ABSTRACT
Structure functions offer a way of indexing the class of coherent systems, but they have not proven to be particularly useful in this role. While every coherent system has its unique φ, these functions are somewhat awkward algebraic objects which are difficult to compute and to compare. The latter problem is exacerbated by the fact that the relabelling of components changes φ to a different though equivalent form. Samaniego (1985) introduced an alternative index that, while narrower in scope than the structure function, is
substantially more useful. Samaniego’s definition of system signatures involves the assumption of i.i.d. component lifetimes. Under this assumption, the signature s of a coherent system of order n is the n-dimensional probability vector whose ith element is si = P(T = Xi:n), where T is the system lifetime and X1:n, X2:n, . . . , Xn:n are the order statistics of the n i.i.d. component lifetimes. Under the i.i.d. assumption, the signature vector is a distributionfree function (that is, does not depend on the common continuous lifetime distribution F of the components) that constitutes a pure measure of the system’s design. The signature can be computed as the ratio a/b, where a is the number of permutations of the component lifetimes for which a particular ordered component failure is fatal to the system and b = n!, the size of the set Pn of all possible permutations of the n failure times. The i.i.d. assumption has the effect of ‘‘levelling the playing field’’ among systems, eliminating anomalies like the fact that a series system with good components can outperform a parallel system with poor components even though the latter system is clearly ‘‘better’’ from a design point of view.
