Asymptotic Behavior of Ultra-Dense Cellular Networks and Its Economic   Impact
  This paper investigates the relationship between base station (BS) density and average spectral efficiency (SE) in the downlink of a cellular network. This relationship has been well known for sparse deployment, i.e. when the number of BSs is small compared to the number of users. In this case the SE is independent of BS density. As BS density grows, on the other hand, it has previously been shown that increasing the BS density increases the SE, but no tractable form for the SE-BS density relationship has yet been derived. In this paper we derive such a closed-form result that reveals the SE is asymptotically a logarithmic function of BS density as the density grows. Further, we study the impact of this result on the network operator's profit when user demand varies, and derive the profit maximizing BS density and the optimal amount of spectrum to be utilized in closed forms. In addition, we provide deployment planning guidelines that will aid the operator in his decision if he should invest in densifying his network or in acquiring more spectrum. 
Jihong Park, Seong-Lyun Kim, and Jens Zander
cs.IT cs.NI math.IT