Dependability Theory and Methods 2. Reliability Block Diagrams

Dependability Theory and Methods2. 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 Bertinoro, March 10-14, 2003 Bertinoro, March 10-14, 2003

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, …. Bertinoro, March 10-14, 2003

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. Bertinoro, March 10-14, 2003

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. Bertinoro, March 10-14, 2003

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. Bertinoro, March 10-14, 2003

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) Bertinoro, March 10-14, 2003

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: Bertinoro, March 10-14, 2003

-  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 Bertinoro, March 10-14, 2003

Availability for Series System • Assuming independent repair for each component, • where Ai is the (steady state or transient) availability of component i Bertinoro, March 10-14, 2003

Series system: an example Bertinoro, March 10-14, 2003

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. Bertinoro, March 10-14, 2003

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. Bertinoro, March 10-14, 2003

The part-count method Bertinoro, March 10-14, 2003

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. Bertinoro, March 10-14, 2003

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.” Bertinoro, March 10-14, 2003

Parallel system Therefore: Bertinoro, March 10-14, 2003

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 Bertinoro, March 10-14, 2003

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) Bertinoro, March 10-14, 2003

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 Bertinoro, March 10-14, 2003

Parallel system: an example Bertinoro, March 10-14, 2003

Partial redundancy: an example Bertinoro, March 10-14, 2003

Availability for parallel system • Assuming independent repair, • where Ai is the (steady state or transient) availability of component i. Bertinoro, March 10-14, 2003

Series-parallel systems Bertinoro, March 10-14, 2003

System vs component redundancy Bertinoro, March 10-14, 2003

Component redundant system: an example Bertinoro, March 10-14, 2003

Is redundancy always useful ? Bertinoro, March 10-14, 2003

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 Bertinoro, March 10-14, 2003

A B x t 0 B A Stand-by redundancy Bertinoro, March 10-14, 2003

A B Stand-by redundancy (exponential components) Bertinoro, March 10-14, 2003

A1 Voter A2 A3 Majority voting redundancy Bertinoro, March 10-14, 2003

A1 Voter A2 A3 2:3 majority voting redundancy Bertinoro, March 10-14, 2003

Dependability Theory and Methods 2. Reliability Block Diagrams