The loss of serving in the dark

We study the following balls and bins stochastic process: There is a buffer with B bins, and there is a stream of balls X = (X₁,X ₂,...,X) i such that Xi is the number of balls that arrive before time i but after time i - 1. Once a ball arrives, it is stored in one of the unoccupied bins. If all the bins are occupied then the ball is thrown away. In each time step, we select a bin uniformly at random, clear it, and gain its content. Once the stream of balls ends, all the remaining balls in the buffer are cleared and added to our gain. We are interested in analyzing the expected gain of this randomized process with respect to that of an optimal gain-maximizing strategy, which gets the same online stream of balls, and clears a ball from a bin, if exists, at any step. We name this gain ratio the loss of serving in the dark. In this paper, we determine the exact loss of serving in the dark. We prove that the expected gain of the randomized process is worse by a factor of ρ + ε from that of the optimal gain-maximizing strategy for any ε > 0, where ρ = maxα>1 αeα/((α - 1)eα + e - 1) ≈ 1:69996 and B = Ω(1/ε³). We also demonstrate that this bound is essentially tight as there are specific ball streams for which the above-mentioned gain ratio tends to ρ. Our stochastic process occurs naturally in many applications. We present a prompt and truthful mechanism for bounded capacity auctions, and an application relating to packets scheduling..