In this paper, we first prove that the Randomly Addressed Polling (RAP) protocol is unstable under the random access channel with heavy traffic. We also show that network stability can be ensured by controlling the arrival rate λ or by expanding the available addresses p on the assumption that there are M finite stations within the coverage of the controller (the base station). From analyses and results, we see the equilibrium of arrival rate is inversely proportional to the product of users (stations) and the exponent of stations. We also see that the maximum throughput can be derived at the point of λ1/M. This maximum performance can be easily obtained under the consideration of RAP protocol's stability. It also implies that the maximum throughput is independent of the available addresses of RAP protocol when pM.