Optimum Search for Objects Having Unknown Arrival Times

In defense against missiles, early detection is crucial. This is the motivation for the mathematical solution of the following general search problem. Objects arrive in accordance with a Poisson process. Having arrived, the object appears (and remains until detected) in box i with probability p_i. A single scan of box i costs Ci (possibly including the cost of false alarms), takes time t_i, and, if the object is present in box i at the beginning of the scan, will detect it with probability Pi. The resultant gain g_i(t) is a nonincreasing function of t, the delay between arrival and the beginning of the detecting look, i = 1, 2, …, n. An asymptotically optimum search procedure is obtained. A number of particular cases of interest are solved.