1. Lecture 2: Combinatorial Modeling CS 7040 Trustworthy System Design, Implementation, and Analysis Spring 2015, Dr. Rozier Adapted from slides by WHS at UIUC

2. Introduction

3. Introduction to Combinatorial Methods One of the simplest validation methods utilizing analytical/numerical techniques that can be used for reliability and availability modeling. Requires certain assumptions…

4. Combinatorial Assumptions Component failures are independent For availability, repairs are independent When these assumptions hold, simple formulas for reliability and availability exist!

5. Defining Reliability

6. Reliability A key to trustworthy systems is the use of reliable components and systems. – Leads to high availability! Reliability: The reliability of a system at time t, R(t), is the probability that system operation is proper throughout the interval [0,t].

7. Reliability Reliability: The reliability of a system at time t, R(t), is the probability that system operation is proper throughout the interval [0,t]. Probability theory and combinatorics can be applied directly to reliability models.

8. Reliability Reliability: The reliability of a system at time t, R(t), is the probability that system operation is proper throughout the interval [0,t]. Let X be a random variable representing the time to failure of a component. The reliability at time t is given by:

9. Reliability Reliability: The reliability of a system at time t, R(t), is the probability that system operation is proper throughout the interval [0,t]. Unreliability can be defined similarly as:

10. Probability Refresher A random variable X is unique determined by its set of possible values,, and and the associated probability distribution (or density) function (pdf), a real-valued function defined for each possible value as a probability that X has the value x

11. Probability Refresher The cumulative distribution function (cdf) of the discrete random variable X is the real valued function defined for each as

12. Probability Refresher The cumulative distribution function (cdf) of the continuous random variable X is the real valued function defined for each as

13. PDFs and CDFs

14. Reliability and Unreliability Reliability: Unreliability:

15. Failure Rates

16. Failure Rate What is the rate at which a component fails at time t? – The probability that a component that has not yet failed, fails in the interval Note: We are not looking at We are seeking

17. Failure Rate

20. is called the failure rate or hazard rate

21. Survival Function In addition to the reliability/hazard function we have the survival function

22. Survival Function In addition to the reliability/hazard function we have the survival function

23. Survival Function In addition to the reliability/hazard function we have the survival function

24. Survival and Hazard Hazard (or Failure) function – instantaneous failure rate at some time t. Survival function – the probability that the time of failure is later than some time t.

25. Typical Failure Rate

26. System Reliability

27. While can give the reliability of a component, how do you compute the reliability of a system?

28. System Reliability System failure can occur when one, all, or some of the components fail. If one makes the independent failure assumption, system failures can be computed quite simply. The independent failure assumption states that all component failures of a system are independent, i.e., the failure of one component does not cause another component to be more or less likely to fail.

29. System Reliability Given this assumption, we can determine: – Minimum failure time of a set of components – Maximum failure time of a set of components – Probability that k of N components have failed at a particular time t.

30. Maximum of n Independent Failure Times Let be independent component failure times. Suppose the system fails at time S if all the components fail. Thus, What is ?

31. Maximum of n Independent Failure Times

32. By independence

33. Maximum of n Independent Failure Times By definition

34. Maximum of n Independent Failure Times

35. Minimum of n Independent Failure Times Let be independent component failure times. A system fails at time S if any of the components fail. Thus, What is ?

36. Minimum of n Independent Failure Times What is ?

37. Minimum of n Independent Failure Times What is ? Trick: If is an event, and is the set complement such that and, then

38. Minimum of n Independent Failure Times

39. By trick

40. Minimum of n Independent Failure Times By independece

41. Minimum of n Independent Failure Times By LOTP

42. Minimum of n Independent Failure Times

43. k of N Let be component failure times that have identical distributions (i.e., ). The system has failed by time S if k or more of the N components have failed by S. P[at least k components failed by time t] = P[exactly k failed OR exactly k+1 failed …] = P[exactly k failed] + P[exactly k+1 failed] …

44. k of N What is P[exactly k failed]? = P[k failed and (N – k) have not] where is the failure distribution of each component

45. k of N in General For non-identical failure distributions, we must sum over all combinations of at least k failures. Let be the set of all subsets of such that each element in is a set of size at least k, i.e.,

46. k of N in General The set represents all the possible failure scenarios. Now is given by

47. Component Building Blocks Complex systems can be analyzed hierarchically. Example: A computer fails if both power supplies fail, or both memories fail, or if the CPU fails. System problem is one of a minimum: the system fails when the first of three subsystems fails…

48. Component Building Blocks Power supply subsystem is a maximum: both must fail Memory supply subsystem is a maximum: both must fail

49. Summary A system comprises N components, where the component failure times are given by the random variables. The system fails at time S with distribution if: ConditionDistribution All components fail One component fails k components fail, identical distributions k components fail, general case

50. Reliability Formalisms

51. There are several popular graphical formalisms to express system reliability. The core of the solvers for these formalisms are the methods we have just examined. We will discuss a subset of these formalisms: – Reliability Block Diagrams – Fault Trees – Reliability Graphs

52. Reliability Formalisms There are several popular graphical formalisms to express system reliability. The core of the solvers for these formalisms are the methods we have just examined. We will discuss a subset of these formalisms: – Reliability Block Diagrams – Fault Trees – Reliability Graphs There is nothing special about these formalisms except their popularity.

53. What is a Graphical Formalism A way to draw visual diagrams with formal underlying mathematical meanings.

54. Reliability Block Diagrams Blocks represent components A system failure occurs if there is no path from source to sink.

55. Reliability Block Diagrams Series: – System fails if any component fails

56. Reliability Block Diagrams Parallel: – System fails if all components fail

57. Reliability Block Diagrams k of N: – System fails if at least k of N components fail.

58. Example A NASA satellite architecture under study is designed for high reliability. The major computer system components include the CPU system, the high-speed network for data collection and transmission, and the low-speed network for engineering and control. The satellite fails if any of the major systems fail. There are 3 computers, and the computer system fails if 2 or more of the computers fail. Failure distribution of a computer is given by There is a redundant (2) high-speed network, and the high-speed network system fails if both networks fail. The distribution of a high-speed network failure is given by The low-speed network is arranged similarly, with a failure distribution of

59. Example

63. Background: Series-Parallel Graphs

64. Series-Parallel Decomposition of NASA Example

65. Fault Trees

66. Fault Tree Example

67. Reliability Graphs

68. Reliability Graph Example

69. Solve by Conditioning

72. Conditioning Fault Trees

73. Reliability/Availability Point Estimates

74. Reliability/Availability Tables

75. Reliability Modeling Process

78. For next time Homework 1! Due next Tuesday Review combinatorics