Multivalued State Component Systems

Consider a system that is composed of n components, each of which is operating at some performance level. We suppose that there exists a nondecreasing function φ such that φ(x₁, ⋯, xₙ) denotes the performance level of the system when the ith component's performance level is xᵢ, i = 1, ⋯, n. We allow both xᵢ and φ(x₁, ⋯, xₙ) to be arbitrary nonnegative numbers and extend many of the important results of the usual binary model to this more general framework. In particular, we obtain a fundamental inequality for Eφ(X₁, ⋯, Xₙ) when φ is binary, which can, among other things, be used to generate a host of inequalities concerning increasing failure rate average distributions including, as a special case, the convolution and system closure theorem. We also define the concept of an increasing failure rate average stochastic process and prove the analog of the closure theorem; and then also do the same for new better than used stochastic processes.