System Reliability Analysis - Concepts and Metrics

1. Systems Reliability, Supportability and Availability Analysis System Reliability Analysis- Concepts and Metrics

2. 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)

3. Reliability Concepts, Principles and Methodology Hardware Software Operator Service Product Production/Manufacturing Processes and Equipment Product and Customer Support Systems

4. What is Reliability? To the user of a product, reliability is problem free operation Reliability is a function of stress To understand reliability, understand stress on hardware - where its going to be used - how its going to be used - what environment it is going to be used in To efficiently achieve reliability, rely on analytical understanding of reliability and less on understanding reliability through testing Field Problems Stress/Design, Parts and Workmanship

5. Definitions of Reliability Reliability is a measure of the capability of a system, equipment or component to operate without failure when in service. Reliability provides a quantitative statement of the chance that an item will operate without failure for a given period of time in the environment for which it was designed. In its simplest and most general form, reliability is the probability of success. To perform reliability calculations, reliability must first be defined explicitly. It is not enough to say that reliability is a probability. A probability of what?

6. More Definitions of Reliability Reliability is defined as the probability that an item will perform its intended unction 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.

7. Definitions of Reliability • Essential elements needed to define reliability are: • What does it do? • System, subsystem, equipment or component functions • What is satisfactory performance? • Figures of merit @ System • Allocations &/or derived @ subsystem, equipment & component • How long does it need to function? Life: required number of operational units (time, sorties, cycles, etc) • What are conditions under which it operates? • Environment • Operation • Maintenance • Support

8. 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

10. Basic Reliability Design and development Basic reliability is a measure of serial reliability or logistics reliability and reflects all elements in a system

11. Basic Reliability Measures Air Force MFHBF - Mean Flight Hours Between Failures MFHBUM - MFHB Unscheduled Maintenance Army MFHBE - Mean Flight Hours Between Events Navy MFHBF - Mean Flight Hours Between Failures MFHBMA - MFHB Maintenance Actions Automotive Industry Number of defects per 100 vehicles

13. Mission Reliability Mission Reliability is defined as the probability that a system will perform its mission essential functions during a specified mission, given that all elements of the system are in an operational state at the start of the mission. Measure Ps or R(t) - Probability of mission success based on: Mission Essential Functions Mission Essential Equipment Mission Operating Environment Mission Length

14. Basic Elements of Reliability Modeling & Analysis Reliability is a probability Therefore a working knowledge of probability, random variables and probability distributions is required for: - Development of reliability models - Performing reliability analyses An understanding of the concepts of probability is required for design and support decisions

15. 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

16. Failure Density Function associated with a continuous random variable T, the time to failure of an item, is a function f, called the probability density function, or in reliability, the failure density. The function f has the following properties: for all values of t and

17. Failure Distribution Function The failure distribution function or, the probability distribution function is the cumulative proportion of the population failing in time t, i.e.,

18. Failure Distribution Function The failure distribution function, F, has the following properties: 1. F is nondecreasing, i.e., if 0  t1 < t2 < , then F(t1)  F(t2), 2. 0  F(t)  1 for all t 3. in general, but here F(0) = 0 4. 5. P(a < T  b) = F(b) - F(a)

19. Remark The time to failure distribution has a special name and symbol in reliability. It is called the unreliability and is denoted by Q, i.e. Q(t) = F(t) = P(T  t)

20. Failure Densities and Distributions f(t) Area = P(t1 < T <t2) t 0 F(t) 1 F(t2) P(t1 < T < t2) = F(t2) - F(t1) F(t1) t 0 t1 t2

21. Percentile The 100pth percentile, 0 < p < 1, of the time to failure probability distribution function, F, is the time, say tp, within which a proportion, p, of the items has failed, i.e., tp is the value of t such that F(tp) = P(T  tp) = p or tp = F-1(p) F(t) p tp

22. f(t) p 0 t tp Reliability In terms of the failure density, f, of an item, the 100pth percentile, tp, is

23. The Reliability Function 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.,

24. Properties of the Reliability Function • R is a non-increasing function, i.e., if 0  t1 < t2 < , then R(t1)  R(t2) 2) 0  R(t)  1 for all t 3) R(t) = 1 at t = 0 4)

25. Properties of the Reliability Function The probability of failure in a given time interval, t1 to t2, can be expressed in terms of either reliability or unreliability functions, i.e., P(t1 < T < t2) = R(t1) - R(t2) = F(t2) – F(t1)

26. Reliability Relationship between failure density and reliability

27. Failure Rate Remark: The failure rate h(t) is a measure of proneness to failure as a function of age, t.

28. Properties of the Failure Rate The (instantaneous) failure rate, h, has the following properties: 1. h(t)  0 , t  0 and 2.

29. The Reliability Function 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

30. Cumulative Failure Rate The cumulative failure rate at time t, H(t), is the cumulative number of failures at time t, divided by the cumulative time, t, i.e., The average failure rate of an item over an interval of time from t1 to t2, where t1 < t2, is the number of failures occurring in the interval (t1, t2), divided by the interval length, t2 - t1

31. 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.

32. MTTF MTTF (Mean Time to Failure) or MTBF (Mean Time Between Failures) may be determined from the time to failure probability density function by use of three equivalent methods: 1. definition of MTBF 2. moment generating functions 3. characteristic function

33. 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).

34. Relationship Between MTTF and Reliability

35. Reliability “Bathtub Curve”

36. Reliability Humor

37. Example • If f(t) = e-t for t  0, • Verify that f(t) is a failure density and derive the mathematical expression for: • b. R(t) • c. MTBF • d. h(t) and H(t) • e. tp • f. Show that P(T > t1 + t2 | T > t1) = P(T > t2)

38. If f(t) = e-t for t  0, a. therefore, f(t) is a failure density b.

39. c. or d.

40. Since • , since

41. f. But so that

42. Following the same argument so therefore

43. B t1 t2 A  B = A t1+t2 A