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

2. 2 An Application of Probability to Reliability Modeling and Analysis

3. 3 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) Reliability Definitions and Concepts

4. 4 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. What is Reliability?

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

6. 6

7. 7 “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 Reliability Humor: Statistics

8. 8 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

9. 9 Relationship between failure density and reliability Reliability

10. 10 Remark: The failure rate h(t) is a measure of proneness to failure as a function of age, t. Relationship Between h(t), f(t), F(t) and R(t)

11. 11 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

12. 12 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. Mean Time to Failure and Mean Time Between Failures

13. 13 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). Relationship Between MTTF and Failure Density

14. 14 Relationship Between MTTF and Reliability

15. 15 Reliability “Bathtub Curve”

16. 16 Reliability Humor

17. 17 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 The Exponential Model: (Weibull Model with β = 1)

18. 18 Weibull W( ,  ),for t  0 Where F(t) is the population proportion failing in time t Exponential E(  ) = W(1,  ) Probability Distribution Function

19. 19 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. The Exponential Model

20. 20 Probability Distribution Function Weibull Exponential Reliability Function

21. 21 Reliability Functions R(t) t t is in multiples of  β=5.0 β=1.0 β=0.5 1.0 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 The Weibull Model - Distributions

22. 22 Weibull Exponential Mean Time Between Failure - MTBF

23. 23 The Gamma Function  Values of the Gamma Function

24. 24 Weibull and, in particular Exponential Percentiles, t p

25. 25 Failure Rate a decreasing function of t if  < 1 Notice 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

26. 26 Failure Rate Note: Only for the Exponential Distribution Cumulative Failure Failure Rates - Exponential

27. 27 Failure Rates h(t) t t is in multiples of  h(t) is in multiples of 1/  32103210 0 1.0 2.0 β=5 β=1 β=0.5 The Weibull Model - Distributions

28. 28 Problem - Four Engine Aircraft Engine Unreliability Q(t) = p = 0.1 Mission success: At least two engines survive Find R S (t) The Binomial Model - Example Application 1

29. 29 Solution - X = number of engines failing in time t R S (t) = P(x  2) = b(0) + b(1) + b(2) = 0.6561 + 0.2916 + 0.0486 = 0.9963 The Binomial Model - Example Application 1

30. 30 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: E 1, E 2,..., E n System Failure occurs upon the first element failure E1E1 E2E2 EnEn Series Reliability Configuration

31. 31 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 E1E1 E2E2 EnEn Series Reliability Configuration with Exponential Distribution

32. 32 Reliability Block Diagram  Identical and independent Elements  Exponential Distributions Element Time to Failure Distribution with failure rate System reliability E1E1 E2E2 EnEn Series Reliability Configuration

33. 33 System mean time to failure Note that  /n is the expected time to the first failure, E(T 1 ), when n identical items are put into service Series Reliability Configuration

34. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 34 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, E 1, E 2,... E n E1E1 E2E2 EnEn

35. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 35 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

36. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 36 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.

37. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 37 Parallel Reliability Configuration Block Diagram System reliability - for a system consisting of n elements, E 1, E 2,... E n if the n elements operate independently of each other and where R i (t) is the reliability of element i, for i=1,2,…,n E1E1 E2E2 EnEn

38. Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08 38 System Reliability Model - Parallel Configuration Product rule for unreliabilities Mean Time Between System Failures

39. 39 Parallel Reliability Configuration s p=R(t)

40. 40 Element time to failure is exponential with failure rate Reliability block diagram: Element Time to Failure Distribution with failure rate for I=1,2. System reliability System failure rate E1E1 E2E2 Parallel Reliability Configuration with Exponential Distribution

41. 41 System Mean Time Between Failures: MTBF S = 1.5  Parallel Reliability Configuration with Exponential Distribution

42. 42 A system consists of five components connected as shown. Find the system reliability, failure rate, MTBF, and MTBM if T i ~E(λ) for i=1,2,3,4,5 E1E1 E2E2 E3E3 E4E4 E5E5 Example

43. 43 This problem can be approached in several different ways. Here is one approach: There are 3 success paths, namely, Success PathEvent E 1 E 2 A E 1 E 3 B E 4 E 5 C Then R s (t)=P s = =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) =P 1 P 2 +P 1 P 3 +P 4 P 5 -P 1 P 2 P 3 -P 1 P 2 P 4 P 5 -P 1 P 3 P 4 P 5 +P 1 P 2 P 3 P 4 P 5 assuming independence and where P i =P(E i ) for i=1, 2, 3, 4, 5 Solution

44. 44 Since P i =e -λt for i=1,2,3,4,5 R s (t) h s (t)

45. 45 MTBF s