An Optimum Policy for Detecting a Fault in a Complex System

Consider a complex system consisting of N modules containing n (1), ..., n ( N ) items or sub-circuits. A very simple device of this nature, for example, would be a hi-fi set, where the amplifier, the cartridge, the speaker(s) and the turntable could be considered to be the modules, and the various parts of each as the basic items in which faults could occur. A more realistic application of the results obtained might be to a radar-missile system, in which surveillance radar, tracking radar, communications system, computer, missile launcher and missile, or parts of them, represent the modules. When the system breaks down, the procedure for detecting the breakdown and hence repairing the system will determine the time taken to repair it. Assuming two different possible models, equations are developed for the optimum policy, i.e., the policy which minimizes the expected amount of time consumed or penalties paid, and these equations are solved for the first model. Model I assumes that over-all tests of each module may be performed, and individual item tests within modules, model II assumes that over-all module tests are not possible, and that penalty costs must be paid whenever the search moves from one module to another. In both models, penalty costs---in time, for example---are associated with the testing of each item and module.