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Dependability Theory and Methods Part 1: Introduction and definitions

Andrea Bobbio Dipartimento di Informatica Universit à del Piemonte Orientale, “ A. Avogadro ” 15100 Alessandria (Italy) bobbio@unipmn.it - http://www.mfn.unipmn.it/~bobbio. Dependability Theory and Methods Part 1: Introduction and definitions. Bertinoro, March 10-14, 2003.

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Dependability Theory and Methods Part 1: Introduction and definitions

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  1. Andrea Bobbio Dipartimento di Informatica Università del Piemonte Orientale, “A. Avogadro” 15100 Alessandria (Italy) bobbio@unipmn.it - http://www.mfn.unipmn.it/~bobbio Dependability Theory and MethodsPart 1: Introduction and definitions Bertinoro, March 10-14, 2003 Bertinoro, March 10-14, 2003

  2. Dependability: Definition Dependability is the property of a system to be dependable in time, i.e. such that reliance can justifiably be placed on the service it delivers. Dependability extends the interest on the system from the design and construction phase to the operational phase (life cycle). Bertinoro, March 10-14, 2003

  3. What dependability theory and practice wants to avoid Bertinoro, March 10-14, 2003

  4. fault forecasting fault tolerance fault removal fault prevention means faults errors failures threats Dependability: Taxonomy reliability availability maintainability safety security measures dependability Bertinoro, March 10-14, 2003

  5. Quantitative analysis The quantitative analysis aims at numerically evaluating measures to characterize the dependability of an item: • Risk assessment and safety • Design specifications • Technical assistance and maintenance • Life cycle cost • Market competition Bertinoro, March 10-14, 2003

  6. Risk assessment and safety The risk associated to an activity is given proportional to the probability of occurrence of the activity and to the magnitute of the consequences. R = P  M A safety critical system is a system whose incorrect behavior may cause a risk to occur, causing undesirable consequences to the item, to the operators, to the population, to the environment. Bertinoro, March 10-14, 2003

  7. Design specifications • Technological items must be dependable. • Some times, dependability requirements (both qualitative and quantitative) are part of the design specifications: • Mean time between failures • Total down time Bertinoro, March 10-14, 2003

  8. Technical assistance and maintenance The planning of all the activity related to the technical assistance and maintenance is linked to the system dependability (expected number of failure in time). • planning spare parts and maintenance crews; • cost of the technical assistance (warranty period); • preventive vs reactive maintenance. Bertinoro, March 10-14, 2003

  9. Market competition • The choice of the consumers is strongly influenced by the perceived dependability. • advertisement messages stress the dependability; • the image of a product or of a brand may depend on the dependability. Bertinoro, March 10-14, 2003

  10. Understanding a system Observation Operational environment Reasoning Predicting the behavior of a system Need a model A model is a convenient abstraction Accuracy based on degree of extrapolation Purpose of evaluation Bertinoro, March 10-14, 2003

  11. Measurement-Based Most believable, most expensive Not always possible or cost effective during system design Methods of evaluation • Model-Based • Less believable, Less expensive • Analytic vs Discrete-Event Simulation • Combinatorial vs State-Space Methods Bertinoro, March 10-14, 2003

  12. Most believable, most expensive; Data are obtained observing the behavior of physical objects. field observations; measurements on prototypes; measurements on components (accelerated tests). Measurement-Based Bertinoro, March 10-14, 2003

  13. Models • Closed-form • Answers • Numerical • Solution • Analytic • Simulation All models are wrong; some models are useful Bertinoro, March 10-14, 2003

  14. Measurements + Models data bank Methods of evaluation Bertinoro, March 10-14, 2003

  15. The probabilistic approach The mechanisms that lead to failure a technological object are very complex and depend on many physical, chemical, technical, human, environmental … factors. The time to failure cannot be expressed by a determin-istic law. We are forced to assume the time to failure as a random variable. The quantitative dependability analysis is based on a probabilistic approach. Bertinoro, March 10-14, 2003

  16. Reliability The reliability is a measurable attribute of the dependability and it is defined as: The reliability R(t) of an item at time t is the probability that the item performs the required function in the interval (0 – t) given the stress and environmental conditions in which it operates. Bertinoro, March 10-14, 2003

  17. LetXbe the random variable representing the time to failure of an item. Basic Definitions: cdf Thecumulative distribution function (cdf) F(t)of the r.v. X is given by: F(t) = Pr { X  t } F(t) represents the probability that the item is already failed at time t (unreliability) . Bertinoro, March 10-14, 2003

  18. Equivalent terminoloy for F(t): CDF (cumulative distribution function) Probability distribution function Distribution function Basic Definitions: cdf Bertinoro, March 10-14, 2003

  19. Basic Definitions: cdf F(t) 1 F(b) F(a) 0 a b t F(0) = 0 lim F(t) = 1 t F(t) = non-decreasing Bertinoro, March 10-14, 2003

  20. LetXbe the random variable representing the time to failure of an item. Basic Definitions: Reliability Thesurvivor function (sf) R(t)of the r.v. X is given by: R (t) = Pr { X > t } = 1 -F(t) R(t) represents the probability that the item is correctly working at time t and gives the reliability function . Bertinoro, March 10-14, 2003

  21. Equivalent terminology for R(t) = 1 -F(t): Reliability Complementary distribution function Survivor function Basic Definitions Bertinoro, March 10-14, 2003

  22. Basic Definitions: Reliability R(t) 1 R(a) 0 a b t R(0) = 1 lim R(t) = 0 t R(t) = non-increasing Bertinoro, March 10-14, 2003

  23. LetXbe the random variable representing the time to failure of an item and let F(t)be a derivable cdf: Basic Definitions: density Thedensity function f(t)is defined as: d F(t) f (t) = ——— dt f (t)dt = Pr { tX < t + dt } Bertinoro, March 10-14, 2003

  24. Basic Definitions: Density f (t) 0 t a b b  f(x) dx = Pr { a < X  b } = F(b) – F(a) a Bertinoro, March 10-14, 2003

  25. Basic Definitions: Density f (t) 1 0 t Bertinoro, March 10-14, 2003

  26. Equivalent terminology: pdf probability density function density function density f(t) = Basic Definitions For a non-negative random variable Bertinoro, March 10-14, 2003

  27. Correct Wrong Quiz 1:The higher the MTTF is, the higher the item reliability is. • The correct answer is wrong !!! Bertinoro, March 10-14, 2003

  28. h(t) t = Conditional Prob. system will fail in (t, t + t) given that it is survived until time t f(t) t = Unconditional Prob. System will fail in (t, t + t) Hazard (failure) rate Bertinoro, March 10-14, 2003

  29. is the conditional probability that the unit will fail in the interval given that it is functioning at time t. is the unconditional probability that the unit will fail in the interval Difference between the two sentences: probability that someone will die between 90 and 91, given that he lives to 90 probability that someone will die between 90 and 91 The Failure Rate of a Distribution Bertinoro, March 10-14, 2003

  30. Bathtub curve h(t) (infant mortality – burn in) (wear-out-phase) CFR Constant fail. rate (useful life) DFR IFR t Increasing fail. rate Decreasing failure rate

  31. Infant mortality (dfr) Also called infant mortality phase or reliability growth phase. The failure rate decreases with time. • Caused by undetected hardware/software defects; • Can cause significant prediction errors if steady-state failure rates are used; • Weibull Model can be used; Bertinoro, March 10-14, 2003

  32. Useful life (cfr) The failure rate remains constant in time (age independent) . • Failure rate much lower than in early-life period. • Failure caused by random effects (as environmental shocks). Bertinoro, March 10-14, 2003

  33. Wear-out phase (ifr) The failure rate increases with age. It is characteristic of irreversible aging phenomena (deterioration, wear-out, fatigue, corrosion etc…) Applicable for mechanical and other systems. (Properly qualified electronic parts do not exhibit wear-out failure during its intended service life) Weibull Failure Model can be used Bertinoro, March 10-14, 2003

  34. Cumul. distribution function: Reliability : Density Function : Failure Rate (CFR): Mean Time to Failure: Exponential Distribution Failure rate is age-independent (constant). Bertinoro, March 10-14, 2003

  35. The Cumulative Distribution Function of an Exponentially Distributed Random Variable With Parameter  = 1 F(t) 1.0 F(t) = 1 - e -  t 0.5 0 1.25 2.50 3.75 5.00 t Bertinoro, March 10-14, 2003

  36. R(t) = e -  t The Reliability Function of an Exponentially Distributed Random Variable With Parameter  = 1 R(t) 1.0 0.5 0 1.25 2.50 3.75 5.00 t Bertinoro, March 10-14, 2003

  37. Exponential Density Function (pdf) f(t) MTTF = 1/  Bertinoro, March 10-14, 2003

  38. Memoryless Property of the Exponential Distribution • Assume X > t. We have observed that the component has not failed until time t • Let Y = X - t , the remaining (residual) lifetime Bertinoro, March 10-14, 2003

  39. Thus Gt(y) is independent of t and is identical to the original exponential distribution of X The distribution of the remaining life does not depend on how long the component has been operating An observed failure is the result of some suddenly appearing failure, not due to gradual deterioration Memoryless Property of the Exponential Distribution (cont.) Bertinoro, March 10-14, 2003

  40. 1. They will always fail at the same time 2. They have the same probability of failing at time ‘t’ during operation 3. When these two components are operating simultaneously, the component which has been operational for a shorter duration of time will survive longer Quiz 3:If two components (say, A and B) have independent identical exponentially distributed times to failure, by the “memoryless” property, which of the following is true? Bertinoro, March 10-14, 2003

  41. Distribution Function: Density Function: Reliability: Weibull Distribution Bertinoro, March 10-14, 2003

  42. Weibull Distribution : shape parameter; : scale parameter. Failure Rate: Dfr Cfr Ifr Bertinoro, March 10-14, 2003

  43. Failure Rate of the Weibull Distribution with Various Values of  Bertinoro, March 10-14, 2003

  44. Weibull Distribution for Various Values of  Cdf density Bertinoro, March 10-14, 2003

  45. We use a truncated Weibull Model Infant mortality phase modeled by DFR Weibull and the steady-state phase by the exponential Failure Rate Models Figure 2.34 Weibull Failure-Rate Model 7 6 5 4 3 2 1 0 Failure-Rate Multiplier 0 2,190 4,380 6,570 8,760 10,950 13,140 15,330 17,520 Operating Times (hrs) Bertinoro, March 10-14, 2003

  46. This model has the form: where: steady-state failure rate is Weibull shape parameter Failure rate multiplier = Failure Rate Models (cont.) Bertinoro, March 10-14, 2003

  47. There are several ways to incorporate time dependent failure rates in availability models The easiest way is to approximate a continuous function by a piecewise constant step function Failure Rate Models (cont.) Discrete Failure-Rate Model 7 6 5 4 3 2 1 0 Failure-Rate Multiplier 0 2,190 4,380 6,570 8,760 10,950 13,140 15,330 17,520 Operating Times (hrs) Bertinoro, March 10-14, 2003

  48. Here the discrete failure-rate model is defined by: Failure Rate Models (cont.) Bertinoro, March 10-14, 2003

  49. A lifetime experiment X 1 1 X 2 2 X 3 3 X 4 4 X N N t = 0 N i.i.d components are put in a life test experiment. Bertinoro, March 10-14, 2003

  50. A lifetime experiment X 1 1 X 2 2 X 3 3 4 X 4 X N N Bertinoro, March 10-14, 2003

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