On Searching for Events of Limited Duration

Given a set of events, an observer wishes to detect as many of these as possible. The events arise at several discrete points according to independent Poisson processes, and the lifetimes of individual occurrences are independent and identically distributed random variables. The specific problem is: given that the observer can only visit one point per unit time, in what sequence should he make his visits so as to maximize the steady-state fraction of events he detects? We obtain some results about the optimal search policy and find the best policy precisely in some circumstances.