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Reliability Block Diagram

- Combinatorial Models
- RBD for Series Systems
- RBD for Parallel Systems

MODELING TAXONOMY

Simulation

Modeling

Non-State-Space Method

Analytic modeling

State-Space Method

Non-State-Space Modeling Techniques TAXONOMY

Non-State-Space method

Performance models

Dependability models

Reliability Graphs

Queuing models

Fault Tree models

Reliability Block Diagram models

Combinatorial Approach

- If a system consisting of n components, and every

component is either working or failed, then we

can simply list out of all the possible

combinations and calculate the probability for

each combination.

Combinatorial Method

- Use probabilistic techniques to enumerate the

different ways in which a system can remain

operational - The reliability of a system is derived in terms

of the reliabilities of the individual components

of the system (thus the term combinatorial)

Complexity Concerns

- How many possible combinations of the status of

these n components? - What can be done to manage the complexity?
- During model construction
- Need a more intelligent way to describe the

systems failure behavior - Series and parallel RBD (Reliability Block

Diagram) approach - During model solution
- Need more efficient ways of calculations, rather

than counting individual probabilities

Structured Combinatorial Approach

- Reliability block diagrams
- Integrate certain probability events into a

module, which contains the info - A probability of failure
- A failure rate
- A distribution of time to failure
- Steady-state and instantaneous unavailability
- Organize the modules in a structured way,

according to the effects of each modules failure - Statistical independence Assumption
- Failures independence
- Repairs independence

Structured Combinatorial models

- Reliability block diagrams, Fault trees and

Reliability graphs - Integrate certain probability events into a

module - Organize the modules in a structured way,

according to the effects of each modules failure - Commonly used in reliability, availability, or

safety assessment - These model types are similar in that they

capture conditions that make a system fail in

terms of the structural relationships between the

system components.

RBD Features

- Easy to use
- Assuming statistical independence
- Failures independence
- Repairs independence
- Each component can have attached to it
- A probability of failure
- A failure rate
- A distribution of time to failure
- Steady-state and instantaneous unavailability

RBD Features continue

- Easy specification,
- Fast computation
- Relatively good algorithms are available for

solving such models so that 100 component systems

can be handled computationally (consider the case

where you need to handle 2100 probability events,

or simply the homework problem you just

experienced)

Series Systems

- No redundancy
- Each component is needed to make the system work
- If any one of the components fails, the system

fails - Example

The purpose of this example is to show how to

construct a simple series RBD model and solve it

using Excel

RDB Example for a Series System

- System Block Diagram for Example

Reliability Block Diagram Model Reliability

Calculation

- RBD for Example

Processor

Monitor

Keyboard

Let ?1 be the failure rate for Monitor Assume

exponential distribution for the failures,

thenRmonitor(t) e -?1 t Similarly,

Rprocessor(t) e -?2 t and Rkeyboardv(t) e

-?3 t

Rsystem (t) Rmonitor (t) Rprocessor (t)

Rkeyboard (t) e -?1 t e -?2 t e -?3 t

e (?1 t ?2 t ?3 t) e (?1?2?3) t

When exponential failure distribution is

assumed, the failure rate of a series system is

the sum of individual components failure rates

Real-Time Exercise

- Use Excel Spreadsheet to construct the above

Series RBD - Show the trend of reliability with regard to the

time factor - Show the relationship between reliability and the

failure rate

SS-Availability Calculation

Let ?1, ?2, ?3 be the failure rates and ?1, ?2,

?3 be the repair rates for the monitor, processor

and keyboard. Then

- ASS-Monitor
- ASS-processor
- ASS-keyboard

ASS-system-series

Parallel Systems

- A basic parallel system only one of the N

identical components is required for the system

to function - Example

Example Basic Parallel System

- System Block Diagram

The purpose here is to show the parallel RBD and

the corresponding reliability/availability

calculations.

RDB example Parallel System

- Reliability Block Diagram

RDB using Hierarchical Composition/Decomposition

The Highest level (overall system level)

Computer

Computer

or

1 of 2

1 of 2

Usually indicate two different components

On the Computer level

Monitor

Processor

Keyboard

Reliability Calculation

- The Unreliability of the parallel system can be

computed as the probability that all N components

fail. - Assume all N components are having the same

failure rate ?, and the probability that a

component is failed at time t is Pfail(t) - Rparallel(t) 1- ?i1 to N Pfail(t)
- If exponential distribution is used for Pfail(t),

derive the formula for Rparallel(t)

Independence Assumption

- Where in the above equation that the independence

assumption is made? - Just to remind you

- Failure/Repair Dependencies are often assumed
- RBD usually does not handle the dependency such

as - Event-dependent failure
- Shared repair

Availability Calculation

- ASS-Monitor
- ASS-processor
- ASS-keyboard

ASS-system-parallel

Monitor

Processor

Keyboard

Monitor

Processor

Keyboard

Exercise (using Excel)

- ?monitor 1? 10-4 failures per hour
- ?processor 1? 10-5 failures per hour
- ?keyboard 4? 10-4 failures per hour
- ? 2 repair per hour for all components
- For series system, ASS is
- For parallel system (with 12 redundancy), ASS is

Parallel/Series System Example

Processor 1

Keyboard 1

Monitor 1

Bus 1

Bus 2

Computer 2

Keyboard 1

Monitor 1

What is the corresponding RBD ?

Corresponding RBD

Assuming Buses are perfect

Monitor

Processor

Keyboard

Keyboard

Monitor

Processor

Compare to the RBD below, which one has better

reliability?

Monitor

Processor

Keyboard

Monitor

Processor

Keyboard

Numerical Comparison(1)

Monitor

Processor

Keyboard

Keyboard

Processor

Monitor

Component Pw Pf Pw (1 of

2) Monitor 0.99 0.01 0.9999 Keyboard 0.9

0.1 0.99 Processor 0.999 0.001

0.999999 Psystem-w

0.98990001

Numerical Comparison (2)

Monitor

Processor

Keyboard

Monitor

Processor

Keyboard

Component Pw Pf Pw-single Psystem-w 0.890109 0

.987923968 Monitor 0.99 0.01 Keyboard 0.9 0.1 P

rocessor 0.999 0.001

Does this analysis result make sense?

Modeling Steps

- Model construction
- Model parameterization
- Model solution
- Result interpretation
- Model validation

N Modular Redundancy

- M of N System
- M of the total of N identical modules are

required to function, M ? N - TMR (Triple Modular Redundancy) is a famous

example, where M is 2 and N is 3

Example 6 RBD for TMR

Module 1

Voter

Module 2

Module3

Module 3

Single point of failure

Module2

Voter

Module1

2 3

Reliability Calculation for TMR

Module3

Module2

Voter

Module1

- Cases for the TMR to be working
- all of the 3 modules are working
- any 2 modules are working, and 1 module is

failed - Look at it from another way
- Cases for the TMR to be failed
- all 3 modules are failed
- any one module is working, however, the rest 2

are not working - Remember, the voter is a Single-Point-Of-Failure

2 3

Module voter TMR System Pw 0.999 0.999 0.9999

97 0.998997005

From this chart, you can see the effect that a

single point of failure made ismuch more

significant than that of a component with

redundancy

Bottom Line

- RBD provides the vehicles for analysts to

construct models easier than the combinatorial

approach - The fundamental math is the same
- The reliability/availability calculation methods

are provided by the tool - RMODEL

Hierarchical Composition Method

- Given a detailed description of a system, too

many components are displayed, which makes the

modeling task difficult which creates unnecessary

complexity - Abstract the detailed description into a higher

level description hierarchical composition

method

Hierarchical Composition/Decomposition

- The size of the model grows with the size of the

system. - Issue of Fidelity vs. Complexity
- The hierarchical composition method

Hudson Professor of Electrical and Computer

Engineering Duke University Phone (919)

660-5269Fax (919) 660-5293Email

kst_at_ee.duke.edu

Trivedi

Exercise A simple Aircraft Control System

- Use the system block diagram given in the

handout, construct the corresponding RBD - Abstract the system block diagram into a higher

level block diagram - From the higher level system block diagram,

construct the corresponding RBD - Each block in the higher level RBD has its own

RBD underneath

Failure rates

- ?sensor 1? 10-6 failures per hour
- ?actuator 1? 10-5 failures per hour
- ?computer 4? 10-4 failures per hour
- ?bus 1? 10-6 failures per hour

Homework

- Plot the system reliability as a function of
- The failure rate of Computer
- The failure rate of Actuator
- The failure rate of Bus
- The failure rate of Sensor

JPAL Example Demonstration

Homework

- Text P. 35 Problem 2
- Text P. 36-37 Problem 5

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