A. BobbioBertinoro, March 10-14, Dependability Theory and Methods 2. Reliability Block Diagrams Andrea Bobbio Dipartimento di Informatica Università del Piemonte Orientale, “A. Avogadro” Alessandria (Italy) - Bertinoro, March 10-14, 2003

A. BobbioBertinoro, March 10-14, Model Types in Dependability Combinatorial models assume that components are statistically independent: poor modeling power coupled with high analytical tractability.  Reliability Block Diagrams, FT, …. State-space models rely on the specification of the whole set of possible states of the system and of the possible transitions among them.  CTMC, Petri nets, ….

A. BobbioBertinoro, March 10-14, 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.

A. BobbioBertinoro, March 10-14, 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.

A. BobbioBertinoro, March 10-14, 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.

A. BobbioBertinoro, March 10-14, Series system of n components. Components are statistically independent Define event E i = “component i functions properly.” Series system in RBD A1A1A2A2AnAn P(E i ) is the probability “component i functions properly”  the reliability R i (t) (non repairable)  the availability A i (t) (repairable)

A. BobbioBertinoro, March 10-14, Reliability of Series system Series system of n components. Components are statistically independent Define event E i = "component i functions properly.” A1A1A2A2AnAn Denoting by R i (t) the reliability of component i Product law of reliabilities:

A. BobbioBertinoro, March 10-14, 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: R s (t) = e - s t with s =  i i=1 n R i (t) = e - i t 1 1 MTTF = —— = ———— s  i i=1 n

A. BobbioBertinoro, March 10-14, Availability for Series System Assuming independent repair for each component, where A i is the (steady state or transient) availability of component i

A. BobbioBertinoro, March 10-14, Series system: an example

A. BobbioBertinoro, March 10-14, Series system: an example

A. BobbioBertinoro, March 10-14, 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.

A. BobbioBertinoro, March 10-14, 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.

A. BobbioBertinoro, March 10-14, The part-count method

A. BobbioBertinoro, March 10-14, 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.

A. BobbioBertinoro, March 10-14, Parallel redundancy A system consisting of n independent components in parallel. It will fail to function only if all n components have failed. E i = “The component i is functioning” E p = “the parallel system of n component is functioning properly.” A1A1 AnAn

A. BobbioBertinoro, March 10-14, Parallel system Therefore :

A. BobbioBertinoro, March 10-14, Parallel redundancy F i (t) = P (E i ) Probability component i is not functioning (unreliability) R i (t) = 1 - F i (t) = P (E i ) Probability component i is functioning (reliability) A1A1 AnAn — F p (t) =  F i (t) i=1 n R p (t) = 1 - F p (t) = 1 -  (1 - R i (t)) i=1 n

A. BobbioBertinoro, March 10-14, component parallel system For a 2-component parallel system: F p (t) = F 1 (t) F 2 (t) R p (t) = 1 – (1 – R 1 (t)) (1 – R 2 (t)) = = R 1 (t) + R 2 (t) – R 1 (t) R 2 (t) A1A1 A2

A. BobbioBertinoro, March 10-14, component parallel system: constant failure rate For a 2-component parallel system with constant failure rate: R p (t) = A1A1 A2 e - 1 t + e - 2 t – e - ( ) t MTTF = —— + —— – ————

A. BobbioBertinoro, March 10-14, Parallel system: an example

A. BobbioBertinoro, March 10-14, Partial redundancy: an example

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

A. BobbioBertinoro, March 10-14, Series-parallel systems

A. BobbioBertinoro, March 10-14, System vs component redundancy

A. BobbioBertinoro, March 10-14, Component redundant system: an example

A. BobbioBertinoro, March 10-14, Is redundancy always useful ?

A. BobbioBertinoro, March 10-14, Stand-by redundancy A B The system works continuously during 0 — t if: a)Component A did not fail between 0 — t b)Component A failed at x between 0 — t, and component B survived from x to t. x 0 t A B

A. BobbioBertinoro, March 10-14, Stand-by redundancy A B x 0 t A B

A. BobbioBertinoro, March 10-14, A B Stand-by redundancy (exponential components)

A. BobbioBertinoro, March 10-14, Majority voting redundancy A1A1 A2 A3 Voter

A. BobbioBertinoro, March 10-14, :3 majority voting redundancy A1A1 A2 A3 Voter