Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations

Suppose that an unknown number of objects arrive sequentially according to a Poisson process with random intensity λ on some fixed time interval [0, T]. We assume a gamma prior density Gsλ(r. 1/a) for λ. Furthermore, we suppose that all arriving objects can be ranked uniquely among all preceding arrivals. Exactly one object can be selected. Our aim is to find a stopping time (selection time) which maximizes the time during which the selected object will stay relatively best. Our main result is the following. It is optimal to select the ith object that is relatively best and arrives at some time s ^(r) _i onwards. The value of s ^(r) _i can be obtained for each r and i as the unique root of a deterministic equation.