1. Stracener_EMIS 7305/5305_Spr08_02.05.08 1 Systems Reliability Modeling & Analysis Series and Active Parallel 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_02.05.08 2 System Reliability Models Series Configurations Parallel or Redundant Configurations Active Parallel r-out-of-n Standby Series-Parallel and Parallel-Series Configurations General

3. Stracener_EMIS 7305/5305_Spr08_02.05.08 3 System Reliability Models The reliability definitions, concepts and models presented apply at any level of a system, from a single discrete component up to and including the entire system. Systems reliability deals with the reliability of the end-item system and is based on the system configuration and component failure rates as well intended service usage There are two basic types of reliability configurations Series Parallel or Redundant

4. Stracener_EMIS 7305/5305_Spr08_02.05.08 4 Terminology and Notation Path: A physical means for accomplishing a given function. Element: The basic system level under discussion. An element may be a Component, an Assembly, an Equipment, a Line Replaceable Unit (LRU), a Subsystem or a System Block: A logical representation of an Element. Reliability Block Diagram: A logical representation of a System, Subsystem, or Assembly in terms of its Elements.

5. Stracener_EMIS 7305/5305_Spr08_02.05.08 5 Series Reliability Configuration

6. Stracener_EMIS 7305/5305_Spr08_02.05.08 6 Series Reliability Configuration Simplest and most common structure in reliability analysis. Functional operation of the system depends on the successful operation of all system components Note: The electrical or mechanical configuration may differ from the reliability configuration Reliability Block Diagram Series configuration with n elements: E 1, E 2,..., E n System Failure occurs upon the first element failure E1E1 E2E2 EnEn

7. Stracener_EMIS 7305/5305_Spr08_02.05.08 7 Series Reliability Configuration: Math Models System Reliability R s (t) = P(A 1 ) P(A 2 |A 1 ) P(A 3 |A 1 A 2 )... P(A n |A 1 A2... A n-1 ) Where R S (t) is system reliability, i.e. The probability of system success for time t, given that the system was ‘up’ at t = 0 and P(A i |A 1 A 2... A i-1 ) is the conditional probability of event A occurring (i.e., element E i survives for time t), given that events A 1, A 2,... And A i-1 have occurred (i.e. Elements E 1, E 2,... and E i-1 have survived for time t, for i = 1, 2,..., n Product Rule of System Reliability if the n elements are independent

8. Stracener_EMIS 7305/5305_Spr08_02.05.08 8 System Reliability Models - Series Reliability Configuration Simplest and most common structure in reliability analysis. Functional operation of the system depends on the successful operation of all system components Note: The electrical or mechanical configuration may differ from the reliability configuration Block Diagram For Series Reliability Configuration with n elements: E 1, E 2,..., E n Since a single path exists, the failure of any element in the system interrupts the path and causes the system to fail. E1E1 E2E2 EnEn

9. Stracener_EMIS 7305/5305_Spr08_02.05.08 9 System Reliability Models - Series Configuration: Product Rule R s (t) = P(A 1 ) P(A 2 |A 1 ) P(A 3 |A 1 A 2 )... P(A n |A 1 A2... A n-1 ) Where R S (t) is system reliability, i.e. The probability of system success for time t, given that the system was ‘up’ at t = 0 and P(A i |A 1 A 2... A i-1 ) is the conditional probability of event A occurring (i.e., element Ei survives for time t), given that events A 1, A 2,... And A i-1 have occurred (i.e. Elements E 1, E 2,... And E i-1 have survived for time t, for i = 1, 2,..., n if the n elements are independant

10. Stracener_EMIS 7305/5305_Spr08_02.05.08 10 System Reliability Models - Series Configuration General time to element failure distributions System reliability Where H i (t) is the cumulative failure rate of element i, for i = 1, 2,... n System mean time to failure

11. Stracener_EMIS 7305/5305_Spr08_02.05.08 Series Reliability Configuration with Exponential Distribution Reliability Block Diagram Exponential distributions of element time to failure T i ~ E(  i ) for i = 1, 2,... n System reliability Where the system failure rate is System mean time to failure E1E1 E2E2 EnEn

12. Stracener_EMIS 7305/5305_Spr08_02.05.08 12 Series Reliability Configuration with Exponential Distribution Reliability Block Diagram Exponential distributions of element time to failure T i ~ E(  ) for i = 1, 2,... n System reliability Which is the same as the expected time to the first failure, E(T 1 ), when n identical items are put into service System mean time to failure E1E1 E2E2 EnEn

13. Stracener_EMIS 7305/5305_Spr08_02.05.08 13 Series Reliability Configuration with Weibull Distribution Reliability Block Diagram Weibull distribution of element time to failure T i ~ W(  i,  i ) for i = 1, 2,... n System reliability System failure rate E1E1 E2E2 EnEn

14. Stracener_EMIS 7305/5305_Spr08_02.05.08 14 System Reliability Models - Series Configuration System cumulative failure rate System mean time to failure rate

15. Stracener_EMIS 7305/5305_Spr08_02.05.08 15 System Reliability Models - Series Configuration Weibull distribution of element time to failure T i ~ W( ,  ) for i = 1, 2,... n System reliability System failure rate

16. Stracener_EMIS 7305/5305_Spr08_02.05.08 16 System Reliability Models - Series Configuration System cumulative failure rate System mean time to failure rate

17. Stracener_EMIS 7305/5305_Spr08_02.05.08 17 Example: Series Reliability Configuration

18. Stracener_EMIS 7305/5305_Spr08_02.05.08 18 Parallel Reliability Configuration

19. Stracener_EMIS 7305/5305_Spr08_02.05.08 19 Parallel Reliability Configuration – Basic Concepts Definition - a system is said to have parallel reliability configuration if the system function can be performed by any one of two or more paths Reliability block diagram - for a parallel reliability configuration consisting of n elements, E 1, E 2,... E n E1E1 E2E2 EnEn

20. Stracener_EMIS 7305/5305_Spr08_02.05.08 20 Parallel Reliability Configuration Redundant reliability configuration - sometimes called a redundant reliability configuration. Other times, the term ‘redundant’ is used only when the system is deliberately changed to provide additional paths, in order to improve the system reliability Basic assumptions All elements are continuously energized starting at time t = 0 All elements are ‘up’ at time t = 0 The operation during time t of each element can be described as either a success or a failure, i.e. Degraded operation or performance is not considered

21. Stracener_EMIS 7305/5305_Spr08_02.05.08 21 Parallel Reliability Configuration System success - a system having a parallel reliability configuration operates successfully for a period of time t if at least one of the parallel elements operates for time t without failure. Notice that element failure does not necessarily mean system failure.

22. Stracener_EMIS 7305/5305_Spr08_02.05.08 22 Parallel Reliability Configuration Block Diagram System reliability - for a system consisting of n elements, E 1, E 2,... E n if the n elements operate independently of each other and where R i (t) is the reliability of element i, for i=1,2,…,n E1E1 E2E2 EnEn

23. Stracener_EMIS 7305/5305_Spr08_02.05.08 23 System Reliability Model - Parallel Configuration Product rule for unreliabilities Mean Time Between System Failures

24. Stracener_EMIS 7305/5305_Spr08_02.05.08 24 Parallel Reliability Configuration s p=R(t) s

25. Stracener_EMIS 7305/5305_Spr08_02.05.08 25 Parallel Reliability Configuration with Exponential Distribution Element time to failure is exponential with failure rate Reliability block diagram: Element Time to Failure Distribution with failure rate for i=1,2. System reliability System failure rate E1E1 E2E2

26. Stracener_EMIS 7305/5305_Spr08_02.05.08 26 Parallel Reliability Configuration with Exponential Distribution System Mean Time Between Failures: MTBF S = 1.5 

27. Stracener_EMIS 7305/5305_Spr08_02.05.08 27 Parallel Reliability Configuration Exponential distributions of element time to failure T i ~ E(  i ) for i = 1, 2 System reliability System failure rate

28. Stracener_EMIS 7305/5305_Spr08_02.05.08 28 Notice that h S (t) is an increasing function of t even though the failure rate of each element is constant System mean time between failures Parallel Reliability Configuration - continued

29. Stracener_EMIS 7305/5305_Spr08_02.05.08 29 System Reliability Models - Parallel Configuration Exponential distributions of element time to failure T i ~ E(  i ) for i = 1, 2,... n System reliability System mean time between failure MTBF S =

30. Stracener_EMIS 7305/5305_Spr08_02.05.08 30 System Reliability Models - Parallel Configuration Exponential distributions of element time to failure T i ~ E(  ) for i = 1, 2,... n System reliability System mean time between failures

31. Stracener_EMIS 7305/5305_Spr08_02.05.08 31 System Reliability Models - Parallel Configuration Weibull distribution of element time to failure T i ~ W(  i,  i ) for i = 1, 2 System reliability

32. Stracener_EMIS 7305/5305_Spr08_02.05.08 32 System Reliability Models - Parallel Configuration Weibull distribution of element time to failure T i ~ W( ,  ) for i = 1, 2 System reliability System failure rate

33. Stracener_EMIS 7305/5305_Spr08_02.05.08 33 System Reliability Models - Parallel Configuration Weibull distribution of element time to failure T i ~ W( ,  ) for i = 1, 2 System mean time between failures

34. Stracener_EMIS 7305/5305_Spr08_02.05.08 34 n elements m elements n elements m elements System Reliability Models - Parallel Configuration