  Load balancing is a common approach in web server farms or inventory routing
problems. An important issue in such systems is to determine the server to
which an incoming request should be routed to optimize a given performance
criteria. In this paper, we assume the server's scheduling disciplines to be
heterogeneous. More precisely, a server implements a scheduling discipline
which belongs to the class of limited processor sharing (LPS-$d$) scheduling
disciplines. Under LPS-$d$, up to $d$ jobs can be served simultaneously, and
hence, includes as special cases First Come First Served ($d=1$) and Processor
Sharing ($d=\infty$).
  In order to obtain efficient heuristics, we model the above load-balancing
framework as a multi-armed restless bandit problem. Using the relaxation
technique, as first developed in the seminal work of Whittle, we derive
Whittle's index policy for general cost functions and obtain a closed-form
expression for Whittle's index in terms of the steady-state distribution.
Through numerical computations, we investigate the performance of Whittle's
index with two different performance criteria: linear cost criterion and a cost
criterion that depends on the first and second moment of the throughput. Our
results show that \emph{(i)} the structure of Whittle's index policy can
strongly depend on the scheduling discipline implemented in the server, i.e.,
on $d$, and that \emph{(ii)} Whittle's index policy significantly outperforms
standard dispatching rules such as Join the Shortest Queue (JSQ), Join the
Shortest Expected Workload (JSEW), and Random Server allocation (RSA).
