On the equivalence of systems of different sizes, with applications to system comparisons. (English) Zbl 1344.60084

Lindqvist, Bo H.; Samaniego, Francisco J.; Huseby, Arne B.

Adv. Appl. Probab. 48, No. 2, 332-348 (2016).

Summary: The signature of a coherent system is a useful tool in the study and comparison of lifetimes of engineered systems. In order to compare two systems of different sizes with respect to their signatures, the smaller system needs to be represented by an equivalent system of the same size as the larger system. In this paper, we show how to construct equivalent systems by adding irrelevant components to the smaller system. This leads to simpler proofs of some current key results, and throws new light on the interpretation of mixed systems. We also present a sufficient condition for equivalence of systems of different sizes when restricting to coherent systems. In cases where for a given system there is no equivalent system of smaller size, we characterize the class of lower-sized systems with a signature vector which stochastically dominates the signature of the larger system. This setup is applied to an optimization problem in reliability economics.

MSC:

60K10	Applications of renewal theory (reliability, demand theory, etc.)
62N05	Reliability and life testing
90B25	Reliability, availability, maintenance, inspection in operations research

Keywords:

coherent system; system signature; \(k\)-out-of-\(n\) system; mixed system; reliability polynomial; irrelevant components; cut set; critical path vector; stochastic order; reliability economics