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Most of the early input-queued switch research focused on establishingthroughput optimality of the max-weight scheduling policy, with some recentresearch showing that max-weight scheduling is optimal with respect to totalexpected delay asymptotically in the heavy-traffic regime. However, thequestion of delay-optimal scheduling in input-queued switches remains open ingeneral, as does the question of delay-optimal scheduling under more generalobjective functions. To gain fundamental insights into these very difficultproblems, we consider a fluid model of $n \times n$ input-queued switches withassociated fluid-flow costs, and we derive an optimal scheduling control policyto an infinite horizon discounted control problem with a general linearobjective function of fluid cost. Our optimal policy coincides with the$c\mu$-rule in certain parameter domains. More generally, due to theinput-queued switch constraints, the optimal policy takes the form of thesolution to a flow maximization problem, after we identify the Lagrangianmultipliers of some key constraints through carefully designed algorithms.Computational experiments demonstrate the benefits of our optimal schedulingpolicy over variants of max-weight scheduling within fluid models ofinput-queued switches.

Optimal Scheduling Control in Fluid Models of General $n\times n$ Input-Queued Switches