Optimal plans for aggregation

We consider the following problem, which arises in the context of distributed Web computations. An aggregator aims to combine specific data from n sources. The aggregator contacts all sources at once. The time for each source to return its data to the aggregator is independent and identically distributed according to a known distribution. The aggregator at some point stops waiting for data and returns an answer depending only on the data received so far. If the aggregator returns the aggregated information from k of the n sources at time t it obtains a reward Rk(t) that grows with k and decreases with t. The goal of the aggregator is to maximize its expected reward.We prove that for certain broad families of distributions and broad classes of reward functions, the optimal plan for the aggregator has a simple form and hence can be easily computed.