Optimizing costs of age replacement policies

This paper continues earlier work on the best implementation procedure for an age replacement policy. Under an age replacement policy, a stochastically failing unit is replaced at failure or after being in service for x units of time, whichever comes first. Sequentially estimating [phi], the optimal replacement time, produces substantial cost savings. In this paper the rate of convergence of the actual costs to the theoretical optimal cost is studied. For any sequential procedure satisfying some mild measurability conditions, it is shown that with probability one the rate of convergence of the cost can be described based on the rate of convergence of the estimator of [phi]. Further, a sequential procedure is described whose cost converges to the optimal cost more rapidly than known competing procedures. For this procedure, the rate of convergence of the costs is further described by a result which states that an average actual cost per unit, when suitably standardized, converges in distribution to a normal random variable.