Fixed Point Analysis of Single Cell IEEE 802.11e WLANs: Uniqueness and Multistability

We consider the vector fixed point equations arising out of the analysis of the saturation throughput of a single cell IEEE 802.11e (EDCA) wireless local area network with nodes that have different backoff parameters, including different arbitration interframe space (AIFS) values. We consider balanced and unbalanced solutions of the fixed point equations arising in homogeneous (i.e., one with the same backoff parameters) and nonhomogeneous networks. By a balanced fixed point, we mean one where all coordinates are equal. We are concerned, in particular, with: 1) whether the fixed point is balanced within a class, and 2) whether the fixed point is unique. Our simulations show that when multiple unbalanced fixed points exist in a homogeneous system then the time behavior of the system demonstrates severe short term unfairness (or multistability). We provide a condition for the fixed point solution to be balanced, and also a condition for uniqueness. We then extend our general fixed point analysis to capture AIFS based differentiation and the concept of virtual collision when there are multiple queues per station; again a condition for uniqueness is established. For the case of multiple queues per node, we find that a model with as many nodes as there are queues, with one queue per node, provides an excellent approximation. Implications for the use of the fixed point formulation for performance analysis are also discussed.