Transmission capacities for overlaid wireless networks with channel inversion based power control

We study the effect of channel inversion based power control on the transmission capacities of two coexisting wireless networks (a primary network vs. a secondary network) that operate in the same geographic region and share the same spectrum. The primary (PR) network has a higher priority to access the spectrum without particular considerations for the secondary (SR) network, where the SR network limits its interference to the PR network by carefully controlling its node density. Considering a general power-law wireless channels with path-loss exponent a > 2 and small-scale Rayleigh fading, by using stochastic geometry theory, we derive the transmission capacities for both of the two networks and quantify their tradeoff via asymptotic analysis. Our results show that when the transmitter densities for both of the PR and SR networks are vanishingly small, with power control the sum transmission capacity of the two networks (i.e., the overall spectrum efficiency per unit area) will be increased compared to that without power control, which is different from the recent result observed by Weber et al in for a single network case when the transmitter density is relatively large.