RBDs-Redundancia-Disponibilidad_(slide).ppt

A. Bobbio Reggio Emilia, June 17-18, 2003 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@unipmn.it - http://www.mfn.unipmn.it/~bobbio/IFOA
IFOA, Reggio Emilia, June 17-18, 2003

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, ….

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.

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.

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.

Series system of n components.
Components are statistically independent
Define event Ei = “component i functions properly.”
Series system in RBD
)
(
)...
(
)
(
)
...
(
)
"
"
(
2
1
2
1 n
n E
P
E
P
E
P
E
E
E
P
P






properly
g
functionin
is
system
series
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)

Reliability of Series system
Series system of n components.
Components are statistically independent
Define event Ei = "component i functions properly.”
)
(
)...
(
)
(
)
...
(
)
"
"
(
2
1
2
1 n
n E
P
E
P
E
P
E
E
E
P
P






properly
ng
functioni
is
system
series
A1 A2 An



n
i
i
s t
R
t
R
1
)
(
)
(
Denoting by R i(t) the reliability of component i
Product law of reliabilities:

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:
Rs(t) = e
-  s t
with s =  i
i=1
n
Ri (t) = e
-  i t
1 1
MTTF = —— = ————
s
 i
i=1
n

Availability for Series System
Assuming independent repair for each component,
where Ai is the (steady state or transient) availability
of component i










n
i
i
s
n
i i
i
i
n
i
i
s
t
A
t
A
MTTR
MTTF
MTTF
A
A
1
1
1
)
(
)
(
or
,

Series system:
an example

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.

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.

The part-count method

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.

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.”
A1
An
.
.
.
.
.
.

Parallel system
"
failed
has
system
parallel
The
"

p
E
"
failed
have
components
n
All
"

__
__
2
__
1 ... n
E
E
E 



)
...
(
)
(
__
__
2
__
1
__
n
p E
E
E
P
E
P 


 )
(
)...
(
)
(
__
__
2
__
1 n
E
P
E
P
E
P

Therefore:
)
(
1
)
( p
p E
P
E
P 


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)
A1
An
.
.
.
.
.
.
—
Fp (t) =  Fi (t)
i=1
n
Rp (t) = 1 - Fp (t) = 1 -  (1 - Ri (t))
i=1
n

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)
A1
A2

2-component parallel system:
constant failure rate
For a 2-component parallel system
with constant failure rate:
Rp (t) =
A1
A2
e
- 1 t
+ e
-  2 t
– e
- ( 1 +  2 ) t
1 1 1
MTTF = —— + —— – ————
1 2 1 + 2

Parallel system:
an example

Partial
redundancy:
an example

Availability for parallel system
Assuming independent repair,
where Ai is the (steady state or transient) availability of
component i.















n
i
i
p
n
i i
i
i
n
i
i
p
t
A
t
A
or
MTTR
MTTF
MTTR
A
A
1
1
1
))
(
1
(
1
)
(
1
)
1
(
1

Series-parallel
systems

System vs component redundancy

Component redundant system:
an example

Is redundancy always useful ?

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

Stand-by redundancy
A
B
x
0 t
A B

A
B
Stand-by redundancy
(exponential
components)

Majority voting
redundancy
A1
A2
A3
Voter

2:3 majority voting redundancy
A1
A2
A3
Voter

RBDs-Redundancia-Disponibilidad_(slide).ppt

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