  In this paper, using the stochastic geometry, we develop a tractable uplink
modeling framework for the outage probability of both the single and multi-tier
millimeter wave (mmWave) cellular networks. Each tier's mmWave base stations
(BSs) are randomly located and they have particular spatial density, antenna
gain, receiver sensitivity, blockage parameter and pathloss exponents. Our
model takes account of the maximum power limitation and the per-user power
control. More specifically, each user, which could be in line-of-sight (LOS) or
non-LOS to its serving mmWave BS, controls its transmit power such that the
received signal power at its serving BS is equal to a predefined threshold.
Hence, a truncated channel inversion power control scheme is implemented for
the uplink of mmWave cellular networks. We derive closed-form expressions for
the signal-to-interference-and-noise-ratio (SINR) outage probability for the
uplink of both the single and multi-tier mmWave cellular networks. Furthermore,
we analyze the case with a dense network by utilizing the simplified model,
where the LOS region is approximated as a fixed LOS disc. The results show that
imposing a maximum power constraint on the user significantly affects the SINR
outage probability in the uplink of mmWave cellular networks.
