A. Bobbio Bertinoro, March 10-14, 2003 1

Dependability Theory and Methods

2. Reliability Block Diagrams

Andrea BobbioDipartimento 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

A. Bobbio Bertinoro, March 10-14, 2003 2

Model Types in DependabilityModel Types in DependabilityCombinatorial 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 Bertinoro, March 10-14, 2003 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.

A. Bobbio Bertinoro, March 10-14, 2003 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.

A. Bobbio Bertinoro, March 10-14, 2003 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.

A. Bobbio Bertinoro, March 10-14, 2003 6

Series system of n components.

Components are statistically independent

Define event Ei = “component i functions properly.”

Series system in RBD

)()...()( )...(

)""(

2121 nn EPEPEPEEEP

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)

A. Bobbio Bertinoro, March 10-14, 2003 7

Reliability of Series system

Series system of n components.

Components are statistically independent

Define event Ei = "component i functions properly.”

)()...()( )...(

)""(

2121 nn EPEPEPEEEP

P

properly ng functioni is system series

A1 A2 An

n

iis tRtR

1

)()(

Denoting by R i(t) the reliability of component i

Product law of reliabilities:

A. Bobbio Bertinoro, March 10-14, 2003 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:

Rs(t) = e- s t with s = i

i=1

n

Ri (t) = e- i t

1 1MTTF = —— = ———— s i

i=1

n

A. Bobbio Bertinoro, March 10-14, 2003 9

Availability for Series System

Assuming independent repair for each component,

where Ai is the (steady state or transient) availability of component i

n

iis

n

i ii

in

iis

tAtA

MTTRMTTF

MTTFAA

1

11

)()(

or ,

A. Bobbio Bertinoro, March 10-14, 2003 10

Series system: an example

A. Bobbio Bertinoro, March 10-14, 2003 11

Series system: an example

A. Bobbio Bertinoro, March 10-14, 2003 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.

A. Bobbio Bertinoro, March 10-14, 2003 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.

A. Bobbio Bertinoro, March 10-14, 2003 14

The part-count method

A. Bobbio Bertinoro, March 10-14, 2003 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.

A. Bobbio Bertinoro, March 10-14, 2003 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.”

A1

An

...

...

A. Bobbio Bertinoro, March 10-14, 2003 17

Parallel system

"failedhassystemparallelThe"pE

"failedhavecomponentsnAll"____

2

__

1 ... nEEE

)...()(____

2

__

1

__

np EEEPEP )()...()(____

2

__

1 nEPEPEP

Therefore:

)(1)( pp EPEP

A. Bobbio Bertinoro, March 10-14, 2003 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)

A1

An

...

...

—

Fp (t) = Fi (t) i=1

n

Rp (t) = 1 - Fp (t) = 1 - (1 - Ri (t)) i=1

n

A. Bobbio Bertinoro, March 10-14, 2003 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)

A1

A2

A. Bobbio Bertinoro, March 10-14, 2003 20

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 1MTTF = —— + —— – ———— 1 2 1 + 2

A. Bobbio Bertinoro, March 10-14, 2003 21

Parallel system: an example

A. Bobbio Bertinoro, March 10-14, 2003 22

Partial redundancy:

an example

A. Bobbio Bertinoro, March 10-14, 2003 23

Availability for parallel system

Assuming independent repair,

where Ai is the (steady state or transient) availability of component i.

n

iip

n

i ii

in

iip

tAtAor

MTTRMTTF

MTTRAA

1

11

))(1(1)(

1)1(1

A. Bobbio Bertinoro, March 10-14, 2003 24

Series-parallel systems

A. Bobbio Bertinoro, March 10-14, 2003 25

System vs component redundancy

A. Bobbio Bertinoro, March 10-14, 2003 26

Component redundant system: an example

A. Bobbio Bertinoro, March 10-14, 2003 27

Is redundancy always useful ?

A. Bobbio Bertinoro, March 10-14, 2003 28

Stand-by redundancyA

B

The system works continuouslyduring 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 .

x0 tA B

A. Bobbio Bertinoro, March 10-14, 2003 29

Stand-by redundancyA

B

x0 tA B

A. Bobbio Bertinoro, March 10-14, 2003 30

A

B

Stand-by redundancy (exponential

components)

A. Bobbio Bertinoro, March 10-14, 2003 31

Majority voting redundancy

A1

A2

A3

Voter

A. Bobbio Bertinoro, March 10-14, 2003 32

2:3 majority voting redundancy

A1

A2

A3

Voter