An Explicit Closed-Form Formula for Profit-Maximizing k-Out-Of-n Systems Subject to Two Kinds of Failures

Abstract

This paper derives and analyzes an explicit closed-form formula for the optimal k in k-out-of-n systems consisting of i.i.d. components. The system can be in one of two possible modes with a pre-specified probability. The components are subject to failure in each of the two modes. The costs of the two kinds of system failures are generally not identical. Since the formula is explicit, it permits a calculation of the optimal k directly in terms of the parameters of the system. In addition, it yields many results concerning both the bounds of the optimal k and the effects of a change in parameters on the optimal k and on the optimized value of the system's expected profit.