  To solve the access-balancing problem in distributed storage systems, we
introduce a new combinatorial model, called MinVar model for fractional
repetition (FR) codes. Since FR codes are based on graphs or set systems, our
MinVar model is characterized by the property that the variance among the sums
of block-labels incident to a fixed vertex is minimized. This characterization
is different from Dau and Milenkovic's MaxMinSum model, while the minimum sum
of labels is maximized. We show that our MinVar model is meaningful by
distinguishing labelings with different variances but with the same MaxMin
value for some FR codes. By reformulating the MinVar model to an equivalent
vertex-labeling problem of graphs, we find several families of optimal FR codes
with balanced access frequency, and provide fundamental results for both
problems. It is interesting that MinVar model is closely related to the concept
of magic-labeling in graph theory.
