Ergodicity of forward times of the renewal process in a block-based inspection model using the delay time concept

The delay time concept and the techniques developed for modelling and optimising plant inspection practice have been reported in many papers and case studies. For a system subject to a few major failure modes, component based delay time models have been developed under the assumptions of an age-based inspection policy. An age-based inspection assumes that an inspection is scheduled according to the age of the component, and if there is a failure renewal, the next inspection is always, say τ times, from the time of the failure renewal. This applies to certain cases, particularly important plant items where the time since the last renewal or inspection is a key to schedule the next inspection service. However, in most cases, the inspection service is not scheduled according to the need of a particular component, rather it is scheduled according to a fixed calendar time regardless whether the component being inspected was just renewed or not. This policy is called a block-based inspection which has the advantage of easy planning and is particularly useful for plant items which are part of a larger system to be inspected. If a block-based inspection policy is used, the time to failure since the last inspection prior to the failure for a particular item is a random variable. This time is called the forward time in this paper. To optimise the inspection interval for block-based inspections, the usual criterion functions such as expected cost or down time per unit time depend on the distribution of this forward time. We report in this paper the development of a theoretical proof that a limiting distribution for such a forward time exists if certain conditions are met. We also propose a recursive algorithm for determining such a limiting distribution. A numerical example is presented to demonstrate the existence of the limiting distribution.