Chains-into-bins processes

The study of balls-into-bins processes or occupancy problems has a long history. These processes can be used to translate realistic problems into mathematical ones in a natural way. In general, the goal of a balls-into-bins process is to allocate a set of independent objects (tasks, jobs, balls) to a set of resources (servers, bins, urns) and, thereby, to minimize the maximum load. In this paper, we analyze the maximum load for the chains-into-bins problem, which is defined as follows. There are n bins, and m objects to be allocated. Each object consists of balls connected into a chain of length ℓ, so that there are mℓ balls in total. We assume the chains cannot be broken, and that the balls in one chain have to be allocated to ℓ consecutive bins. We allow each chain d   independent and uniformly random bin choices for its starting position. The chain is allocated using the rule that the maximum load of any bin receiving a ball of that chain is minimized. We show that, for d⩾2d⩾2 and m⋅ℓ=O(n)m⋅ℓ=O(n), the maximum load is ((lnlnm)/lnd)+O(1) with probability 1−O˜(1/md−1).