# Reliability of Technical Systems

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1 Reliability of Technical Systems

2 Main Topics 1. Short Introduction, Reliability Parameters: Failure Rate, Failure Probability, etc. 2. Some Important Reliability Distributions 3. Component Reliability 4. Introduction, Key Terms, Framing the Problem 5. System Reliability I: Reliability Block Diagram, Structure Analysis (Fault Trees), State Model. 6. System Reliability II: State Analysis (Markovian chains) 7. System Reliability III: Dependent Failure Analysis 8. Data Collection, Bayes Theorem, Static Redundancy 9. Combined Redundancy, Dynamic Redundancy; Advanced Methods for Systems Modeling and Simulation I: Petri Nets 10. Advanced Methods for Systems Modeling and Simulation II: Object-oriented modeling and MC modeling 11. Human Reliability Analysis 12. Software Reliability, Fault Tolerance 13. Case study: Building a Reliable System 2

3 Combined Series-Parallel System Most technical systems contain components both in both series and parallel. Such systems need to be broken down in series and parallel subsystems. Finally the reliability of the system may be obtained based on the relationship among the subsystems. Try to compute the reliability of system : RA=[1-(1-R1)(1-R2)] RB = RA R3 RC= R4R5 RS=[1-(1-RB)(1-RC)]R6 3

4 Combined Series-Parallel System Low Level Redundancy: Each component comprising the system may have one or more parallel components. - Assume each component has reliability R A B A B R low 2 = [ 1 (1 R) ] = (2R R 2 2 ) 2 High Level Redundancy: The entire system may be placed in parallel with one or more identical system. - Assume each component has reliability R A A B B R high = [ 1 (1 R ) ] 2R R = 4

5 Combined Series-Parallel System By comparing the two reliability, it may be said that the reliability of low level redundancy is greater than the reliability of the high-level redundancy. It may be seen below: (R low R high ) = (2R R 2 ) 2 (2R 2 R 4 ) = 2R 2 (R 2-2R +1) = 2R 2 (R-1) 2 0 5

6 Dynamic Redundancy 6

7 Dynamic Redundancy Activation of Redundancy Use of redundant means/resources for fault tolerance. Recall, static redundancy denotes redundant means/resources, which perform the specified function during the whole time of operation. Example: 2-out-of-3 system. Dynamic redundancy: denotes redundant means/resources, which perform the specified function after fault occurrence (and detection and a possible exceptional operation). For dynamic structural redundancy we distinguish: Primary component Spare or stand-by component so-called hot stand-by so-called cold stand-by 7

8 Dynamic Redundancy 8

9 Dynamic Redundancy 9

10 Dynamic Redundancy 10

11 Redundancy 11

12 Redundancy Application of the State Model: Calculation of the availability or similar measures (Example) System consists of a primary component K1 and a spare component K2. System is faultless (b=1) or not (b=0) Each state is defined by (K1, K2, b). Z2 : models the failure after occurrence of a fault where error processing fails. Z3 : models the successful reconfiguration when system becomes faultless again. Z4 : models an undetected fault in the spare component. Z5 : models the failure of the system when both components fail Assume same failure rate λ, and repair rate µ for both components. ŋ stands for reconfiguration rate. What is the probability of the faultless operation: P1+ P3 + P4? 12

13 Redundancy Application of the State Model: Calculation of the availability or similar measures (Example) 13

14 Redundancy Application of the State Model: Calculation of the availability or similar measures (Solution) 14

15 Redundancy 15

16 16

17 Advantages of Redundacy: Improves system reliability Can be implemented in different ways Disadvantages: Cost and complexity increase Inefficient by common mode failures Security weak points are multiplied 17

18 18

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