  Storage architectures ranging from minimum bandwidth regenerating encoded
distributed storage systems to declustered-parity RAIDs can be designed using
dense partial Steiner systems in order to support fast reads, writes, and
recovery of failed storage units. In order to ensure good performance,
popularities of the data items should be taken into account and the frequencies
of accesses to the storage units made as uniform as possible. A proposed
combinatorial model ranks items by popularity and assigns data items to
elements in a dense partial Steiner system so that the sums of ranks of the
elements in each block are as equal as possible. By developing necessary
conditions in terms of independent sets, we demonstrate that certain Steiner
systems must have a much larger difference between the largest and smallest
block sums than is dictated by an elementary lower bound. In contrast, we also
show that certain dense partial $S(t, t+1, v)$ designs can be labeled to
realize the elementary lower bound. Furthermore, we prove that for every
admissible order $v$, there is a Steiner triple system $(S(2, 3, v))$ whose
largest difference in block sums is within an additive constant of the lower
bound.
