The reliability of coherent systems based on system signatures is considered. A system of \(n\) components is described by a state vector \(X=(x_1,\dots,x_n)\), where for each \(i\), \(x_i=1\) if the \(i\)-th component is working and \(x_i=0\) if it is not working. The structure function \(\varphi(X) \in \{0,1\}\) describes the state of the whole system, so that \(\varphi(X)=1\) if the system is working and \(\varphi(X)=0\) otherwise. A coherent system has a monotone structure function (\(\varphi(X) \leq \varphi(Y)\) if \(X\leq Y\)) essentially depending on all its components.
Assume that the lifetimes of the coherent system’s \(n\) components are i.i.d. according to the continuous distribution \(F\). The system signature, which was introduced by the author [IEEE Trans. Reliab. 34, 69–72 (1985; Zbl 0585.62169)], is a vector \(S=(s_1,\dots,s_n)\), where \(s_i\) is equal to the probability that the \(i\)-th component failure causes the whole system to fail (\(i=1,\dots,n\)). The fundamental property of the system signature is that the distribution of the lifetime of the whole system depends on the common distribution function \(F\) and on the system signature \(S\) only. Moreover, it allows to compare lifetimes of different systems. So, it is a powerful tool which can be used in all physical structures and stochastic systems where reliability is an important consideration (e.g., automobiles, bridges, electronic networks, airplanes, etc.)
Since the introduction of system signatures, the properties of this technical concept have been examined, tested and proven in a wide variety of systems applications. Based on the practical and research success in building reliability into systems with system signatures, this is the first book treatment of the approach. It is, therefore, the purpose of this book to provide guidance on how reliability problems might be structured, modelled and solved. Over the past ten years the broad applicability of system signatures has become apparent and the tool’s utility in coherent systems and communications networks was firmly established. The book compared actual system lifetimes where the tool has been and has not been used. These comparisons, which have been done over the years, demonstrate the practical, feasible and fruitful use of the tool in building reliable systems. Finally, new results and future directions for system signatures are also explored.