  Given a set $V$ of $v$ servers along with $b$ files (data), each file is
replicated (placed) on exactly $k$ servers and thus a file can be represented
by a set of $k$ servers. Then we produce a data placement consisting of $b$
subsets of $V$ called blocks, each of size $k$. Each server has some
probability to fail and we want to find a placement that minimizes the variance
of the number of available files. It was conjectured that there always exists
an optimal data placement (with variance better than any other placement for
any value of the probability of failure). An optimal data placement for triple
replication with $b$ blocks (of size three) on a $v$-set was proved to exist by
Wei et al. if $v$ and $b$ are not excluded by two conditions. This article
concentrates on the parameters $v, b$ satisfying the two conditions and
characterizes the combinatorial properties of the corresponding optimal data
placements. Nearly well-balanced triple systems (NWBTSs) are defined to produce
optimal data placements. Many constructions for NWBTSs are developed, mainly by
constructing candelabra systems with various desirable partitions. The main
result of this article is that there always exist optimal data placements for
triple replication with $b$ blocks on a $v$-set possibly except when $v\equiv
4$ (mod 24) or $v=50,74$, and $\frac{\lambda v(v-1)}{6}-\frac{v}{6} < b <
\frac{\lambda v(v-1)}{6}+\frac{v}{6}$ for an odd integer $\lambda$.
