We consider the sequential allocation of m balls (jobs) into n bins (servers) by allowing each ball to choose from some bins sampled uniformly at random. The goal is to maintain a small gap between the maximum load and the average load.

In this paper, we present a general framework that allows us to analyze various allocation processes that slightly prefer allocating into underloaded, as opposed to overloaded bins. Our analysis covers several natural instances of processes, including:

The Caching process (a.k.a. memory protocol) as studied by Mitzenmacher, Prabhakar and Shah (2002).

The Packing process: At each round we only take one bin sample. If the load is below some threshold (e.g., the average load), then we place as many balls until the threshold is reached; otherwise, we place only one ball.

The Twinning process: At each round, we only take one bin sample. If the load is below some threshold, then we place two balls; otherwise, we place only one ball.

The Thinning process as recently studied by Feldheim and Gurel-Gurevich (2021).

As we demonstrate, using an interplay between several potential functions our general framework implies for all these processes a gap of O(log n) for any number of balls m ≥ n.