Dependability & Maintainability
Theory and Methods
3. Reliability Block Diagrams
Andrea Bobbio
Dipartimento di Informatica
Università del Piemonte Orientale, “A. Avogadro”
15100 Alessandria (Italy)
[email protected] - http://www.mfn.unipmn.it/~bobbio/IFOA
A. Bobbio
IFOA, Reggio
Emilia,
June 2003
17-18, 2003
Reggio Emilia,
June 17-18,
1
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. Bobbio
Reggio Emilia, June 17-18, 2003
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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.
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Reggio Emilia, June 17-18, 2003
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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. Bobbio
Reggio Emilia, June 17-18, 2003
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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.
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Reggio Emilia, June 17-18, 2003
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Series system in RBD
Series system of n components.
A1
A2
An
Components are statistically independent
Define event Ei = “component i functions properly.”
P(" series system is functionin g properly" )
 P( E1  E2  ...  En )  P( E1 )  P( E2 )...P( En )
P(Ei) is the probability “component i functions properly”
 the reliability R i(t) (non repairable)
 the availability A i(t) (repairable)
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Reggio Emilia, June 17-18, 2003
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Reliability of Series system
Series system of n components.
A1
A2
An
Components are statistically independent
Define event Ei = "component i functions properly.”
P(" series system is functioni ng properly " )
 P( E1  E2  ...  En )  P( E1 )  P( E2 )...P( En )
Denoting by R i(t) the reliability of component i
n
Product law of reliabilities:
Rs (t )   Ri (t )
i 1
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Reggio Emilia, June 17-18, 2003
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Series system with time-independent
failure rate
Let  i be the time-independent failure rate of
component i.
-  it
Then: Ri (t) = e
The system reliability Rs(t) becomes:
Rs(t) = e
-  st
n
with
s =  i
i=1
1
1
MTTF = —— = ————
s  ni
i=1
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Reggio Emilia, June 17-18, 2003
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Availability for Series System
Assuming independent repair for each component,
n
n
MTTFi
As   Ai  
, or
i 1
i 1 MTTFi  MTTRi
n
As (t )   Ai (t )
i 1
where Ai is the (steady state or transient) availability
of component i
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Reggio Emilia, June 17-18, 2003
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Series system:
an example
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Series system:
an example
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Improving the Reliability of a
Series System
Sensitivity analysis:
Si=
 Rs
———— =
 Ri
Rs
————
Ri
The optimal gain in system reliability is obtained by
improving the least reliable component.
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Reggio Emilia, June 17-18, 2003
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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 timeindependent failure rate.
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The part-count method
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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.
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Reggio Emilia, June 17-18, 2003
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Parallel redundancy
A1
A system consisting of n
independent components in parallel.
It will fail to function only if all n
..
.
..
.
An
components have failed.
Ei = “The component i is functioning”
Ep = “the parallel system of n component is
functioning properly.”
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Reggio Emilia, June 17-18, 2003
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Parallel system
E p  " The parallel system has failed "
 " Alln components have failed "
__
__
__
 E1  E2  ...  En
Therefore:
__
__
__
__
__
__
__
P( E p )  P( E1  E2  ...  En )  P( E1 ) P( E2 )...P( En )
P( E p )  1  P( E p )
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Reggio Emilia, June 17-18, 2003
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Parallel redundancy
—
Fi (t) = P (Ei) Probability component i
A1
..
.
is not functioning (unreliability)
Ri (t) = 1 - Fi (t) = P (Ei) Probability
..
.
An
component i is functioning (reliability)
n
Fp (t) =

i=1
Fi (t)
n
Rp (t) = 1 - Fp (t) = 1 -  (1 - Ri (t))
i=1
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Reggio Emilia, June 17-18, 2003
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2-component parallel system
For a 2-component parallel system:
A1
Fp (t) = F1 (t) F2 (t)
A2
Rp (t) = 1 – (1 – R1 (t)) (1 – R2 (t)) =
= R1 (t) + R2 (t) – R1 (t) R2 (t)
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Reggio Emilia, June 17-18, 2003
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2-component parallel system:
constant failure rate
For a 2-component parallel system
A1
with constant failure rate:
A2
Rp (t) = e
- 1 t
+e
1
1
MTTF = —— + ——
1
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2
-  2t
–
– e
- ( 1 +  2 ) t
1
————
 1 + 2
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Parallel system:
an example
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Partial
redundancy:
an example
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Availability for parallel system
Assuming independent repair,
n
n
MTTRi
Ap  1   (1  Ai )  1  
i 1
i 1 MTTFi  MTTRi
n
or Ap (t )  1   (1  Ai (t ))
i 1
where Ai is the (steady state or transient) availability of
component i.
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Reggio Emilia, June 17-18, 2003
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Series-parallel
systems
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System vs component redundancy
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Component redundant system:
an example
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Is redundancy always useful ?
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Stand-by redundancy
A
The system works continuously
during 0 — t if:
B
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
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A
t
B
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Stand-by redundancy
A
x
0
A
t
B
B
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Stand-by redundancy
(exponential
components)
A
B
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Majority voting
redundancy
A1
Voter
A2
A3
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2:3 majority voting redundancy
A1
Voter
A2
A3
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