Dependability: a generic term encompassing the concepts of reliability, availability,
maintainability, security, etc. It is also simply designated by the term
“reliability”, which underlines its quantitative aspect.
The aim of reliability is the study of systems (sets of elements – hardware,
software, human resources – that interact with a view to accomplishing a mission)
that are subjected to physical processes such as the processes of failure,
repair, and stresses.
The component is a part of a system which is non-resolvable within the
framework of the study for which sufficient qualitative information (functioning,
modes of failure, etc.) as well as quantitative information (failure rate, repair
rate, etc.) has been provided in this study. The notion of component is relative
and depends on the study. For example, an aircraft, within the framework of a
study dealing with flight safety, represents the system, whereas for the airline
company it represents a component.
Fault Trees [FT]-quantitative modeling of a system.
– a deductive way of finding faults in a system
– represents a failure mode of a subsystem or component
– if not studying system faults, it is called Event Tree
– to obtain the map of a complex system quantitatively
– to improve reliability
– to find faults of a system
– systematic modeling of a system structure
– to model human errors
Minimal cut sets-the minimal path for studying the reliability of a system [simplified]
Classification of Reliability
1. Component Reliability
2. Reliability Statistics
3. Reliability of Systems
Classes of Systems
1. System components
2. System’s structure function
3. Maintenance Systems
4. Binary systems
5. Multi performance systems
6. Non-repairable systems
7. Repairable
8. The mission
3 Stages of FT Modeling
1. Construction of the tree
2. Qualitative analysis
3. Quantitative analysis
Note: the construction should be exhaustive in nature representing all the significant causes for the failure of the system
3 Questions To Ask
1. What are the failure modes of the component and of the system?
2. How does the system break down?
3. What is the probability of the system breaking down?
The Qualitative Analysis
– deals with methods for obtaining the minimal sets: minimal cut sets and minimal paths
– what is the top event?
– what is undesirable?
Reliability Phase Diagram
(a) non-repairable system
(b) repairable system
1: state of good functioning
2: state of breakdown
Multi-Component Systems
Whereas in the case of single-component systems the failure in the component
implies the failure of the system, this is no longer the case for
multi-component systems, at least not automatically. The failure in a multi component
system arises in the wake of the failure in the sub-sets of well-defined
components. For example, the failure in one of the four cylinders of a car does
not jeopardize its overall working: it works badly, but it still works. On the
other hand, the failure of the drive shaft leads to the immobilization of the car.
Let us consider the brake system of a vehicle, made up of front
and rear brakes. If the rear brakes are down, we can still continue braking.
On the other hand, if both the brakes are down (front and rear), we can no
longer brake and the braking system is declared to be failed. If the front brakes
be indicated by x1 and the rear brakes by x2, then the braking system of the
vehicle can be described by the function:
Elementary structure systems
The distinction between elementary structure systems and complex structure
systems is very useful in matters of reliability. The elementary structure
systems are concerned with series structure and the parallel structure or, more
precisely, the structure k-over-n, which is a generalization of the preceding two
structures. Any system giving a k-over-n structure through successive reductions
is called a elementary structure system. In the contrary case, it is known
as a complex structure system. The structural function of an elementary structure
system is obtained directly using the functions of structures, as described
in the following section. On the contrary, for the complex structure systems,
we require the notions of minimal cut set or of minimal path.
Series system
A system is said to be in series if its functioning is subjected to the simultaneous
functioning of all its components. If just one of its components is failed,
then the system will have broken down.
Parallel system
The functioning of this system is assured, if at least one of its components
is in good state. The system will be failed if and only if all its components are
simultaneously failed.
Parallel-series system
This is formed by r blocks mounted in series, and each block i constitutes
a parallel system of ij components j =1, …, r.
Series-parallel system
This is formed by r blocks mounted in parallel, and each block i constitutes
a series system of ij components j =1, …, r.
Systems with complex structure
A system that cannot be directly classified into the preceding cases of elementary
structures is called a system with complex structure. For the study
of systems with complex structure, the concepts of minimal paths and cuts (in
the case of monotone structures) must be introduced. The system in Figure 2.6
is a system with complex structure. This system cannot be directly classified
into modules with traditional structures. If the input/output of this system are
located at the extremities of component 3, it will then involve a system with
elementary structure.
Path: a sub-set of components whose simultaneous good functioning will
assure the good working of the system, which is independent of the states of
the other components.
Minimal path: a path which does not contain another path.
Cut set: a sub-set of components whose simultaneous failure leads to the
system failure, which is independent of the states of the other components.
Minimal cut set: a cut set that does not contain another cut set.
Reliability bounds
The exact calculation for the reliability of the systems is very costly (NPhard
problem); that is why the calculation of the bounds, instead of for exact
value, is suitable. We have four types of bounds: the bounds of inclusion–
exclusion, the minimal sets bounds, the “min–max” bounds and the trivial
bounds. Other bounds can be obtained by improving the above-cited bounds
or through the construction of mixed bounds or by a modularization of the
system. It has to be noted that there is a relationship of order among those
four types of bounds. The best bound depends on the problem.
Construction of Fault Trees
Operators
Special Operators
Events
Transfer Triangles
Graphic Symbols
FT is represented by means of 3 types of graphic symbols
1. Logic Gates or Operators
2. The Events
3. The Transfer Triangles
Primary operators-when all inputs are basic events
Top Event: A
Intermediate Events, B,C, D, E, F
Basic Events: 1,2,3,4,5,6,7
Non-Elementary Events: 5,6 [transfer triangles]
Event 2 and 3 are repeated events
Operator OR: G1, G3, G4, G5, G6
Operator AND: G2
G5 and G6 are primary operators because all their input are basic events
OR Operator-Series System
AND Operator-Parallel System
– active redundancy
Priority AND operator
-passive redundancy
-component2 is in standby but starts when component1 fails
