Dependability

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_at_unipmn.it - http//www.mfn.unipmn.it/bob
  bio/IFOA

IFOA, Reggio Emilia, June 17-18, 2003
2
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, .
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.

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.

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.

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 availability A i(t) (repairable)

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 by R i(t) the reliability of component i
Product law of reliabilities
8
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
i1
1 1 MTTF
?s
n
? ?i
i1
9
Availability for Series System

• Assuming independent repair for each component,
• where Ai is the (steady state or transient)
  availability of component i

10
Series system an example
11
Series system an example
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.
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.
14
The part-count method
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.

16
Parallel redundancy
A system consisting of n independent components
in parallel. It will fail to function only if all
n components have failed.
Ei The component i is functioning Ep the
parallel system of n component is functioning
properly.
17
Parallel system
Therefore
18
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)
i1
n
Rp (t) 1 - Fp (t) 1 - ? (1 - Ri (t))
i1
19
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)
20
2-component parallel system constant failure rate
For a 2-component parallel system with constant
failure rate
- ? 2 t
- (? 1 ? 2 ) t
Rp (t)
e
e
1 1
1 MTTF
?1 ?2 ?1 ?2
21
Parallel system an example
22
Partial redundancy an example
23
Availability for parallel system

• Assuming independent repair,
• where Ai is the (steady state or transient)
  availability of component i.

24
Series-parallel systems
25
System vs component redundancy
26
Component redundant system an example
27
Is redundancy always useful ?
28
Stand-by redundancy
A
The system works continuously during 0 t if
B

• Component A did not fail between 0 t
• Component A failed at x between 0 t , and
  component B survived from x to t .

x
t
0
B
A
29
Stand-by redundancy
30
Stand-by redundancy (exponential components)
31
Majority voting redundancy
32
23 majority voting redundancy