The performance of multi-hop CSMA/CA networks has in most cases been evaluated via simulations, or analytically using a perfect collision channel model. Using such methods, one can neither have access to the capture nature of the terminals nor evaluate the performance of the network in a fading environment. In this paper a new analytical framework that takes into account these capture and fading effects is presented. The network investigated in this paper has an infinite number of nodes which is assumed to be spatially Poisson distributed. The approach is to develop analytical expressions for the transition probabilities between states of nodes in the network by approximating these states as a Markov chain. A fast algorithm to solve these equations is also given. Numerical results show that the model agrees with simulation results for both omni-directional networks and networks employing beamforming.

An asynchronous unicast ad hoc network is considered, where each node i is equipped with a receive/transmit beam-former pair (W/sub i/, g/sub i/) designed under a quality-of-service (QoS) SNR constraint. It is first shown that the minimum sum-power beamformers for the network satisfy a weak duality condition, in which the pairs ((g/sub i//sup opt/)*, (W/sub i//sup opt/)*) achieve the same sum power as the primal network. However, the optimum receive beamformer w/sub i//sup opt/ is not in general equal to (g/sub i//sup opt/)*, in contrast to the case of cellular and time-division duplexing networks. Iterative minimum mean-square error (IMMSE) beamforming algorithms are then proposed in which w/sub i/ = g/sub i/* is enforced. These algorithms are shown to be instances of the Power Algorithm in which gi is the maximizing eigenvector of an SNR-related objective matrix. The IMMSE algorithm can also be viewed as a noncooperative beamforming game, in which the payoff includes normalized SNR, and the tax is related to interference caused at other nodes. The existence of fixed points (Nash equilibria) is proved for IMMSE. Furthermore, fixed points of IMMSE are shown to satisfy the first-order necessary conditions for optimization using a network Lagrangian. The IMMSE game is modified to yield a sequential distortionless-response beamforming algorithm, which is shown to be convergent using a Total Interference Function. Extensive simulation results illustrate that IMMSE yields better power efficiency than a greedy noncooperative SNR-maximizing game.