Optimum Checking Schedules for Systems Subject to Random Failure - Keller, Joseph B. (Management Science, 1974)
An optimum checking schedule is one that minimizes the expected cost, which is the sum of the cost of checking and the expected loss due to an undetected failure. The problem is made tractable by supposing that checking is so frequent that it can be described by a continuous density n(t) of checks per unit time. The optimum n(t) is found by the methods of the calculus of variations. An explicit result is given when the loss is proportional to the duration of an undetected failure. A minimax solution is also given for the case in which the probability of failure is not known.