The document discusses redundancy allocation and how it is used to increase reliability in complex systems where high reliability is needed. It defines reliability and explains different redundancy techniques like using components in series, parallel, standby, and k-out-of-n configurations. These redundancy methods allow the system to continue functioning even if one or more components fail by having backup or redundant components. The document also provides an example calculation of reliability for different component configurations and concludes by discussing how redundancy is useful but also costly, and future work could analyze optimal redundancy designs with minimum cost.

1. Redundancy AllocationRedundancy Allocation
Abir Naskar :: 13MA60R21
Computer Science and Data Processing
Department of Mathematics
IIT KHARAGPUR

2. INTRODUCTIONINTRODUCTION
➔
What is Reliability ?What is Reliability ?
Probability that system will perform without failure during a time under specific environmental condition.
➔
Why we need it ?Why we need it ?
Provides quantitative basis for evaluating product reliability.
Invented during world war 2.
A measurement of probalility or capability for a system to perform a assigned task.
➔
What and Why RedundancyWhat and Why Redundancy :
Using same components mainly `in parallel.
Used mainly for complex system where high reliability is needed.
Way of increase reliability of the system.

3. Failure Rate and MTBFFailure Rate and MTBF
➔ Number of failure during an unit time t is failure rate.
Failure rate function provides relationship between time and failure frequency.
➔ MTBF is reciprocal of failure rate.
Mean time between consequent failure during operational time.
➔ Below graph is called bathtub curve stating relation between failure rate and time t.

4. Reliability and ExponentialReliability and Exponential
DistributionDistribution
➔
ReliabilityReliability
R(t) = 1-Q(t) = P(X>t)R(t) = 1-Q(t) = P(X>t)
➔
NonreliabilityNonreliability
Q(t) = 1-R(t) = P(X<t)Q(t) = 1-R(t) = P(X<t)
➔
Why Exponential ?Why Exponential ?
●
Memory less property.Memory less property.
●
Only one parameter, easy to compute.Only one parameter, easy to compute.

5. Series SystemSeries System
➔ Components are used in a series.
➔ Works good (i.e without failure) if all components works good.
➔ Failure of single component causes full system failure.
➔ Events can be independent, dependent, mutually exclusive.
➔ Here it assumed to be independent.

6. Parallel SystemParallel System
➔ We use the components in parallel mode.
➔ System will not fail if at-least one component works.
➔ Used when very high reliability required, and designer has to
duplicate some components or whole circuit.
➔ Make your system more reliable.

7. Series vs ParallelSeries vs Parallel
➔
Reliability( component in series ) ≤ Reliability( single component ) ≤
Reliability( components in parallel ).

8. Stand-by SystemStand-by System
➔ Special case of parallel system.
➔ One or more components are in stand-by mode to support single
running component.
➔ If we have n+1 components then upto n failure the system will work,
for n+1th failure the whole system fails.
➔ Need failure sensing and switch over devices which are assumed to
be 100 percent reliable
➔ Assumption that all stand-by components has same failure rate.

9. k-out-of-n systemk-out-of-n system
➔ The system of n components works in good condition (i.e does not
fail) if and only if at-least k of the n components works good. It is
called k-out-of n:G system.
➔ The system of n components fails if and only if at-least k of the n
components fail. It is called k-out-of n:F system.
➔ k-out-of-n:G and (n-k+1)-out-of-n:F systems are equivalent.

10. ExampleExample
➔ Let a system consists 2 propulsions (p) and 2 generator (g).
➔ Let the reliability of propulsion and generator be .999965 and .9945 resp.
So for single power unit reliability is: .999965 × .9945 = .99446519
If two power unit is in series then reliability is: .99446519 × .99446519 = .98896
For two unit in parallel the reliability is: 1-(1-.99446519)² = .99996937

11. Conclusion and Future workConclusion and Future work
➔ Redundancy is very much useful in our daily life to increase reliability as well
as availability. For example in our IIT we use several proxy server, if by
chance one fails then we can use another. Several server systems store
their data in more than one place to increase the availability, operational
speed and to save their self from the data loss situation.
➔ But redundancy does not come in free of cost. Sometimes it costs huge. I
want to analyze that how the maximum reliability can be achieved using
optimum cost. The designing part of a particular system for specific
configuration and reliability with optimal cost will also be my point of view.

12. REFERENCEREFERENCE
➔ Kececioglu, D. B. (2002). Reliability Engineering Handbook (Vol. 1, pp. 1-41). Pennsylvania:
DEStech Publication, ISBN: 1-932078-00-2.
➔ Trivedi, K. S. (1982). Probability Statistics with Reliability, Queuing and Computer Science
Application (pp. 283-290, 309-324). Englewood: Prectice-Hall, ISBN: 0-13-711564-4.
➔ Bazovsky, I (2004). Reliability Theory and Practice. Mineola: Dover Publication, ISBN:
0-486-43867-8.
➔ Zuo, M. J., Huang, J. and Kuo, W (2002). Multi-State k-out-of-n Systems. London:
Springer-Verlag.
➔ Pham, H. (2002). Reliability of Systems with Multiple Failure Modes. London: Springer-Verlag.
Special Thanks To. Prof. N. Gupta

13. THANK YOU....