Number-dependent replacement model with random lead-time for systems under maintenance continuously

This article considers a number-dependent replacement policy where a system has two types of failures and is replaced at the nth type I failure (minor failure) or first type II failure (catastrophic failure), whichever occurs first where type I and type II failures are age dependent. Type I failures can be removed by restoration without any cost, since the maintenance work is executed continuously during the operation of the system. However, type II failure can be removed only through a replacement at a replacement cost. A spare unit for replacement can be delivered upon order and is available only when the random lead-time is finished. A model is developed for the average cost per unit time and is based on the stochastic behaviour of the assumed system and reflects the cost of storing a spare as well as system downtime. The optimal number for a minimum-cost policy is described and discussed. It is shown that the optimal number n* which minimises the cost rate is given by a unique solution of the equation under certain conditions.