  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.
