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Dependability Theory and Methods 2. 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. Bertinoro, March 10-14, 2003.

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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

slide2

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

slide3

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
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 rbds1
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 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
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

series system with time independent failure rate

-  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
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
Series system: an example

Bertinoro, March 10-14, 2003

series system an example1
Series system: an example

Bertinoro, March 10-14, 2003

improving the reliability of a series system
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
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 method1
The part-count method

Bertinoro, March 10-14, 2003

redundant systems
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

parallel redundancy

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
Parallel system

Therefore:

Bertinoro, March 10-14, 2003

parallel redundancy1

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

2 component parallel system

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

2 component parallel system constant failure rate

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
Parallel system: an example

Bertinoro, March 10-14, 2003

partial redundancy an example
Partial redundancy: an example

Bertinoro, March 10-14, 2003

availability for parallel system
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
Series-parallel systems

Bertinoro, March 10-14, 2003

system vs component redundancy
System vs component redundancy

Bertinoro, March 10-14, 2003

component redundant system an example
Component redundant system: an example

Bertinoro, March 10-14, 2003

is redundancy always useful
Is redundancy always useful ?

Bertinoro, March 10-14, 2003

stand by redundancy
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

stand by redundancy1

A

B

x

t

0

B

A

Stand-by redundancy

Bertinoro, March 10-14, 2003

majority voting redundancy

A1

Voter

A2

A3

Majority voting redundancy

Bertinoro, March 10-14, 2003

2 3 majority voting redundancy

A1

Voter

A2

A3

2:3 majority voting redundancy

Bertinoro, March 10-14, 2003