All elements are ‘up’ at time t = 0 • The operation during time t of each element can be described • as either a success or a failure, i.e. Degraded operation or • performance is not considered

  35. Parallel Reliability Configuration System success - a system having a parallel reliability configuration operates successfully for a period of time t if at least one of the parallel elements operates for time t without failure. Notice that element failure does not necessarily mean system failure.

  36. E1 E2 En • Parallel Reliability Configuration • Block Diagram • System reliability - for a system consisting of n elements, E1, E2, ... En if the n elements operate independently of each other and where Ri(t) is the reliability of element i, for i=1,2,…,n

  37. System Reliability Model - Parallel Configuration • Product rule for unreliabilities • Mean Time Between System Failures

  38. Parallel Reliability Configuration s p=R(t)

  39. Parallel Reliability Configuration with Exponential Distribution • Element time to failure is exponential with failure rate  • Reliability block diagram: • Element Time to Failure Distribution • with failure rate for I=1,2. E1 E2 • System reliability • System failure rate

  40. Parallel Reliability Configuration with Exponential Distribution • System Mean Time Between Failures: • MTBFS = 1.5 

  41. Example A system consists of five components connected as shown. Find the system reliability, failure rate, MTBF, and MTBM if Ti~E(λ) for i=1,2,3,4,5 E2 E1 E3 E4 E5

  42. Solution This problem can be approached in several different ways. Here is one approach: There are 3 success paths, namely, Success PathEvent E1E2 A E1E3 B E4E5 C Then Rs(t)=Ps= =P(A)+P(B)+P(C)-P(AB)-P(AC)-P(BC)+P(ABC) =P(A)+P(B)+P(C)-P(A)P(B)-P(A)P(C)-P(B)P(C)+ P(A)P(B)P(C) =P1P2+P1P3+P4P5-P1P2P3-P1P2P4P5 -P1P3P4P5+P1P2P3P4P5 assuming independence and where Pi=P(Ei) for i=1, 2, 3, 4, 5

  43. Since Pi=e-λt for i=1,2,3,4,5 Rs(t) hs(t)

  44. MTBFs

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