Statistical Models on Maintenance

Abstract

This chapter discusses a variety of approaches to performing maintenance. The first section describes the importance of preparing for maintenance correctly, by collecting data on unit lifetimes and estimating the reliability of the units statistically using quantities such as their mean lifetimes, failure rates and failure distributions. Suppose that the time that the unit has been operational is known (or even just the calendar time since it was first used), and its failure distribution has been estimated statistically. The second section of the chapter shows that the time to failure is approximately given by the reciprocal of the failure rate, and the time before preventive maintenance is required is simply given by the pth percentile point of the failure distribution. Standard replacement policies, such as age replacement, in which a unit undergoes maintenance before it reaches a certain age, and periodic replacement, where the unit undergoes maintenance periodically, are also presented. Suppose that the failure of a unit can only be recorded at discrete times (so the unit completes a specific number of cycles before failure). In the third section, the age replacement and periodic replacement models from the previous section are converted into discrete models. Three replacement policies in which the unit undergoes maintenance after a specific number of failures, episodes of preventive maintenance or repairs, are also presented. The optimum number of units for a parallel redundant system is derived for when each unit fails according to a failure distribution and fails upon some shock with a certain probability. Suppose that the unit fails when the total amount of damage caused by shocks has exceeded a certain failure level. The fourth section describes the replacement policy in where the unit undergoes maintenance before failure for a cumulative damage model. The optimum damage level at which the unit should be replaced when it undergoes minimal repair upon failure is also derived analytically. The last part introduces the repair limit policy, where the unit is replaced instead of being repaired if the repair time is estimated to be more than a certain time limit, as well as the inspection with human error policy, where units are checked periodically and failed units are only detected and replaced upon inspection. Finally, the maintenance of a phased array radar is analyzed as an example of the practical use of maintenance models. Two maintenance models are considered in this case, and policies that minimize the expected cost rates are obtained analytically and numerically.