  Most of the early input-queued switch research focused on establishing
throughput optimality of the max-weight scheduling policy, with some recent
research showing that max-weight scheduling is optimal with respect to total
expected delay asymptotically in the heavy-traffic regime. However, the
question of delay-optimal scheduling in input-queued switches remains open in
general, as does the question of delay-optimal scheduling under more general
objective functions. To gain fundamental insights into these very difficult
problems, we consider a fluid model of $n \times n$ input-queued switches with
associated fluid-flow costs, and we derive an optimal scheduling control policy
to an infinite horizon discounted control problem with a general linear
objective function of fluid cost. Our optimal policy coincides with the
$c\mu$-rule in certain parameter domains. More generally, due to the
input-queued switch constraints, the optimal policy takes the form of the
solution to a flow maximization problem, after we identify the Lagrangian
multipliers of some key constraints through carefully designed algorithms.
Computational experiments demonstrate the benefits of our optimal scheduling
policy over variants of max-weight scheduling within fluid models of
input-queued switches.
