Fixed point theorem and its limitation for derivation of Wi-Fi networks performance

The performance analysis of the distributed access mechanism of the IEEE 802.11 standard also known as Wi-Fi has been extensively studied for more than a decade. Several mathematical models that describe this mechanism have been developed. Some of them using Markov chain and others based on geometrical distribution. One important point often omitted is the way to solve the problem numerically. Indeed, the set of equations exhibited by the model is very complex to be solved. A natural way to solve a set of nonlinear equations is to use the fixed point theorem. In this paper, we present an application of the fixed point theorem method to solve the problem of throughput derivation. After an involved series of mathematical derivations, we show why this method is limited since it requires extreme condition on the error probability.