Convergence Analysis and Its Application in the Fixed Point Formulation of Medium Access in Wireless Network

In the Internet of things, it is of critical importance to fully utilize the potential capacity of the network with efficient medium access control (MAC) mechanisms. In this paper, we study the convergence property of the fixed point formulation of distributed coordination function (DCF), which is widely used for medium access control in wireless networks. We first find that the fixed point could be repelling, which means that it is impossible for an MAC system to converge at its fixed point. Next, we show the existence of periodic points to prove that the fixed point function will oscillate between two periodic points when the fixed point is repelling. We also find that the average of the two periodic points is a close approximation of the fixed point. Based on the findings, we propose an algorithm to compute the fixed point efficiently. Simulation results verify the accuracy and efficiency of our algorithm compared with the previous fixed point computing method.