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
Stracener_EMIS 7305/5305_Spr08_02.07.08
Systems Reliability Models- r-out-of-n Reliability Configuration
Stracener_EMIS 7305/5305_Spr08_02.07.08
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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Stracener_EMIS 7305/5305_Spr08_02.07.08
System Reliability Models - r-out-of-n Configuration
• 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
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Stracener_EMIS 7305/5305_Spr08_02.07.08
System Reliability Models - r-out-of-n Configuration
• 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
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Stracener_EMIS 7305/5305_Spr08_02.07.08
System Reliability Models - r-out-of-n Configuration
Exponential distributions of element time to failureTi ~ E() for i = 1, 2, ... 5 special case: n = 5 and r = 3
•Reliability Block Diagram:
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E1
E2
E3
E4
E5
3 out of 5
Stracener_EMIS 7305/5305_Spr08_02.07.08
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
)(
• System failure rate
t2t
t2t
S e6e1510
ee2130)t(h
ttt eee 543 61510
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Stracener_EMIS 7305/5305_Spr08_02.07.08
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
Stracener_EMIS 7305/5305_Spr08_02.07.08
System Reliability Models - r-out-of-n Reliability Configuration
• System mean time between failures 83.1MTBFS
• System reliability
tttS eeetR 233)(
• System failure rate
t2t
t2t
S ee33
ee213)t(h
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Stracener_EMIS 7305/5305_Spr08_02.07.08
Systems Reliability Models- Standby Configuration
Stracener_EMIS 7305/5305_Spr08_02.07.08
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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Stracener_EMIS 7305/5305_Spr08_02.07.08
System Reliability Models - Standby Configuration
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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Stracener_EMIS 7305/5305_Spr08_02.07.08
System Reliability Models - Standby Configuration
• 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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Stracener_EMIS 7305/5305_Spr08_02.07.08
System Reliability Models - Standby Configuration
• System reliability RS(t) = (1 + t)e-t
• System failure rate hS(t) = 2t/(1 + t)
• System mean time between failures MTTFS = 2
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Stracener_EMIS 7305/5305_Spr08_02.07.08
System Reliability Models - Standby Configuration
tt
21
1tS
121 ee e)t(R
Reliability contributionof the 1st element
Reliability contributionof the 2nd element
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Stracener_EMIS 7305/5305_Spr08_02.07.08
System Reliability Models - Standby Configuration
• 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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Stracener_EMIS 7305/5305_Spr08_02.07.08
System Reliability Models - Standby Configuration
nnnn
tn
ne
121
121
...
...
• System reliability
......
e...)t(R
1n1312
tn32
S
1
.........
......
111
1121
iniiiii
tnii
ie
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Stracener_EMIS 7305/5305_Spr08_02.07.08
System Reliability Models - Standby Configuration
If i = , i = 1, 2, ... n, then
1n
0i
it
S !i
te)t(R and
nn
MTBFS
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Stracener_EMIS 7305/5305_Spr08_02.07.08
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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Stracener_EMIS 7305/5305_Spr08_02.07.08
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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Stracener_EMIS 7305/5305_Spr08_02.07.08
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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