On Stochastic Geometry Modeling of Cellular Uplink Transmission with   Truncated Channel Inversion Power Control
  Using stochastic geometry, we develop a tractable uplink modeling paradigm for outage probability and spectral efficiency in both single and multi-tier cellular wireless networks. The analysis accounts for per user equipment (UE) power control as well as the maximum power limitations for UEs. More specifically, for interference mitigation and robust uplink communication, each UE is required to control its transmit power such that the average received signal power at its serving base station (BS) is equal to a certain threshold $\rho_o$. Due to the limited transmit power, the UEs employ a truncated channel inversion power control policy with a cutoff threshold of $\rho_o$. We show that there exists a transfer point in the uplink system performance that depends on the tuple: BS intensity ($\lambda$), maximum transmit power of UEs ($P_u$), and $\rho_o$. That is, when $P_u$ is a tight operational constraint with respect to [w.r.t.] $\lambda$ and $\rho_o$, the uplink outage probability and spectral efficiency highly depend on the values of $\lambda$ and $\rho_o$. In this case, there exists an optimal cutoff threshold $\rho^*_o$, which depends on the system parameters, that minimizes the outage probability. On the other hand, when $P_u$ is not a binding operational constraint w.r.t. $\lambda$ and $\rho_o$, the uplink outage probability and spectral efficiency become independent of $\lambda$ and $\rho_o$. We obtain approximate yet accurate simple expressions for outage probability and spectral efficiency which reduce to closed-forms in some special cases. 
Hesham ElSawy and Ekram Hossain
cs.IT cs.NI math.IT math.ST stat.TH