A single item subject to random failure is considered. The item is inspected at certain time instants to see whether it is still in working order. Each inspection carries a fixed cost, and there is also a cost per unit time associated with down times. For this scenario, there exist various optimum and near-optimum policies for deterministic inspection schedules. This paper is concerned with the case when the inter-inspection times are random, a situation that may arise in the practical implementation of deterministic inspection policies or in opportunistic inspection schedules. We use a technique which is known as the "Ross Scheme" to derive recurrence relations for the exact computation of the expected length of a renewal cycle and the moments of the total cost in a renewal cycle for random inspection policies. The item lifetime and the inter-inspection times are assumed distributed according to mixtures of Erlang distributions. A numerical example is used to compare the present policy with a (near-)optimal policy. The penalty for random (as opposed to deterministic) inspections is found fairly limited.
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