1. Reliability Engineering
Redundant Structure and Standby Units

2. Cold Standby, Perfect Switching, With Repairs

3. Possible States of a 2-Unit System with Cold Standby and Perfect Switching
Unit A
Unit B 4 O S 3 F 2 1

4. State Space Diagram 4 3 2 1

5. State Equations

6. Eliminating the Failed State

7. Laplace Transform
Substitute s=0
Note that

8. Solution

9. Mean Time to Failure

10. Mean Time to Failure
Take Laplace transform of R(t)
Substitute s=0

11. Mean Time to Failure

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

13. Possible States
System
Unit A
(Main Unit)
Unit B 4 O S 3 F 2 1

14. State Space Diagram 4 3

15. State Equations
Where

16. Steady State Probabilities

17. Availability and Unavailability

18. Eliminate Failed State from State Equations
Where

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

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

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

22. State Space Diagram 4 3

23. Steady State Probabilities

24. Availability and Unavailability

25. Mean Time to Failure

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

27. Possible States of a 2-Unit System with Partly-Loaded Standby and Perfect Switching
Unit A
Unit B 4 O S 3 F 2 1

28. State Space Diagram 4 3 1

29. Steady State Probabilities

30. L Spares, With Replacements and Repairs

31. State Space Diagram 2 2j 2L 1

32. 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.

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

34. Steady-State State Equations

35. Steady-State Availability