Define Reliability
- The probability that a component (or system will function properly. ie. Rs =P(S^s)
- Assume components are independent
- No partial failures
- Known reliabilities per component.
Series System
- The system survives if and only if all components survive and provided the provided the performance of each component does not depend on the performance of any other component.
- Or gate
- e.g. Road with a bridge
Parallel System
- Reliability of the system here is the chance that not all components fail.
- And gate
Network Reliability
- Birnbaum suggested that to maximise the improvement in network reliability (R), one should improve the link a with the highest Reliability Importance (RIa), where Ria = dR/dra, where ra is the reliability of link a.
- In a practical reliability study of a complex system, one of the most time- consuming tasks is to find adequate estimates for the input parameters (failure rates, repair rates, etc.).
- In some cases, we may start with rather rough estimates, calculate Birnbaum’s measure of importance for the various components, or the parameter sensitivities, and then spending the most of the time finding high-quality data for the most important components.
- Components with a very low value of Birnbaum’s measure will have a negligible effect on the system reliability, and extra efforts finding high- quality data for such components may be considered a waste of time.
For a simple with two links in parallel:
1-R = probability that both link 1 and link 2 fail = (prob. link 1 fails) x (prob. link 2 fail) = (1-r1) x (1-r2)
and R = r1 + r2 - r1r2
It follows that: RI1 = 1 - r2, RI2 = 1 - r2
Hence, if link 1 is more reliable than link 2 (i.e. r1 > r2), then RI1 > RI2
Hence, should improve link 1. That is, one should improve the ore reliable (or stronger) link. This is a counter- intuitive result.
Improving Network Reliability
Throughout our discussion we have made many simplifying assumptions. e.g. Independence, parallel & series and reliability and reliability importance.
- Network analysis is more complex than commonly realised.
- One must beware of relying on intuition as shown by the previous example.
- Traditionally system reliability fields like traffic network analysis have put many student off studying it, due to amount of advanced mathematics and computation required.
For a simple network with two links in a series:
R = probability that both link 1 and link 2 survive
= (prob. link 1 survives) x ( prob. link 2 survives) = r1r2
It follows that: RI1 = r2, RI2 = r1
Hence, if link 1 is more reliability than link 2 (i.e. r1 > r2), then RI1 < RI2
Hence, one should improve link 2. That is, one should improve the less reliable (or weaker) link. This is an intuitively sound result.
Reliability Diagrams versus Fault Trees.
Mixed Systems Exercise and Answers - in Notes!
