Well Balanced Designs for Data Placement Jean-Claude Bermond∗

The problem we consider in this article is motivated by data placement, in particular data replica-tion in distributed storage and retrieval systems. We are given a set V of v servers and b files (data, documents). Each file is replicated on exactly k servers. A placement consists in finding a family of b subsets of V (representing the files) called blocks each of size k. Each server has some probability to fail and we want to find a placement which minimizes the variance of the number of available files. It was conjectured that there always exists an optimal placement (with variance better than that of any other placement for any value of the probability of failure). We show that the conjecture is true, if there exists a well balanced design, that is a family of blocks, each of size k, such that each j-element subset of V, 1 ≤ j ≤ k, belongs to the same or almost the same number of blocks (difference at most one). The existence of well balanced designs is a difficult problem as it contains as a subproblem the existence of Steiner systems. We completely solve the case k = 2 and give bounds and constructions for k = 3 and some values of v and b