1. System Reliability Modeling and Analysis-
Systems Engineering Program
Department of Engineering Management, Information and Systems
EMIS 7305/5305
Systems Reliability, Supportability and Availability Analysis
System Reliability Modeling and Analysis-
r-out-of-n and Standby Configurations
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering

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

3. System Reliability Models - r-out-of-n Configuration
Definition - a system containing n elements, out of which at least
r 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 least
r strands are necessary to support the required load 3

4. System Reliability Models - r-out-of-n Configuration
Reliability block diagram E1 E2 En
Assumption - the system consists of
n identical and independent elements
r-out-of-n
Remark - the number of elements, r, surviving time t, is a
random variable with Binomial distribution 4

5. System Reliability Models - r-out-of-n Configuration
System mean time between failures
System reliability
System failure rate
Element reliability Ri(t) = R(t) for i = 1, 2, ... n 5

6. System Reliability Models - r-out-of-n Configuration
Exponential distributions of element time to failure
Ti ~ E() for i = 1, 2, special case: n = 5 and r = 3
Reliability Block Diagram: E1 E2 E3 E4 E5
3 out of 5 6

7. System Reliability Models - r-out-of-n Configuration continued
System failure rate
System mean time between failures 7

8. System Reliability Models - r-out-of-n Reliability Configuration
Element time to failure is exponential with failure rate 
n = 3 and r = 1
Reliability block diagram: E1 E2 E3
1 out of 3

9. System Reliability Models - r-out-of-n Reliability Configuration
System failure rate
System mean time between failures 9

10. Systems Reliability Models-
Standby Configuration

11. System Reliability Models - Standby Configuration
Definition - the standby reliability configuration consists of one
or more elements standing by to take over the system operation on
occurrence of failure of the operating element
Remarks
Usually a standby configuration requires failure sensing
and switching devices to monitor the operating element
and to switch a standby element into operation whenever
a failure is sensed
The standby elements can be completely de-energized
‘cold standby’ or partially energized ‘warm standby’ 11

12. System Reliability Models - Standby Configuration
Reliability block diagram E1 E2 E3 En M S
where E1 is the initial operating element
E2, E3, ... En are initial standby elements
M is the failure sensing device
S is the switching device and 12

13. Reliability block diagram
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 13

14. 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 14

15. Reliability contribution
System Reliability Models - Standby Configuration
Reliability contribution
of the 1st element
of the 2nd element 15

16. 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 16

17. System Reliability Models - Standby Configuration17

18. System Reliability Models - Standby Configuration
If i = , i = 1, 2, ... n, then
and 18

19. System Reliability Models - Standby Configuration Conclusions
As the number of redundant paths increases, the mission reliability
approaches the reliability of the monitor/switching device.
When the failure rates of the path, the switching devices, and the
monitor/switching device are equal, standby redundancy with two
paths results in a mission reliability considerably less than that of a
single non-redundant path.
For systems where the switching device and monitor failure rates
are less than the path failure rate, the greatest increase in reliability
occurs when one redundant path is added to a single path. 19

20. 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 through
redundancy results from the utilization of switching devices and
monitors that are much more reliable than the path being switched. 20

21. Configuration Considerations in Design
Series Configuration - Relative to Redundant Configuration
Simpler
Increases Basic Reliability
Reduces Support Resources
Decreases Mission Reliability
Redundant Configuration - Relative to Series Configuration
More Complex - Increases Weight
Requires More Testability
Increases Support Resources
Decreases Basic Reliablity
Increases Mission Reliability 21