Efficiency comparison between proportional fairness and dominant resource fairness with two different type resources

We study efficiency of proportional fairness (PF) and dominant resource fairness (DRF) in the aspect of resource usage and total throughput when there are two different type resources are available. With a unified convex optimization describing both fair resource allocation schemes, we characterize bottleneck resources (resources in full utilization) of PF and DRF and prove that PF always more efficiently utilizes the available resources with less waste than DRF does. We provide conditions when PF has higher throughput than DRF and give examples showing that PF may have smaller throughput than DRF even though PF has higher resource usage than DRF.