1 / 32

Dependability & Maintainability Theory and Methods 3 . Reliability Block Diagrams

Dependability & Maintainability Theory and Methods 3 . Reliability Block Diagrams. Andrea Bobbio Dipartimento di Informatica Universit à del Piemonte Orientale, “ A. Avogadro ” 15100 Alessandria (Italy) bobbio@unipmn.it - http://www.mfn.unipmn.it/~bobbio/IFOA.

david-rowland
Download Presentation

Dependability & Maintainability Theory and Methods 3 . Reliability Block Diagrams

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams • Andrea Bobbio • Dipartimento di Informatica • Università del Piemonte Orientale, “A. Avogadro” • 15100 Alessandria (Italy) • bobbio@unipmn.it - http://www.mfn.unipmn.it/~bobbio/IFOA IFOA, Reggio Emilia, June 17-18, 2003 Reggio Emilia, June 17-18, 2003

  2. Model Types in Dependability Combinatorial models assume that components are statisticallyindependent: poor modeling power coupled with highanalytical tractability. Reliability Block Diagrams, FT, …. State-space models rely on the specification of the whole set ofpossible states of the system and of the possible transitionsamong them. CTMC, Petri nets, …. Reggio Emilia, June 17-18, 2003

  3. Reliability Block Diagrams • Each component of the system is represented as a block; • System behavior is represented by connecting the blocks; • Failures of individual components are assumed to be independent; • Combinatorial (non-state space) model type. Reggio Emilia, June 17-18, 2003

  4. Reliability Block Diagrams (RBDs) • Schematic representation or model; • Shows reliability structure (logic) of a system; • Can be used to determine dependability measures; • A block can be viewed as a “switch” that is “closed” when the block is operating and “open” when the block is failed; • System is operational if a path of “closed switches” is found from the input to the output of the diagram. Reggio Emilia, June 17-18, 2003

  5. Reliability Block Diagrams (RBDs) • Can be used to calculate: • Non-repairable system reliability given: • Individual block reliabilities (or failure rates); • Assuming mutually independent failures events. • Repairable system availability given: • Individual block availabilities (or MTTFs and MTTRs); • Assuming mutually independent failure and restoration events; • Availability of each block is modeled as 2-state Markov chain. Reggio Emilia, June 17-18, 2003

  6. Series system in RBD • Series system of n components. • Components are statistically independent • Define event Ei = “component i functions properly.” A1 A2 An • P(Ei) is the probability “component i functions properly” • the reliability R i(t)(non repairable) • the availabilityAi(t)(repairable) Reggio Emilia, June 17-18, 2003

  7. Reliability of Series system • Series system of n components. • Components are statistically independent • Define event Ei = "component i functions properly.” A1 A2 An Denoting byR i(t)the reliability of component i Product law of reliabilities: Reggio Emilia, June 17-18, 2003

  8. -  s t Rs(t) = e Series system with time-independent failure rate • Let i be the time-independent failure rate of component i. • Then: • The system reliability Rs(t) becomes: -  i t Ri (t) = e n with s =  i i=1 1 1 MTTF = —— = ———— s n  i i=1 Reggio Emilia, June 17-18, 2003

  9. Availability for Series System • Assuming independent repair for each component, • where Ai is the (steady state or transient) availability of component i Reggio Emilia, June 17-18, 2003

  10. Series system: an example Reggio Emilia, June 17-18, 2003

  11. Series system: an example Reggio Emilia, June 17-18, 2003

  12. Improving the Reliability of a Series System • Sensitivity analysis:  R s R s S i = ———— = ————  R i R i The optimal gain in system reliability is obtained by improving the least reliable component. Reggio Emilia, June 17-18, 2003

  13. The part-count method • It is usually applied for computing the reliability of electronic equipment composed of boards with a large number of components. Components are connected in series and with time-independent failure rate. Reggio Emilia, June 17-18, 2003

  14. The part-count method Reggio Emilia, June 17-18, 2003

  15. Redundant systems • When the dependability of a system does not reach the desired (or required) level: • Improve the individual components; • Act at the structure level of the system, resorting to redundant configurations. Reggio Emilia, June 17-18, 2003

  16. A1 . . . . . . An Parallel redundancy A system consisting of nindependent components in parallel. It will fail to function only if all ncomponents have failed. Ei = “The component i is functioning” Ep= “the parallel system of n component is functioning properly.” Reggio Emilia, June 17-18, 2003

  17. Parallel system Therefore: Reggio Emilia, June 17-18, 2003

  18. A1 . . . . . . An Parallel redundancy — Fi(t) = P (Ei) Probability component i is not functioning (unreliability) Ri(t) = 1 - Fi(t) = P (Ei) Probability component i is functioning (reliability) n Fp(t) = Fi(t) i=1 n Rp(t) = 1 - Fp(t) = 1 -  (1 - Ri(t)) i=1 Reggio Emilia, June 17-18, 2003

  19. A1 A2 2-component parallel system For a 2-component parallel system: Fp(t) = F1(t)F2(t) Rp(t) = 1 –(1 – R1(t)) (1 – R2(t)) = = R1(t) + R2(t) –R1(t) R2(t) Reggio Emilia, June 17-18, 2003

  20. A1 A2 - 1 t e 2-component parallel system: constant failure rate For a 2-component parallel system with constant failure rate: -  2 t - ( 1 + 2 )t + e – e Rp(t) = 1 1 1 MTTF = —— + —— – ———— 121 +2 Reggio Emilia, June 17-18, 2003

  21. Parallel system: an example Reggio Emilia, June 17-18, 2003

  22. Partial redundancy: an example Reggio Emilia, June 17-18, 2003

  23. Availability for parallel system • Assuming independent repair, • where Ai is the (steady state or transient) availability of component i. Reggio Emilia, June 17-18, 2003

  24. Series-parallel systems Reggio Emilia, June 17-18, 2003

  25. System vs component redundancy Reggio Emilia, June 17-18, 2003

  26. Component redundant system: an example Reggio Emilia, June 17-18, 2003

  27. Is redundancy always useful ? Reggio Emilia, June 17-18, 2003

  28. Stand-by redundancy A The system works continuously during 0 — tif: B • Component Adid not fail between 0 — t • Component A failed at x between 0 — t, and component Bsurvived from x to t. x t 0 B A Reggio Emilia, June 17-18, 2003

  29. A B x t 0 B A Stand-by redundancy Reggio Emilia, June 17-18, 2003

  30. A B Stand-by redundancy (exponential components) Reggio Emilia, June 17-18, 2003

  31. A1 Voter A2 A3 Majority voting redundancy Reggio Emilia, June 17-18, 2003

  32. A1 Voter A2 A3 2:3 majority voting redundancy Reggio Emilia, June 17-18, 2003

More Related