Stracener_EMIS 7305/5305_Spr08_02.07.08 System Reliability Modeling and Analysis- r-out-of-n and...

Stracener_EMIS 7305/5305_Spr08_02.07.08

System Reliability Modeling and Analysis-r-out-of-n and Standby 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


Systems Reliability Models- r-out-of-n Reliability Configuration


System Reliability Models - r-out-of-n Configuration

• Definition - a system containing n elements, out of which at leastr are required for system success, is the so called r-out-of-n reliability configuration

• Remark - the r-out-of-n reliability configuration is a general configuration. If r = 1, the configuration is a parallel configuration. If r = n, the configuration is a series configuration.

• Example - a piece of stranded wire, with n strands, which at leastr strands are necessary to support the required load

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• Reliability block diagram

E1

E2

En• Assumption - the system consists ofn identical and independent elements

• Remark - the number of elements, r, surviving time t, is a random variable with Binomial distribution

r-out-of-n

4



• System mean time between failures

0

SS dt)t(RMTBF

• System reliability

)()(1)()( rXPtRtRx

ntR xnx

n

rxS

• System failure rate

• Element reliability Ri(t) = R(t) for i = 1, 2, ... n

)t(R

)t(f)t(h

S

SS

5



Exponential distributions of element time to failureTi ~ E() for i = 1, 2, ... 5 special case: n = 5 and r = 3

•Reliability Block Diagram:

6

E1

E2

E3

E4

E5

3 out of 5


System Reliability Models - r-out-of-n Configuration continued

• System mean time between failures 7833.0MTBFS

• System reliabilityx

tx

t

xS ee

xtR

55

3

15

)(


t2t

t2t

S e6e1510

ee2130)t(h

ttt eee 543 61510

7


System Reliability Models - r-out-of-n Reliability Configuration

Element time to failure is exponential with failure rate n = 3 and r = 1

E1

E2

E3

• Reliability block diagram:

1 out of 3


System Reliability Models - r-out-of-n Reliability Configuration

• System mean time between failures 83.1MTBFS


tttS eeetR 233)(


t2t

t2t

S ee33

ee213)t(h

9


Systems Reliability Models- Standby Configuration


System Reliability Models - Standby Configuration

• Definition - the standby reliability configuration consists of oneor more elements standing by to take over the system operation onoccurrence of failure of the operating element

• Remarks

Usually a standby configuration requires failure sensingand switching devices to monitor the operating elementand to switch a standby element into operation whenevera failure is sensed

The standby elements can be completely de-energized‘cold standby’ or partially energized ‘warm standby’

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Reliability block diagram

E1

E2

E3

En

M S

where E1 is the initial operating elementE2, E3, ... En are initial standby elementsM is the failure sensing deviceS is the switching device and

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• Assumptions:Perfect failure sensing and switching

Zero failure rate during standby Two identical and independent elements Element time to failure is exponential with parameter

E1

E2

• Reliability block diagram

• System success results if E1 survives time t or if E1 fails at time t, and E2 survives time t - t1

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• System reliability RS(t) = (1 + t)e-t

• System failure rate hS(t) = 2t/(1 + t)

• System mean time between failures MTTFS = 2

14



tt

21

1tS

121 ee e)t(R

Reliability contributionof the 1st element

Reliability contributionof the 2nd element

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• Assumptions:Perfect failure sensing and switching

Zero failure rate during standby Independent elements Exponential distributions of element time to failure

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

• System mean time between failures

n

1i 1S

1MTBF

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nnnn

tn

ne

121

121

...

...


......

e...)t(R

1n1312

tn32

S

1

.........

......

111

1121

iniiiii

tnii

ie

17



If i = , i = 1, 2, ... n, then

1n

0i

it

S !i

te)t(R and

nn

MTBFS

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System Reliability Models - Standby Configuration Conclusions

• As the number of redundant paths increases, the mission reliabilityapproaches the reliability of the monitor/switching device.

• When the failure rates of the path, the switching devices, and themonitor/switching device are equal, standby redundancy with twopaths results in a mission reliability considerably less than that of asingle non-redundant path.

• For systems where the switching device and monitor failure ratesare less than the path failure rate, the greatest increase in reliabilityoccurs when one redundant path is added to a single path.

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System Reliability Models - Conclusions continued

• For a given path and switching device failure rate, reliability improvement increases rapidly as the monitor failure rate decreases and the number of redundant paths increases. The same is true if the monitor failure rate is held constant and the switching device failure rate decreases.

• Significant improvement in mission reliability throughredundancy results from the utilization of switching devices and monitors that are much more reliable than the path being switched.

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Configuration Considerations in Design

• Series Configuration - Relative to Redundant ConfigurationSimplerIncreases Basic ReliabilityReduces Support ResourcesDecreases Mission Reliability

• Redundant Configuration - Relative to Series ConfigurationMore Complex - Increases WeightRequires More TestabilityIncreases Support ResourcesDecreases Basic ReliablityIncreases Mission Reliability

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Stracener_EMIS 7305/5305_Spr08_02.07.08 System Reliability Modeling and Analysis- r-out-of-n and...

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