Like the time to failure, the random time to repair (TTR) has the repair density function gt (t) describing the random characteristics of the time required to repair a failed system when the failure occurs at time zero. The repair probability Gt(t) is the probability that the failed system can be restored within a given time period (0, t]:
Gt(t) = P(TTR < t) = f gt(t) dr (5.19)
90
The repair probability Gt (t) is also called the maintainability function (Knezevic, 1993), which is one of the measures for maintainability (Kapur, 1988b). Maintainability is a design characteristic to achieve fast, easy maintenance at the lowest life-cycle cost. In addition to the maintainability function, other types of maintainability measures are derivable from the repair density function (Kraus, 1988; Knezevic, 1993), and they are the mean time to repair (described in Sec. 5.3.3), TTRp, and the restoration success.
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TABLE 5.3 Reliability and Maintainability of Water Distribution Subsystems by Size
*MTBF = mean time between failure; MTTR = mean time to repair; MTBF = MTTF + MTTR. SOURCE : From Schultz and Parr (1981). |
The TTRp is the maintenance time by which 100p percent of the repair work is completed. The value of the TTRp can be determined by solving
r TTRp
P (TTR < TTRp) = gt(t) dT = Gt(TTRp) = p (5.20)
J0
In other words, the TTRp is the pth order quantile of the repair density function. In general, p = 0.90 is used commonly.
Note that the repair probability or maintainability function Gt(t) represents the probability that the restoration can be completed before or at time t. Sometimes one may be interested in the probability that the system can be restored by time t2, given that it has not been repaired at an earlier time t1. This type of conditional repair probability, similar to the conditional reliability of Eq. (5.12),
G (t ) _ G (t )
RS (ti, t2) = P [TTR < t2 | TTR > ti] = , (5.21)
1 — G(ti)
Kraus (1988) pointed out the difference in maintainability and maintenance; namely, maintainability is design-related, whereas maintenance is operation — related. Since the MTTF is a measure of maintainability, it includes those time elements that can be controlled by design. Elements involved in the evaluation of the time to repair are fault isolation, repair or replacement of a failed component, and verification time. Administrative times, such as mobilization time and time to reach and return from the maintenance site, are not included in the evaluation of the time to repair. The administrative times are considered under the context of supportability (see Sec. 5.3.4), which measures the ability of a system to be supported by the required resources for execution of the specified maintenance task (Knezevic, 1993).
