Replacement Policies for an Independent Damage Process

This paper discusses preventive replacement policies for an independent damage process, in which successive shocks occur at random times, and the independent damages caused by shocks are random variables. It is assumed that a unit fails when the damage has exceeded a prespecified level [Formula: see text]. We consider three models: Model 1: the unit is replaced at the [Formula: see text]th shock for damage [Formula: see text], at damage [Formula: see text], or at time [Formula: see text], whichever occurs first. Model 2: the unit is replaced at the [Formula: see text]th shock for damage [Formula: see text], at damage [Formula: see text] or at shock [Formula: see text], whichever occurs first. Model 3: the unit is replaced at the [Formula: see text]th shock for damage [Formula: see text], at damage [Formula: see text], at shock [Formula: see text] or at time [Formula: see text], whichever occurs first. We obtain the expected cost rates for each model and discuss analytical optimal [Formula: see text] and [Formula: see text] to minimize their expected cost rates. Numerical examples are given when the damage has an exponential distribution and shocks occur at a gamma distribution.