Multivalued State Component Systems

Abstract

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 $\phi$ such that $\phi(x_1, \cdots, x_n)$ denotes the performance level of the system when the $i$th component's performance level is $x_i, i = 1, \cdots, n$. We allow both $x_i$ and $\phi(x_1, \cdots, x_n)$ 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 $E\lbrack\phi(X_1, \cdots, X_n)\rbrack$ when $\phi$ 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.