  We develop link rate control policies to minimize the queueing delay of
packets in overloaded networks. We show that increasing link rates does not
guarantee delay reduction during overload. We consider a fluid queueing model
that facilitates explicit characterization of the queueing delay of packets,
and establish explicit conditions on link rates that can minimize the average
and maximum queueing delay in both single-hop and multi-stage (switching)
networks. These min-delay conditions require maintaining an identical ratio
between the ingress and egress rates of different nodes at the same layer of
the network. We term the policies that follow these conditions
rate-proportional policies. We further generalize the rate-proportional
policies to queue-proportional policies, which minimize the queueing delay
asymptotically based on the time-varying queue length while remaining agnostic
of packet arrival rates. We validate that the proposed policies lead to minimum
queueing delay under various network topologies and settings, compared with
benchmarks including the backpressure policy that maximizes network throughput
and the max-link-rate policy that fully utilizes bandwidth. We further remark
that the explicit min-delay policy design in multi-stage networks facilitates
co-optimization with other metrics, such as minimizing total bandwidth,
balancing link utilization and node buffer usage. This demonstrates the wider
utility of our main results in data center network optimization in practice.
