1. Fault Tree Analysis Part 12 – Redundant Structure and Standby Units

2. Active Redundancy The redundancy obtained by replacing the important unit with two or more units operating in parallel.

3. Passive Redundancy The reserve units can also be kept in standby in such a way that the first of them is activated when the original unit fails, the second is activated when the first reserve unit fails, and so on. If the reserve units carry no load in the waiting period before activation, the redundancy is called passive. In the waiting period, such a unit is said to be in cold standby.

4. Partly-Loaded Redundancy The standby units carry a weak load.

5. Cold Standby, Perfect Switching, No Repairs

6. Life Time of Standby System The mean time to system failure

7. Exact Distribution of Lifetime If the lifetimes of the n components are independent and exponentially distributed with the same failure rate λ. It can be shown that T is gamma distributed with parameters n and λ. The survivor function is

8. Approximate Distribution of Lifetime Assume that the lifetimes are independent and identically distributed with mean time to failure μ and standard deviation σ. According to Lindeberg-Levy’s central limit theorem, T will be asymptotically normally distributed with mean nμ and variance nσ^2.

9. Cold Standby, Imperfect Switching, No Repairs

10. 2-Unit System A standby system with an active unit (unit 1) and a unit in cold standby. The active unit is under surveillance by a switch, which activates the standby unit when the active unit fails. Let be the failure rate of unit 1 and unit 2 respectively; Let (1-p) be the probability that the switching is successful.

11. Two Disjoint Ways of Survival 1.Unit 1 does not fail in (0, t], i.e. 2.Unit 1 fails in the time interval (τ, τ+dτ], where 0<τ<t. The switch is able to activate unit 2. Unit 2 is activated at time τ and does not fail in the time interval (τ,t].

12. Probabilities of Two Disjoint Events Event 1: Event 2: Unit 1 fails Switching successful Unit 2 working afterwards

13. System Reliability

14. Mean Time to Failure

15. Partly-Loaded Redundancy, Imperfect Switching, No Repairs

16. Two-Unit System Same as before except unit 2 carries a certain load before it is activated. Let denote the failure rate of unit 2 while in partly- loaded standby.

17. Two Disjoint Ways of Survival 1.Unit 1 does not fail in (0, t], i.e. 2.Unit 1 fails in the time interval (τ, τ+dτ], where 0<τ<t. The switch is able to activate unit 2. Unit 2 does not fail in (0, τ], is activated at time τ and does not fail in the time interval (τ,t].

18. Probabilities of Two Disjoint Events Event 1: Event 2: Unit 1 fails at τ Switching successful Unit 2 still working after τ Unit 2 working in (0, τ]

19. System Reliability

20. Mean Time to Failure

21. Cold Standby, Perfect Switching, With Repairs

22. Possible States of a 2-Unit System with Cold Standby and Perfect Switching System Unit AUnit B 4OS 3FO 2SO 1OF 0FF

23. State Space Diagram 0 1 2 3 4

24. State Equations

25. Eliminating the Failed State

26. Laplace Transform Substitute s=0 Note that

27. Solution

28. Mean Time to Failure

29. Take Laplace transform of R(t) Substitute s=0

30. Mean Time to Failure

31. Cold Standby, Perfect Switching, With Repairs, A Main Operating Unit

32. Possible States System Unit A (Main Unit) Unit B 4OS 3FO 2SO 1OF 0FF

33. State Space Diagram 0 3 4

34. State Equations Where

35. Steady State Probabilities

36. Availability and Unavailability

37. Eliminate Failed State from State Equations Where

38. Treating State 0 as An Absorbing State Take Laplace transform and let s=0 Solution

39. Mean Times to Failure and to Repair Mean time to failure Mean time to repair

40. Cold Standby, Imperfect Switching, With Repairs, A Main Operating Unit

41. State Space Diagram 0 3 4

42. Steady State Probabilities

43. Availability and Unavailability

44. Mean Time to Failure

45. Partly-Loaded Standby, Perfect Switching, With Repairs, A Main Operating Unit

46. Possible States of a 2-Unit System with Partly-Loaded Standby and Perfect Switching System Unit AUnit B 4OS 3FO 2SO 1OF 0FF

47. State Space Diagram 0 1 3 4

48. Steady State Probabilities

49. L Spares, With Replacements and Repairs

50. State Space Diagram 0 1 2 2j2L

51. Notation State 2j (j = 0, 1, …,L): A total of j spare units are in a repair queue, and (L-j) spares are normal. A failed unit in the system is being replaced by a normal spared unit, the system is working. State 2j+1 (j = 0, 1, …, L-1): A total of j spare units are in a repair queue, and (L-j) spares are normal. A failed unit in the system is being replaced by a normal spared unit, the system does not work. State 2L+1: All spares are in a repair queue. A failed unit in the system is under priority repair. This is a type of quasi-replacement.

52. Notation λ: Constant failure rate μ: Constant repair rate ε: Constant replacement rate

53. Steady-State State Equations

54. Steady-State Availability