Stochastic Order for Inspection and Repair Policies

Inspection and repair policies for (n.r+1)-out-of-n systems are compared stochastically with respect to two partial orderings .b1 and .b2 on the set of permutations of {1,2,.,n}. The partial ordering .b1 is finer than the partial ordering .b2. A given permutation . of {1,2,.,n} determines the order in which components are visited and inspected. We assume that the reliability of the ith independent component is given by Pi, where P1.P2...Pn. If . and .. are two permutations such that ..b1.., then we show that the number of inspections necessary to achieve minimal or complete repair is stochastically smaller with . than with ... We also consider three policies for minimal repair when the components are each made up of t "parts" assembled in parallel. It is shown that if ..b2.., then the number of repairs necessary under . is stochastically smaller than the number necessary under .., but that in general this is not true for the finer ordering .b1. The results enable one to make interesting comparisons between various inspection and repair policies, as well as to understand better the relationship between the orderings .b1 and .b2 on the set of permutations.