  We introduce a new class of balanced allocation processes which are primarily
characterized by ``filling'' underloaded bins. A prototypical example is the
Packing process: At each round we only take one bin sample, if the load is
below the average load, then we place as many balls until the average load is
reached; otherwise, we place only one ball. We prove that for any process in
this class the gap between the maximum and average load is $\mathcal{O}(\log
n)$ w.h.p. for any number of balls $m\geq 1$. For the Packing process, we also
provide a matching lower bound. Additionally, we prove that the Packing process
is sample-efficient in the sense that the expected number of balls allocated
per sample is strictly greater than one. Finally, we also demonstrate that the
upper bound of $\mathcal{O}(\log n)$ on the gap can be extended to the Memory
process studied by Mitzenmacher, Prabhakar and Shah (2002).
