1. 16111 presentation by Arindam Das System Reliability Prediction A presentation by Arindam Das

2. 26111 presentation by Arindam Das Contents why Reliability Prediction is important ? what is Reliability (quantitatively) ? models and techniques to evaluate Reliability conclusions

3. 36111 presentation by Arindam Das Why is System Reliability prediction so important? avoid economic loss - online stock trading halted – server down - phone calls not going through – switch down - power-grid failures

4. 46111 presentation by Arindam Das System Reliability prediction - so important? avoid loss of human life - Automobile accidents - Aircraft crashes - Nuclear Plant disasters

5. 56111 presentation by Arindam Das System Reliability prediction - so important? D o reliability prediction in the early stages of product development cycle

6. 66111 presentation by Arindam Das What is Reliability ? ability of a component to function for at least a given time interval  component may be - Processor in a computer - Web server in a Computer network - Engine system in an aircraft - Brake system in a car - Reactor Coolant System in Nuclear Power Plant

7. 76111 presentation by Arindam Das …….i understand ability…… what the heck is this probability ?%#$$!@@?? …... i guess, matholoafers are probing the ability of …… %#$$!@ Reliability of the system– (quantitatively defined as….) probability of the system being up and running for at least a given time-interval

8. 86111 presentation by Arindam Das Reliability Models 1.Reliability Block Diagrams (RBD) - Series-Parallel - Non-series Parallel 2.Fault trees - Without Repeated - Repeated …most often, models are used to evaluate reliability…. state enumeration Conditioning Minpath SDP

9. 96111 presentation by Arindam Das A Reliability Block Diagram : Series-Parallel a computer system – 2 processors, 3 memories P1 P2 M1 M3 M2 y x R( sys ) = R( P1-P2 subsystem) multiplied by R(M1-M2-M3 subsystem)

10. 106111 presentation by Arindam Das Reliability Block Diagram: Series-Parallel (contd) a computer system – 2 processors, 3 memories…….. Failed when both are down … P1 P2 z x R( P1-P2 subsystem) = 1 - U( P1 ) multiplied by U( P2 ) = 1 - ( 1 – R( P1 ) ) multiplied by ( 1 - R( P2 ) )

11. 116111 presentation by Arindam Das Reliability Block Diagram: Series-Parallel (contd) a computer system – 2 processors, 3 memories…….. R(M1-M2-M3 subsystem) = 1 - U( M1 ) multiplied by U( M2 ) multiplied by U( M3 ) = 1 - ( 1 – R(P1) ) multiplied by ( 1 – R(P2) ) multiplied by ( 1 – R(P2) ) M1 M3 M2 yz Failed when both are down …

12. 126111 presentation by Arindam Das A Reliability Block Diagram (RBD): Non-Series-Parallel design P1 P2 M1 M3 M2 a b

13. 136111 presentation by Arindam Das System Reliability Computation: by Conditioning P1 P2 M1 M3 M2 a b P1 P2 P1 P2 M1 M2 M3 up M3 down

14. 146111 presentation by Arindam Das P1 P2 M3 up R( sys | M3 is up ) = 1 - ( 1 – R( P1 ) ). ( 1 – R( P2 ) ) System Reliability: by Conditioning (contd.)

15. 156111 presentation by Arindam Das P1 P2 M1 M2 M3 down R( sys | M3 is down) = 1 - ( 1 – R( P1 ) R( M1 ) ). ( 1 – R( P2 ) R( M2 ) ) System Reliability: by Conditioning (contd.)

16. 166111 presentation by Arindam Das P1 P2 M1 M3 M2 ab P1 P2 P1 P2 M1 M2 M3 up M3 down R( sys ) = R( M3 is up ). R( sys | M3 is up ) + R( M3 is down ). R( sys | M3 is down ) (by theorem of total probability) … by Conditioning

17. 176111 presentation by Arindam Das Reliability - by Enumeration 1.Construct Truth Table 2.Compute probability of each row in Truth Table 3. Add probabilities of all those rows where System column has 1

18. 186111 presentation by Arindam Das Reliability - by Enumeration (contd…) 12345Probability P1P2M1M2M3System 010011R1' R2 R3' R4' R5 010101R1' R2 R3' R4 R5' 101001……. row – up/down - system config rows mutually exclusive R(sys) = R1' R2 R3' R4' R5 + R1' R2 R3' R4 R5' + ………… P1 P2 M1 M3 M2 a b

19. 196111 presentation by Arindam Das Reliability – by Minpath technique 1.Find the System-up event using Minpath 2.Apply Inclusion/Exclusion method OR Sum-of-Disjoint-Product method on System-up event

20. 206111 presentation by Arindam Das Reliability – by Minpath (contd…) Path - a set of components such that if they are all up, the System is up Minpath – a path that has no proper subpaths. { P1, P2, M3 } { P1, M3 } P1 P2 M1 M3 M2 a b

21. 216111 presentation by Arindam Das P1 P2 M1 M3 M2 a b Minpaths { P1, M1 } { P1, M3 } { P2, M3 } { P2, M2 } Reliability – by Minpath (contd…)

22. 226111 presentation by Arindam Das P1 P2 M1 M3 M2 a b System is Up = P1 is Up and M1 is Up Or P1 is Up and M3 is Up Or P2 is Up and M3 is Up Or P2 is Up and M2 is Up Reliability – by Minpath (contd…)

23. 236111 presentation by Arindam Das Reliability – by Minpath (contd…) Inclusion/Exclusion technique: P(A) + P(B) Apply this on System is Up event equation

24. 246111 presentation by Arindam Das Reliability – by Minpath (contd…) Sum-of-Disjoint Product technique: P(A) + P( A' B) Apply this on System is Up event equation

25. 256111 presentation by Arindam Das Reliability – by Minpath (contd…) Inclusion/Exclusion - P(A) + P(B) Inclusion/Exclusion Vs Sum-of-Disjoint ??? Sum-of-Disjoint Product - P(A) + P( A' B) Sum-of-disjoint product has less terms to be worried about…

26. 266111 presentation by Arindam Das P1 P2 M1 M3 M2 y x probability that the system will work for at least 30 days ??  R(P1) = R(P2) = 0.959  R(M1) = R(M2) = R(M3) = 0.795 R(Sys) = 0.9897 Assume - Lifetime of each comp follows exp. distrib. We know - Reliability of each comp is a fn of time. For each comp, we plugin time=30 failure-rate (no. of failures per day) Thus, we get these probs.

27. 276111 presentation by Arindam Das prob. that the system will work for at least 30 days ?? - R( P1 ) = R( P2 ) = 0.959 - R( M1 ) = R( M2 ) = R( M3 ) = 0.795 R( Sys ) = 0.9871 P1 P2 M1 M3 M2 a b

28. 286111 presentation by Arindam Das P1 P2 M1 M3 M2 y x P1 P2 M1 M3 M2 a b R(Sys) = 0.9871 R(Sys) = 0.9897 Which design to choose from ?

29. 296111 presentation by Arindam Das Failure or and P1'P2'M1'M2'M3' P1 P2 M1 M3 M2 yx RBDFault-Tree Fault Tree representation - series-parallel

30. 306111 presentation by Arindam Das Failure or and P1'P2'M1'M2'M3' and or P1 P2 M1 M3 M2 a b RBDFault-Tree Fault Tree representation - non-series-parallel

31. 316111 presentation by Arindam Das Conclusion Reliability prediction for a system – very important Other techniques  Binary-Decision Diagram We assumed o Component fails independent of each other To handle dependent failures - Markov Chain based techniques

32. 326111 presentation by Arindam Das References Kishor Trivedi, Probability and Statistics with Reliability, Queuing and Computer Science Applications, 2 nd Edition ( www.ee.duke.edu/~kst )