Cost Elasticities of Reliability and MTTF for k-out-of-n Systems

Abstract

The application of the concept of cost elasticity of reliability was extended from the parallel system to the partially redundant or k-out-of-n:G(F) system (or k-out-of-n system, for short). An expression for the cost elasticity of reliability was derived for a general k-out-of-n system. The expression yielded acceptable results for a wide range of values for k, n, and component reliability. For systems of practical interest characterized by good components, the expression became highly susceptible to round-off errors, and catastrophic cancellations took place. These numerical problems seemed unavoidable as they were inherently associated with the definition of the cost-elasticity-of-reliability metric itself. We introduced another metric, the cost elasticity of the Mean Time To Failure (MTTF), which measures the relative change in the life expectancy that can be obtained for a given relative change in cost. We believe the cost-elasticity-of-MTTF metric is a more tangible and a more cumulative measure than the cost-elasticity-of-reliability metric. We derived a very simple expression for the cost elasticity of MTTF for a k-out-of-n system and showed that it is a function of only k and n, i.e. it is independent of component characteristics such as component failure rate or component reliability. This expression is insensitive to round-off errors since it is a purely additive formula. We provided charts for the cost elasticity of MTTF that can be used to assess the cost incurred in achieving a certain life expectancy for a k-out-of-n system. These charts can be used with any coherent system, since the MTTF for a coherent system can be approximated by that of a k-out-of-n system.