System Reliability Theory. Marvin Rausand

and scientist Réne Descartes (1596–1650) and the English mathematician and physicist Sir Isaac Newton (1642–1726).The paradigm is based on Newton's three laws of forces and motion, his theories on universal gravitation, and the unifying theory that is called Newtonian mechanics. Another important basis for the paradigm is Descartes' theory of reductionism and his division between mind and matter, between mental and physical processes. Reductionism implies that any system (or problem) can be adequately understood by reducing it, or decomposing it, to a set of its constituent components and by carefully and individually studying each component. When all the components on the lowest level have been carefully studied, a synthesis process can be started. By combining the knowledge about the components that feed into a module on the upper, next level, the paradigm implies that all important properties of this module can be deduced from the properties of its constituent components. This is then continued until the system level is reached (see Figure 2.7).The Newtonian–Cartesian paradigm sees the world as a number of discrete, unchanging objects in an empty space. These objects interact in a linear, cause and effect manner. The time is linear and universal and not affected by speed or gravitation. The system behavior is deterministic, such that a particular cause leads to a unique effect. The paradigm supports the analysis of systems with a finite number of (mainly) independent parts that interact in a well‐defined manner with relatively few interconnections.The Newtonian–Cartesian paradigm is also called the Newtonian paradigm and the mechanistic paradigm.The Newtonian–Cartesian paradigm has had an enormous success and most of our current knowledge about physical systems are based on this paradigm. Much more information about the Newtonian–Cartesian paradigm can be found by visiting a good library or searching the Internet.

       Complicated systems. A complicated system has a high number of components with a fair degree of interrelationships and interdependencies between the components. By using current knowledge (e.g. by involving subject experts), we are able to understand the relevant system properties and to analyze it.

       Complex systems. In a complex system, the behavior of at least some of the components or the interactions between them do not comply with the requirements of the Newtonian–Cartesian paradigm. A complex system cannot be adequately understood and analyzed by traditional approaches because the system is something more than a sum of its components.

Figure 2.7 System analysis and synthesis.

      An emergent property is a system property that cannot be deduced from the properties of the system components. In many cases, emergent properties lead to unexpected system behavior that may be dangerous. A system is usually not designed or built to be complex, but may develop into a complex system through changes, coupling, and emergence.

There is a considerable disagreement about how to delimit the concept of emergence. Some authors interpret emergence very widely and say that “properties” such as reliability, quality, and safety are emergent properties of a system.

      Simple and complicated systems can be studied based on the Newtonian–Cartesian paradigm, whereas complex systems cannot be adequately studied within this paradigm. A new worldview called the complexity paradigm is therefore being developed.

      All the examples in this book are related to simple systems, but the theory and methods presented may also be applied to complicated systems and many aspects of complex systems. Complex systems as such are not studied in this book.

      Remark 2.1 (Classical methods waste of time?)

      Finally, you may wonder if the effort you make to learn the theory and methods described in this book is a waste of time when your study object is complex. According to Einstein and Infeld (1938), the development of new theory may be compared with climbing a mountain. When you have come to a certain height, you get a better overview, but you may realize that you need another strategy to reach the summit. To have reached the present height is an achievement that gives a good understanding of the further climbing efforts.

2.8 System Structure Modeling

      An early step of a system reliability study is to establish a model of the system structure. The model defines the system boundary and the elements of the system (i.e. inside the system boundary) and the interactions between these elements. We also make assumptions about how the system is operated and the environmental conditions and constraints that may affect the system elements and their behavior. A range of system modeling techniques are presented in later chapters. Here, we delimit the presentation to a rather simple approach – reliability block diagrams.

      2.8.1 Reliability Block Diagram

This section describes how a system function (SF) can be modeled by a reliability block diagram (RBD). An RBD is a success‐oriented graph with a single source (a) and a single terminal (b). The nodes of the RBD are called blocks or functional blocks. Each block represents a component function (or a combination of two or more functions). We assume that the blocks are numbered , where is known. This numbering is for convenience. In practical applications, a combination of letters and digits are often used to identify component functions. An RBD with blocks is called an RBD of order .

      Each block is either functioning or failed, but the terms up and down are also used. Intermediate states are not allowed. To block (for ) is connected a binary state variable , defined as follows:

      (2.1)

      Observe that means that the specified function of block is up. It does not mean that all the functions of the component associated with block are up.

Blocks are drawn as squares or rectangles, as shown is Figure 2.8 for component function . Connection between the end points (a) and (b) in Figure 2.8 means that block is functioning (i.e. ). It is possible to enter more information into the block and include a brief description of the required component function. An example is shown in Figure 2.9, where the component is a safety shutdown valve that is installed in a pipeline. A label is used to identify the block.