Stracener_EMIS 7305/5305_Spr08_02.05.08 1 Systems Reliability Modeling & Analysis Series and Active...

Stracener_EMIS 7305/5305_Spr08_02.05.08

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Systems ReliabilityModeling & Analysis

Series and Active Parallel Configurations

Dr. Jerrell T. Stracener, SAE Fellow

Leadership in Engineering

EMIS 7305/5305Systems Reliability, Supportability and Availability Analysis

Systems Engineering ProgramDepartment of Engineering Management, Information and Systems


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System Reliability Models

•Series Configurations•Parallel or Redundant Configurations

Active Parallel

r-out-of-n

Standby•Series-Parallel and Parallel-Series Configurations•General


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System Reliability Models

• The reliability definitions, concepts and models presented apply at any level of a system, from a single discrete component up to and including the entire system.

• Systems reliability deals with the reliability of the end-item system and is based on the system configuration and component failure rates as well intended service usage

• There are two basic types of reliability configurationsSeriesParallel or Redundant


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Terminology and Notation

• Path: A physical means for accomplishing a given function.

• Element: The basic system level under discussion. An element may be a Component, an Assembly, an Equipment, a Line Replaceable Unit (LRU), a Subsystem or a System

• Block: A logical representation of an Element.

• Reliability Block Diagram: A logical representation of a System, Subsystem, or Assembly in terms of its Elements.


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Series Reliability Configuration


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Series Reliability Configuration

• Simplest and most common structure in reliability analysis.

• Functional operation of the system depends on the successful operation of all system components Note: The electrical or mechanical configuration may differ from the reliability configuration

Reliability Block Diagram

• Series configuration with n elements: E1, E2, ..., En

• System Failure occurs upon the first element failure

E1 E2 En


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Series Reliability Configuration: Math Models

System Reliability

Rs(t) = P(A1) P(A2|A1) P(A3|A1A2) ... P(An|A1A2 ... An-1)

Where RS(t) is system reliability, i.e. The probability of systemsuccess for time t, given that the system was ‘up’ at t = 0 andP(Ai|A1 A2 ... Ai-1) is the conditional probability of event A occurring (i.e., element Ei survives for time t), given that eventsA1, A2, ... And Ai-1 have occurred (i.e. Elements E1, E2, ... and Ei-1 have survived for time t, for i = 1, 2, ..., n

•Product Rule of System Reliability

if the n elements are independent

n

1ii

n

1iiS )t(R)A(P)t(R


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System Reliability Models - Series Reliability Configuration

• Simplest and most common structure in reliability analysis.

• Functional operation of the system depends on the successfuloperation of all system components Note: The electrical or mechanical configuration may differ from the reliability configuration

• Block Diagram For Series Reliability Configuration with n elements: E1, E2, ..., En

Since a single path exists, the failure of any element in thesystem interrupts the path and causes the system to fail.

E1 E2 En


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System Reliability Models - Series Configuration: Product Rule

• Rs(t) = P(A1) P(A2|A1) P(A3|A1A2) ... P(An|A1A2 ... An-1)

Where RS(t) is system reliability, i.e. The probability of systemsuccess for time t, given that the system was ‘up’ at t = 0 andP(Ai|A1 A2 ... Ai-1) is the conditional probability of event A occurring (i.e., element Ei survives for time t), given that eventsA1, A2, ... And Ai-1 have occurred (i.e. Elements E1, E2, ... And Ei-1 have survived for time t, for i = 1, 2, ..., n

if the n elements are independant

n

1ii

n

1iiS )t(R)A(P)t(R


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System Reliability Models - Series Configuration

General time to element failure distributions• System reliability

n

1ii )t(Ht

S e)t(R

Where Hi(t) is the cumulative failure rate of element i, for i = 1, 2, ... n

• System mean time to failure

0

)(1 dteMTTFn

ii tHt

S


Series Reliability Configuration with Exponential Distribution

•Reliability Block Diagram

•Exponential distributions of element time to failure

Ti ~ E(i) for i = 1, 2, ... n

• System reliability

tS

Se)t(R

SS

SMTTF

1

Where the system failure rate is


n

1iiS )t(

E1 E2 En


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Series Reliability Configuration with Exponential Distribution


•Exponential distributions of element time to failure

Ti ~ E() for i = 1, 2, ... n

•System reliability tnS e)t(R

Which is the same as the expected time to the first failure, E(T1),when n identical items are put into service

nMTTFS


E1 E2 En


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Series Reliability Configuration with Weibull Distribution


•Weibull distribution of element time to failure

Ti ~ W(i,i) for i = 1, 2, ... n

• System reliability in

1i i

t

S e)t(R

• System failure rate

n

1i

1i

i

iS t)t(h

i

E1 E2 En


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• System cumulative failure rate

0

SS dt)t(RMTTF

• System mean time to failure rate

t

0

n

1ii

1

S dt)t(ht

1t)t(H

i

i


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Weibull distribution of element time to failure Ti ~ W(,) for i = 1, 2, ... n


tn

S e)t(R


1

S

tn)t(h


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• System cumulative failure rate

1

1

nMTTF

/1S

• System mean time to failure rate

1

S

nt)t(H


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Example: Series Reliability Configuration


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Parallel Reliability Configuration


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Parallel Reliability Configuration – Basic Concepts

• Definition - a system is said to have parallel reliability configuration if the system function can be performed by any one of two or more paths

• Reliability block diagram - for a parallel reliability configuration consisting of n elements, E1, E2, ... En

E1

E2

En


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• Redundant reliability configuration - sometimes called a redundant reliability configuration. Other times, the term ‘redundant’ is used only when the system is deliberately changed to provide additional paths, in order to improve the system reliability

• Basic assumptions

All elements are continuously energized starting at time t = 0

All elements are ‘up’ at time t = 0

The operation during time t of each element can be describedas either a success or a failure, i.e. Degraded operation orperformance is not considered


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System success - a system having a parallel reliability configuration operates successfully for a period of time t if at least one of the parallel elements operates for time t without failure. Notice that element failure does not necessarily mean system failure.


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• Block Diagram

• System reliability - for a system consisting of n elements, E1, E2, ... En

n

jiij

ji

n

1iiS )t(R)t(R)t(R)t(R

n

ii

nk

n

kjiijk

ji tRtRtRtR1

1 )()1...()()()(

if the n elements operate independently of each other and where Ri(t) is the reliability of element i, for i=1,2,…,n

E1

E2

En


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System Reliability Model - Parallel Configuration

• Product rule for unreliabilities

n

iiS tRtR

1

)(11)(

•Mean Time Between System Failures

0

SS (t)dtRMTBF


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s

p=R(t)

s


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Parallel Reliability Configuration with Exponential Distribution

Element time to failure is exponential with failure rate

• Reliability block diagram:

• Element Time to Failure Distribution

with failure rate for i=1,2.



t

t

S e2

e12)t(h

ttS eetR 22)(

E1

E2

θE~Tiθ

1λ


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Parallel Reliability Configuration with Exponential Distribution

• System Mean Time Between Failures:

MTBFS = 1.5


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Exponential distributions of element time to failure

Ti ~ E(i) for i = 1, 2



t)(tt

t)(21

t2

t1

S 2121

2121

eee

e)(ee)t(h

tttS

2121 eee)t(R


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Notice that hS(t) is an increasing function of t even though thefailure rate of each element is constant

• System mean time between failures

21

2121SMTBF

Parallel Reliability Configuration - continued


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System Reliability Models - Parallel Configuration

Exponential distributions of element time to failureTi ~ E(i) for i = 1, 2, ... n


n

1i

tS

ie11)t(R

• System mean time between failure MTBFS =

n

1ii

1n

kjik j i kji

jij i ji

n

1i i

1)1(...

111


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Exponential distributions of element time to failureTi ~ E() for i = 1, 2, ... n


ntS )e1(1)t(R


n

1iS i

11MTBF


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Weibull distribution of element time to failure

Ti ~ W(i,i) for i = 1, 2

•System reliability

2β

2

1β

1

2β

2

1β

1θ

t

θ

t

θ

t

θ

t

S eee)t(R


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Weibull distribution of element time to failure Ti ~ W(,) for i = 1, 2

• System reliabilityββ

θ

t2

θ

t

S ee2)t(R


t

t1

S

e2

e1t2)t(h


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Weibull distribution of element time to failure

Ti ~ W(,) for i = 1, 2


1

S 2

12

11MTBF


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nelements

melements

nelements

melements

nmpR 11

mnpR 11


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