On modular decompositions of system signatures

Considering a semicoherent system made up of $n$ components having i.i.d. continuous lifetimes, Samaniego defined its structural signature as the $n$-tuple whose $k$-th coordinate is the probability that the $k$-th component failure causes the system to fail. This $n$-tuple, which depends only on the structure of the system and not on the distribution of the component lifetimes, is a very useful tool in the theoretical analysis of coherent systems. It was shown in two independent recent papers how the structural signature of a system partitioned into two disjoint modules can be computed from the signatures of these modules. In this work we consider the general case of a system partitioned into an arbitrary number of disjoint modules organized in an arbitrary way and we provide a general formula for the signature of the system in terms of the signatures of the modules. The concept of signature was recently extended to the general case of semicoherent systems whose components may have dependent lifetimes. The same definition for the $n$-tuple gives rise to the probability signature, which may depend on both the structure of the system and the probability distribution of the component lifetimes. In this general setting, we show how under a natural condition on the distribution of the lifetimes, the probability signature of the system can be expressed in terms of the probability signatures of the modules. We finally discuss a few situations where this condition holds in the non-i.i.d. and nonexchangeable cases and provide some applications of the main results

An efficient algorithm for exact computation of system and survival signatures using binary decision diagrams

System and survival signatures are important and popular tools for studying and analysing the reliability of systems. However, it is difficult to compute these signatures for systems with complex reliability structure functions and large numbers of components. This paper presents a new algorithm that is able to compute exact signatures for systems that are far more complex than is feasible using existing approaches. This is based on the use of reduced order binary decision diagrams (ROBDDs), multidimensional arrays and the dynamic programming paradigm. Results comparing the computational efficiency of deriving signatures for some example systems (including complex benchmark systems from the literature) using the new algorithm and a comparison enumerative algorithm are presented and demonstrate a significant reduction in computation time and improvement in scalability with increasing system complexity

On computing signatures of coherent systems

It is difficult to compute the signature of a coherent system with a large number of components. This paper derives two basic formulas for computing the signature of a system which can be decomposed into two subsystems (modules). As an immediate application, we obtain the formula for computing the signature of systemwise redundancy in terms of the signatures of the original system and the backup one. The formula for computing the signature of a componentwise redundancy system is also derived. Some examples are given to illustrate the power of the main results.Module Redundancy Parallel Series k-out-of-n system