  In large storage systems, files are often coded across several servers to
improve reliability and retrieval speed. We consider a system of $n$ servers
storing files using a Maximum Distance Separable code (cf. \cite{li2016mean}).
Specifically, each file is stored in equally sized pieces across $L$ servers
such that any $k$ pieces can reconstruct the original file. File requests are
routed using the Batch Sampling routing scheme. I.e. when a request for a file
is received, a centralized 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 study the long time behavior of this class of load
balancing mechanisms. In particular, it is shown that the ODE system that
describes the mean field limit of the occupancy measure process has a unique
fixed point which is stable. This fixed point corresponds to a distribution on
$\mathbb{N}_0$ of queue lengths with tails that decay super-exponentially.
Upper and lower bounds on the decay rate are provided. Finally, we show that
the unique invariant measure of the Markov occupancy measure process converges
to the Dirac measure concentrated at the unique fixed point of the ODE system,
establishing the interchangeability of the $t\to\infty$ and $n\to\infty$
limits.
