1. Stracener_EMIS 7305/5305_Spr08_02.28.08 1 System Reliability Analysis - Multi State Models and General Configurations Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7305/5305 Systems Reliability, Supportability and Availability Analysis Systems Engineering Program Department of Engineering Management, Information and Systems

2. Stracener_EMIS 7305/5305_Spr08_02.28.08 2 Multi State Reliability Models

3. Stracener_EMIS 7305/5305_Spr08_02.28.08 3 Multistate Reliability Models In all cases previously discussed the element analyzed could assume only one of two mutually exclusive states, success or failure, operational or nonoperational. Many components may exhibit three or more mutually exclusive states. Conceptually multistate reliability models are merely extensions of two state models. In practice, the mathematical complexities make application of multistate reliability models restricted to specialized cases. 3

4. Stracener_EMIS 7305/5305_Spr08_02.28.08 4 Multistate Reliability Models System Low Resistance High Resistance E 1 2 E DIODES Each diode has three states: good, shorted, and opened. If event x represents good, event x S shorted, and event x O opened, the three states are mutually exclusive and 4

5. Stracener_EMIS 7305/5305_Spr08_02.28.08 5 Multistate Reliability Models The system tie sets (success combinations) are x 1 x 2, x 1 x 20, x 2 x 10 The 9 System States 1 - X 1 X 2 2 - X 1 X 20 3 - X 1 X 2S 4 - X 2 X 10 5 - X 2 X 1S 6 - X 1S X 2S 7 - X 1S X 20 8 - X 10 X 2S 9 - X 10 X 20 - S - F - S - F System Reliability 5

6. Stracener_EMIS 7305/5305_Spr08_02.28.08 6 General Reliability Configuration

7. Stracener_EMIS 7305/5305_Spr08_02.28.08 7 System Reliability Evaluation Techniques In general, the reliability configuration of a system will not consist of only the special configurations we have considered. More general techniques are required for developing reliability models for general reliability configurations Generally, in system reliability analyses the reliability configuration will not be only one of the special configurations we have considered therefore, we need a more general technique for developing the reliability model for a general system 7

8. Stracener_EMIS 7305/5305_Spr08_02.28.08 8 System Reliability Evaluation Techniques Some of the methods for development of a system reliability model are: Direct or inspection method Decomposition method Event-space method Path tracing method Conditional probability (Bayes Theorem) State space approach (Markov Processes) Cut-set and tie-set methods Delta-star techniques 8

9. Stracener_EMIS 7305/5305_Spr08_02.28.08 9 Direct or Inspection method Develop the system reliability equation by following the shortest dependent route through the reliability block diagram.

10. Stracener_EMIS 7305/5305_Spr08_02.28.08 10 Direct or Inspection method - example Apply the direct or inspection method to develop the system reliability equation for the following reliability block diagram: R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R9R9 R 10 R 12 R 13 R8R8 R 11

11. Stracener_EMIS 7305/5305_Spr08_02.28.08 11 Direct or Inspection method - example solution For the system to function: 1. R 1 R 2 R 3 must function R 1 R 2 R 3 2. R 12 R 13 may or may not function R 1 R 2 R 3 [R 12 R 13 + (1 - R 12 R 13 ) 3. If R 12 R 13 fails R 11 must function R 1 R 2 R 3 [R 12 R 13 + (1 - R 12 R 13 )(R 11 ) 4. R 9 R 10 may or may not function R 1 R 2 R 3 [R 12 R 13 + (1 - R 12 R 13 )(R 11 ) [R 9 R 10 + (1 - R 9 R 10 )

12. Stracener_EMIS 7305/5305_Spr08_02.28.08 12 Direct or Inspection method - example solution 5. If R 9 R 10 fails R 4 R 5 must function R 1 R 2 R 3 [R 12 R 13 + (1 - R 12 R 13 )(R 11 ) [R 9 R 10 + (1 - R 9 R 10 )(R 4 R 5 ) 6. R 8 may or may not function R 1 R 2 R 3 [R 12 R 13 + (1 - R 12 R 13 )(R 11 ) [R 9 R 10 + (1 - R 9 R 10 )(R 4 R 5 )[(R 8 +(1-R 8 ) 7. If R 8 fails R 6 R 7 must function R(t) = R 1 R 2 R 3 [R 12 R 13 + (1 - R 12 R 13 )(R 11 ) [R 9 R 10 + (1 - R 9 R 10 )(R 4 R 5 )[(R 8 +(1-R 8 )(R 6 R 7 ))]]]

13. Stracener_EMIS 7305/5305_Spr08_02.28.08 13 Decomposition into Series and Parallel Decompose the configuration into series and parallel configurations At each decomposition the reliability models for the series or parallel configuration are utilized The system reliability model is obtained by taking the product of the reliability models for the series and parallel configurations

14. Stracener_EMIS 7305/5305_Spr08_02.28.08 14 Decomposition into Series and Parallel - example The reliability block diagram for a system consisting of seven elements A, B 1, C, D, E, and E 2 is below A C B1B1 D E1E1 E2E2 B2B2

15. Stracener_EMIS 7305/5305_Spr08_02.28.08 15 Decomposition into Series and Parallel - example The associated element reliabilities are: P(A) = 0.95 P(B 1 ) = P(B 2 ) = 0.95 P(C) = 0.98 P(D) = 0.90 P(E 1 ) = P(E 2 ) = 0.90 Find the system reliability.

16. Stracener_EMIS 7305/5305_Spr08_02.28.08 16 Decomposition into Series and Parallel - solution step 1 where: P(B) = P(B 1  B 2 ) = P(B 1 ) + P(B 2 ) - P(B1)P(B 2 ) A C B D E1E1 E2E2

17. Stracener_EMIS 7305/5305_Spr08_02.28.08 17 Decomposition into Series and Parallel - solution step 2 where: P(E) = P(E 1 )P(E 2 ) A C B D E

18. Stracener_EMIS 7305/5305_Spr08_02.28.08 18 Decomposition into Series and Parallel - solution step 3 where: P(F) = P(D) + P(E) - P(D)P(E) A C B F

19. Stracener_EMIS 7305/5305_Spr08_02.28.08 19 Decomposition into Series and Parallel - solution step 4 where: P(S 1 ) = P(A)P(B) P(S 2 ) = P(C)P(D) S1S1 S2S2

20. Stracener_EMIS 7305/5305_Spr08_02.28.08 20 Decomposition into Series and Parallel - solution step 5 where: P(S) = P(S 1 ) + P(S 2 ) - P(S 1 )P(S 2 ) S

21. Stracener_EMIS 7305/5305_Spr08_02.28.08 21 Decomposition into Series and Parallel - solution Therefore, R(t)= P(S) = P(S 1 ) + P(S 2 ) - P(S 1 )P(S 2 ) = P(A)P(B) + P(C)P(F) - P(A)P(B)P(C)P(D) = P(A)P(B) + P(C)[P(D) + P(E) - P(D)P(E)] - P(A)P(B)P(C)[P(D) + P(E) - P(D)P(E)] = P(A)P(B) + P(C)P(D) + P(C)P(E) - P(C)P(D)P(E) - P(A)P(B)P(C)P(D) - P(A)P(B)P(C)P(E) + P(A)P(B)P(C)P(D)P(E)

22. Stracener_EMIS 7305/5305_Spr08_02.28.08 22 Decomposition into Series and Parallel - solution Therefore, R(t)= P(A)P(B) + P(C)P(D) + P(C)P(E 1 )P(E 2 ) - P(C)P(D)P(E 1 )P(E 2 ) - P(A)P(B)P(C)P(D) - P(A)P(B)P(C)P(E 1 )P(E 2 ) + P(A)P(B)P(C)P(D)P(E 1 )P(E 2 ) = P(A)[P(B 1 ) + P(B 2 ) - P(B 1 )P(B 2 )] + P(C)P(D) + P(C)P(E 1 )P(E 2 ) + P(C)P(D)P(E 1 )P(E 2 ) - P(A)P(C)P(D)[P(B 1 ) + P(B 2 ) - P(B 1 )P(B 2 )] - P(A)P(C)P(E 1 )P(E 2 )[P(B 1 ) + P(B 2 ) - P(B 1 )P(B 2 )] + P(A)P(C)P(D)P(E 1 )P(E 2 )[P(B 1 ) + P(B 2 ) - P(B 1 )P(B 2 )] R(t)= 0.9981

23. Stracener_EMIS 7305/5305_Spr08_02.28.08 23 Event Space Method List all possible logical occurrences in the system Separated the list into favorable and unfavorable events. The list should be prepared so that all events are mutually exclusive. The probability of success, i.e., the reliability, is merely the sum of the occurrence probability of each successful event. As an alternative approach the reliability could be computed by first finding the probability of failure, which is given by the sum of the occurrence probabilities of each of the unsuccessful events

24. Stracener_EMIS 7305/5305_Spr08_02.28.08 24 Event Space Method - Example Find the reliability of a system with a complex structure whose reliability block diagram is given by A B C D E The element reliability is P(A) = P(B) = P(C) = P(D) = P(E) = p

25. Stracener_EMIS 7305/5305_Spr08_02.28.08 25 Event Space Method - Example Solution List all 2 5 = 32 system states (the event space) using the number of combinations of n-things-taken-r-at-a- time formula: r = 0 E 1 = ABCDE r = 1 E 2 = ABCDE E 3 = ABCDE E 4 = ABCDE E 5 = ABCDE E 6 = ABCDE

26. Stracener_EMIS 7305/5305_Spr08_02.28.08 26 Event Space Method - Example Solution r = 3 E 17 = ABCDE E 18 = ABCDE E 19 = ABCDE E 20 = ABCDE E 21 = ABCDE E 22 = ABCDE E 23 = ABCDE E 24 = ABCDE E 25 = ABCDE E 26 = ABCDE r = 4 E 27 = ABCDE E 28 = ABCDE E 29 = ABCDE E 30 = ABCDE E 31 = ABCDE r = 5 E 32 = ABCDE r = 2 E 7 = ABCDE E 8 = ABCDE E 9 = ABCDE E 10 = ABCDE E 11 = ABCDE E 12 = ABCDE E 13 = ABCDE E 14 = ABCDE E 15 = ABCDE E 16 = ABCDE

27. Stracener_EMIS 7305/5305_Spr08_02.28.08 27 Event Space Method - Example Solution Determine the impact on system reliability of each event by inspecting the reliability block diagram and construct lists of favorable and unfavorable events Favorable Events E 1 E 9 E 22 E 2 E 10 E 23 E 3 E 11 E 25 E 4 E 12 E 5 E 13 E 6 E 14 E 7 E 15 E 8 E 19 Unfavorable Events E 16 E 24 E 29 E 17 E 26 E 30 E 18 E 27 E 31 E 20 E 28 E 32 E 21

28. Stracener_EMIS 7305/5305_Spr08_02.28.08 28 Event Space Method - Example Solution Probability of System Success R(t)= P(E 1  E 2  E 3  E 4  E 5  E 6  E 7  E 8  E 9  E 10  E 11  E 12  E 13  E 14  E 15  E 19  E 22  E 23  E 25 ) = p 5 + 5p 4 (1 - p) + 9p 3 (1 - p) 2 + 4p 2 (1 - p) 3 = p 5 - p 4 - 3p 3 + 4p 2 OR R(t)= 1 - P(E 16  E 17  E 18  E 21  E 24  E 26  E 27  E 28  E 29  E 30  E 31  E 32 ) = 1 -[p 3 (1-p) 2 + 6p 2 (1-p) 3 + 5p(1-p) 4 + (1-p) 3 ] = p 5 - p 4 - 3p 3 + 4p 2

29. Stracener_EMIS 7305/5305_Spr08_02.28.08 29 Path Tracing Method Identify all successful paths Each successful path forms a favorable event The system reliability is then the probability of at least one successful path The Path Tracing method is simpler than the event tracing method in that it eliminates the lengthy tabulation of the Event Space method

30. Stracener_EMIS 7305/5305_Spr08_02.28.08 30 Path Tracing Method Disadvantages of the Path Tracing method include: - All possible success paths must be identified - The events are not generally mutually exclusive - Complex systems result in complicated mathematical expressions which may be difficult to simplify

31. Stracener_EMIS 7305/5305_Spr08_02.28.08 31 Path Tracing Method - Example Given the following reliability block diagram E3E3 E1E1 E2E2 E4E4 E5E5 a. Find the system reliability using the path tracing method, if p i = R i (t), for i = 1, …, 5 b. Find R S (t) = h S (t), and MTBF, if p i = R i (t) = e - t, for i = 1, …, 5

32. Stracener_EMIS 7305/5305_Spr08_02.28.08 32 Path Tracing Method - Example Solution a.Success PathEvent 1 2A 1 4 5A 2 1 3 5A 3 4 3 2A 4 R S (t) P(A 1 ) + P(A 2 ) + P(A 3 ) + P(A 4 ) - P(A 1 A 2 ) - P(A 1 A 3 ) - P(A 1 A 4 ) - P(A 2 A 3 ) - P(A 2 A 4 ) - P(A 3 A 4 ) + P(A 1 A 2 A 3 ) + P(A 1 A 2 A 4 ) + P(A 2 A 3 A 4 ) + P(A 1 A 3 A 4 ) - P(A 1 A 2 A 3 A 4 )

33. Stracener_EMIS 7305/5305_Spr08_02.28.08 33 Path Tracing Method - Example Solution = P 1 P 2 + P 4 P 5 + P 1 P 3 P 5 + P 2 P 3 P 4 + P 1 P 2 P 4 P 5 - P 1 P 2 P 3 P 5 - P 1 P 2 P 3 P 4 - P 1 P 3 P 4 P 5 - P 2 P 3 P 4 P 5 - P 1 P 2 P 3 P 4 P 5 + P 1 P 2 P 3 P 4 P 5 + P 1 P 2 P 3 P 4 P 5 + P 1 P 2 P 3 P 4 P 5 + P 1 P 2 P 3 P 4 P 5 - P 1 P 2 P 3 P 4 P 5 = P 1 P 2 + P 4 P 5 + P 1 P 3 P 5 + P 2 P 3 P 4 - P 1 P 2 P 4 P 5 - P 1 P 2 P 3 P 5 - P 1 P 2 P 3 P 4 - P 1 P 3 P 4 P 5 - P 2 P 3 P 4 P 5 - P 1 P 2 P 3 P 4 P 5