  In large storage systems, files are often coded across several servers to
improve reliability and retrieval speed. We study load balancing under the
Batch Sampling routing scheme for a network of $n$ servers storing a set of
files using the Maximum Distance Separable (MDS) code (cf. Li, Ramamoorthy, and
Srikant (2016)). Specifically, each file is stored in equally sized pieces
across $L$ servers such that any $k$ pieces can reconstruct the original file.
When a request for a file is received, the dispatcher routes the job into the
$k$-shortest queues among the $L$ for which the corresponding server contains a
piece of the file being requested. We establish a law of large numbers and a
central limit theorem as the system becomes large (i.e. $n\to\infty$). For the
central limit theorem, the limit process take values in $\mathbf{\ell}_2$, the
space of square summable sequences. Due to the large size of such systems, a
direct analysis of the $n$-server system is frequently intractable. The law of
large numbers and diffusion approximations established in this work provide
practical tools with which to perform such analysis. The Power-of-$d$ routing
scheme, also known as the supermarket model, is a special case of the model
considered here.
