We consider the vector fixed point equations arising out of a saturation throughput analysis of a single cell IEEE 802.11e (EDCA) WLAN. We study 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 of users, and 2) whether the fixed point is unique. For IEEE 802.11 type WLANs, we provide a condition, based on the backoff parameters, for the vector fixed point solution to be balanced within a class, and also a condition for uniqueness of the solution. We then provide an extension of our general fixed point analysis to capture AIFS based differentiation and multiple virtual queues (supported in IEEE 802.11e EDCA); again a condition for uniqueness is established and simulations validate the analysis. The fixed point solutions are used to obtain insights into the throughput differentiation provided by different initial back-offs, persistence factors, AIFS and multiple virtual queues, for finite number of nodes, and for differentiation parameter values similar to those in the standard. Our simulations show that when multiple unbalanced fixed points exist then the time behavior of the system demonstrates severe short term unfairness (or multistability). Implications for the use of the fixed point formulation for performance analysis will be discussed.