Refined asymptotic analysis of two basic order-replacement models for a spare unit

Two basic order-replacement policies for a single spare unit are analysed in this paper. A unit is supplied by order with a deterministic lead time and, when in service, it is subject to random failure. For the first model, as soon as a new spare unit is delivered, it is taken into service even if the unit which it replaces is still functional. Then, the total cost, to be minimized by appropriately selecting the ordering time, comprises ordering and shortage costs. For the second model, replacement takes place of a failed unit only, thus possibly necessitating also inventory costs. The asymptotic analysis of these models based on the renewal reward theorem and involving the minimization of the long-term expected cost rate is well known. In this paper, we perform a refined analysis of these models based on an approximation of the finite-time-horizon expected cost rate function. An integral-equation approach in conjunction with the key renewal theorem is used. We also give a novel and most general account of this approach, the idea of which has been used in the literature for the analysis of repair-replacement models. The present framework subsumes this latter class of models. The degree of gain in accuracy by the refined method as compared with the usual method is assessed by way of two numerical examples implemented in MAT-LAB. The results show that the refined analysis is highly accurate in the practical range of parameter values. A bibliography on ordering policies for a single spare unit is also provided.