  In this paper, we theoretically study the proportional fair (PF) scheduler in
the context of ultra-dense networks (UDNs). Analytical results are obtained for
the coverage probability and the area spectral efficiency (ASE) performance of
dense small cell networks (SCNs) with the PF scheduler employed at base
stations (BSs). The key point of our analysis is that the typical user is no
longer a random user as assumed in most studies in the literature. Instead, a
user with the maximum PF metric is chosen by its serving BS as the typical
user. By comparing the previous results of the round-robin (RR) scheduler with
our new results of the PF scheduler, we quantify the loss of the multi-user
diversity of the PF scheduler with the network densification, which casts a new
look at the role of the PF scheduler in UDNs. Our conclusion is that the RR
scheduler should be used in UDNs to simplify the radio resource management
(RRM).
