Statistics for Maintenance Modelling: Repairable Systems

# Framework¤

Let T1, T2, T3, ... be the random variables of the times of failure of the system, Xi, i = 1, 2,… be the time between the
(i - 1)th and ith failure. We may take 'time' to be operating time, or real time if the 'down' times are considered to be negligible.

Observing the system over time, we are looking for any change in the nature of the Xi. There will be no change if the Xi are independent and identically distributed. Departure from this condition may constitute trend and the detection and estimation of trends is of major interest. Even if there is no long-term trend there may be some other structure of practical importance, for example inadequate repairs may produce a run of short failure times among longer times. So there may be dependence or autocorrelation between the Xi.

In repairable systems one generally has the times of successive failures of a single system, where the intervals between failures may be neither independent nor identically distributed. This is a key feature which results in the use of different statistical methods for repairable systems from those needed in component reliability.

## The Nature of Time

In the above discussion we have used time in its most general sense. In any particular application, time needs to be defined carefully and appropriately. For a continuously operating refrigerator in reasonably constant conditions real time may be appropriate. More commonly, however, operating time may be more appropriate than real time. On the other hand, some non-negative quantity other than time itself may be suitable. For instance, the mileage of a motor vehicle may be more useful for assessing reliability than real time and easier to record than operating time.