We analyze networks with wireless nodes distributed randomly in space, transmitting simultaneously in the same channel to their nearest base-stations using a simple power control algorithm. Base-stations are on a hexagonal grid and have N optimally-phased antennas. We derive the asymptotic mean upstream spectral efficiency (validated by simulation) as a function of N, wireless-node density, base-station separation, and path-loss-exponent which controls signal attenuation with distance. These results indicate that mean per-link spectral efficiency is constant if the number of receive antennas or density of base-stations is increased linearly with wireless node density. They also improve our understanding of systems like city-wide wireless Internet deployments and mobile telephony with small cells for which the spatial distribution of nodes causing out-of-cell interference is important. The results are also readily extendable to random code-division-multiple-access systems with spreading factor N.
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