On the Equivalence of Systems of Different Sizes

The signature of a coherent system with independent and identically distributed component lifetimes has been found to be a useful tool in the study and comparison of lifetimes of engineered systems. A key result is the representation of a system’s survival distribution in terms of its signature vector, which leads to several results on stochastic comparison of system lifetimes. In order to compare two coherent 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. Here equivalence between systems means that their lifetime distributions are identical for any component distribution. While such equivalent systems are usually represented as mixtures of coherent systems (so called mixed systems), in the present paper we demonstrate that they can be obtained in a simpler fashion by addition of irrelevant components to the smaller system, thereby representing them by monotone systems. In addition to making the formulas for signatures of equivalent systems more transparent, the new representation aids in the usual interpretation of mixed systems. We also consider the opposite problem of whether, for a given mixed system, there can be found equivalent systems of smaller sizes. While there is always an equivalent mixed system of larger size, there need not be equivalent systems of smaller sizes. We finally study the problem of equivalence of systems of different sizes when we restrict to coherent systems. A sufficient condition for equivalence of coherent systems of sizes respectively n and n+ 1, for general n, is given; it follows, as a special case, that any k-out-of-n -system with 1 < k < n has an equivalent coherent system of size n + 1. The proof is based on first adding an irrelevant component to the smaller system, and then obtaining an equivalent coherent system by manipulating the minimal cut sets of the original system.